https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Stovall&feedformat=atomUW-Math Wiki - User contributions [en]2021-02-27T07:17:24ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18894Analysis Seminar2020-02-05T16:03:05Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Friday, Jan 31, 4 pm, B239, Colloquium<br />
| Lillian Pierce<br />
| Duke University<br />
|[[#Lillian Pierce | On Bourgain’s counterexample for the Schrödinger maximal function ]]<br />
| Andreas, Simon<br />
|-<br />
|Feb 4<br />
| Ruixiang Zhang<br />
| UW Madison<br />
|[[#Ruixiang Zhang | Local smoothing for the wave equation in 2+1 dimensions ]]<br />
| Andreas<br />
|-<br />
|Feb 11<br />
| Zane Li<br />
| Indiana University<br />
|[[#Zane Li | A bilinear proof of decoupling for the moment curve ]]<br />
| Betsy<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Michel Alexis<br />
| Local<br />
|[[#Michel Alexis | The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]<br />
| Denisov<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#William Green | Dispersive estimates for the Dirac equation ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Brian Street<br />
| University of Wisconsin-Madison<br />
|[[#linktoabstract | Title ]]<br />
| Local<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
===Lillian Pierce===<br />
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b><br />
<br />
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. <br />
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”<br />
<br />
===Ruixiang Zhang===<br />
<br />
<b> Local smoothing for the wave equation in 2+1 dimensions </b><br />
<br />
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.<br />
<br />
===Zane Li===<br />
<br />
<b> A bilinear proof of decoupling for the moment curve</b><br />
<br />
We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.<br />
<br />
<br />
===William Green===<br />
<br />
<b> Dispersive estimates for the Dirac equation </b><br />
<br />
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations<br />
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.<br />
<br />
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations. We will survey recent work on time-decay estimates for the solution operator. Specifically the mapping properties of the solution operator between L^p spaces. As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution. We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay. The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18893Analysis Seminar2020-02-05T16:01:53Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Friday, Jan 31, 4 pm, B239, Colloquium<br />
| Lillian Pierce<br />
| Duke University<br />
|[[#Lillian Pierce | On Bourgain’s counterexample for the Schrödinger maximal function ]]<br />
| Andreas, Simon<br />
|-<br />
|Feb 4<br />
| Ruixiang Zhang<br />
| UW Madison<br />
|[[#Ruixiang Zhang | Local smoothing for the wave equation in 2+1 dimensions ]]<br />
| Andreas<br />
|-<br />
|Feb 11<br />
| Zane Li<br />
| Indiana University<br />
|[[#Zane Li | A bilinear proof of decoupling for the moment curve ]]<br />
| Betsy<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Michel Alexis<br />
| Local<br />
|[[#Michel Alexis | The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]<br />
| Denisov<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#William Green | Dispersive estimates for the Dirac equation ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Brian Street<br />
| University of Wisconsin-Madison<br />
|[[#linktoabstract | Title ]]<br />
| Local<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
===Lillian Pierce===<br />
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b><br />
<br />
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. <br />
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”<br />
<br />
===Ruixiang Zhang===<br />
<br />
<b> Local smoothing for the wave equation in 2+1 dimensions </b><br />
<br />
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.<br />
<br />
<br />
===William Green===<br />
<br />
<b> Dispersive estimates for the Dirac equation </b><br />
<br />
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations<br />
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.<br />
<br />
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations. We will survey recent work on time-decay estimates for the solution operator. Specifically the mapping properties of the solution operator between L^p spaces. As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution. We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay. The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18821Analysis Seminar2020-01-29T17:57:45Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Friday, Jan 31, 4 pm, B239, Colloquium<br />
| Lillian Pierce<br />
| Duke University<br />
|[[#Lillian Pierce | On Bourgain’s counterexample for the Schrödinger maximal function ]]<br />
| Andreas, Simon<br />
|-<br />
|Feb 4<br />
| Ruixiang Zhang<br />
| UW Madison<br />
|[[#Ruixiang Zhang | Local smoothing for the wave equation in 2+1 dimensions ]]<br />
| Andreas<br />
|-<br />
|Feb 11<br />
| Zane Li<br />
| Indiana University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Speaker<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Host<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#William Green | Dispersive estimates for the Dirac equation ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Brian Street<br />
| University of Wisconsin-Madison<br />
|[[#linktoabstract | Title ]]<br />
| Local<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
===Lillian Pierce===<br />
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b><br />
<br />
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. <br />
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”<br />
<br />
===Ruixiang Zhang===<br />
<br />
<b> Local smoothing for the wave equation in 2+1 dimensions </b><br />
<br />
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.<br />
<br />
<br />
===William Green===<br />
<br />
<b> Dispersive estimates for the Dirac equation </b><br />
<br />
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations<br />
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.<br />
<br />
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations. We will survey recent work on time-decay estimates for the solution operator. Specifically the mapping properties of the solution operator between L^p spaces. As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution. We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay. The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18820Analysis Seminar2020-01-29T17:56:02Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Friday, Jan 31, 4 pm, B239, Colloquium<br />
| Lillian Pierce<br />
| Duke University<br />
|[[#Lillian Pierce | On Bourgain’s counterexample for the Schrödinger maximal function ]]<br />
| Andreas, Simon<br />
|-<br />
|Feb 4<br />
| Ruixiang Zhang<br />
| UW Madison<br />
|[[#Ruixiang Zhang | Local smoothing for the wave equation in 2+1 dimensions ]]<br />
| Andreas<br />
|-<br />
|Feb 11<br />
| Zane Li<br />
| Indiana University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Speaker<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Host<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#William Green | Dispersive estimates for the Dirac equation ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Brian Street<br />
| University of Wisconsin-Madison<br />
|[[#linktoabstract | Title ]]<br />
| Local<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
===Lillian Pierce===<br />
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b><br />
<br />
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. <br />
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”<br />
<br />
===Ruixiang Zhang===<br />
<br />
<b> Local smoothing for the wave equation in 2+1 dimensions </b><br />
<br />
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18595Analysis Seminar2020-01-07T02:47:18Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Zane Li<br />
| Indiana University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Dmitry Chelkak<br />
| Ecole Normale, Paris<br />
|[[#linktoabstract | Title ]]<br />
| Denisov<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18594Analysis Seminar2020-01-05T22:08:05Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Tent. reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Dmitry Chelkak<br />
| Ecole Normale, Paris<br />
|[[#linktoabstract | Title ]]<br />
| Denisov<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18589Analysis Seminar2019-12-30T19:26:38Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Dmitry Chelkak<br />
| Ecole Normale, Paris<br />
|[[#linktoabstract | Title ]]<br />
| Denisov<br />
|-<br />
|Mar 3<br />
| William Green<br />
| Rose-Hulman Institute of Technology<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18558Analysis Seminar2019-12-13T03:28:49Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Côte d'Azur<br />
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#Bingyang Hu | Sparse bounds of singular Radon transforms]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#Francesco di Plinio | Maximal directional integrals along algebraic and lacunary sets]]<br />
| Shaoming<br />
|-<br />
|Nov 13 (Wednesday)<br />
| Xiaochun Li <br />
| UIUC<br />
|[[#Xiaochun Li | Roth's type theorems on progressions]]<br />
| Brian, Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#Joao Ramos | Fourier uncertainty principles, interpolation and uniqueness sets ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Sergey Denisov<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Feb 25<br />
| Dmitry Chelkak<br />
| Ecole Normale, Paris<br />
|[[#linktoabstract | Title ]]<br />
| Denisov<br />
|-<br />
|Mar 3<br />
| tent reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Mar 10<br />
| Yifei Pan<br />
| Indiana University-Purdue University Fort Wayne<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|<b>Monday, Apr 13</b><br />
|Yumeng Ou<br />
|CUNY, Baruch College<br />
|[[#linktoabstract | TBA ]]<br />
|Zhang<br />
|-<br />
|Apr 14<br />
| Tamás Titkos<br />
| BBS University of Applied Sciences & Rényi Institute<br />
|[[#linktoabstract | Distance preserving maps on spaces of probability measures ]]<br />
| Street<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
<br />
<br />
===Joao Ramos===<br />
<br />
Title: Fourier uncertainty principles, interpolation and uniqueness sets<br />
<br />
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that <br />
<br />
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$ <br />
<br />
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. <br />
<br />
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ <br />
<br />
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Xiaochun Li===<br />
<br />
Title: Roth’s type theorems on progressions<br />
<br />
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
<br />
<br />
===Francesco di Plinio===<br />
<br />
<b>Maximal directional integrals along algebraic and lacunary sets </b><br />
<br />
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===Laurent Stolovitch===<br />
<br />
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b><br />
<br />
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.<br />
<br />
<br />
===Bingyang Hu===<br />
<br />
<b>Sparse bounds of singular Radon transforms</b><br />
<br />
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18046Analysis Seminar2019-09-27T17:26:44Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18045Analysis Seminar2019-09-27T17:25:42Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18024Analysis Seminar2019-09-26T14:00:44Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18023Analysis Seminar2019-09-26T14:00:16Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
==Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature==<br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18022Analysis Seminar2019-09-26T13:59:51Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
==Concentration and Growth of Laplace Eigenfunctions==<br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
==Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature==<br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18021Analysis Seminar2019-09-26T13:58:56Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature<br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18013Analysis Seminar2019-09-25T17:35:15Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature<br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18012Analysis Seminar2019-09-25T17:33:55Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18011Analysis Seminar2019-09-25T17:00:32Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|tbd | tbd<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18008Analysis Seminar2019-09-25T13:54:46Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|tbd | tbd<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| tent. reserved.<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=17775Analysis Seminar2019-09-06T18:00:22Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|tbd | tbd<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Spring2020&diff=17603Colloquia/Spring20202019-08-02T01:29:12Z<p>Stovall: /* Fall 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
| tentatively reserved<br />
|<br />
|<br />
| Betsy<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
|<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|(Moduli Spaces Conference)<br />
|<br />
|Boggess, Sankar<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|Caroline Turnage-Butterbaugh (Carleton College)<br />
|<br />
|Marshall<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=17575Analysis Seminar2019-07-22T12:00:49Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#linktoabstract | Title ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday)<br />
| Yakun Xi<br />
| Uni Rochester<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Brian<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 8<br />
| tent. reserve<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| Bassam Shayya<br />
| American University of Beirut <br />
|[[#linktoabstract | Title ]]<br />
| Andreas, Betsy<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|Oct 29<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=17500Analysis Seminar2019-06-17T15:06:45Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#linktoabstract | Title ]]<br />
| Andreas, David<br />
|-<br />
|Sept 17<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 8<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 15<br />
| Bassam Shayya<br />
| American University of Beirut <br />
|[[#linktoabstract | Title ]]<br />
| Andreas, Betsy<br />
|-<br />
|Oct 22<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 29<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 31<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 7<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=17499Analysis Seminar2019-06-17T15:06:20Z<p>Stovall: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#linktoabstract | Title ]]<br />
| Andreas, David<br />
|-<br />
|Sept 17<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 8<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 15<br />
| Bassam Shayya<br />
| American University of Beirut <br />
|[[#linktoabstract | Title ]]<br />
| Andreas, Betsy<br />
|-<br />
|Oct 22<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 29<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| Stovall<br />
|-<br />
|Nov 12<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 19<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
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[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15490Analysis Seminar2018-05-01T15:18:58Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1 '''at 3:30pm'''<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 2 in B239 at 4pm'''<br />
| Keith Rush<br />
| senior data scientist with the Milwaukee Brewers<br />
| [[#Keith Rush | Guerilla warfare: ruling the data jungle]]<br />
|-<br />
| '''May 7''' in '''B223'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
===Keith Rush===<br />
<br />
Title: Guerilla warfare: ruling the data jungle<br />
<br />
Abstract: Einstein said ‘As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.’ In this epistemological chaos, the world turns to those experienced with mathematical truth to apply their reasoning powers in the uncertain domain of existence. This talk will describe the fact and fiction of this business reality, the pitfalls (intellectual, moral, and social) and the opportunities. I will discuss the state of business analytics today, at least in sports, the relationship of a pure mathematician to it, and what it is like to help lead the charge as applied mathematics eats the world.<br />
<br />
===Ebru Toprak===<br />
<br />
Title: Dispersive estimates for massive Dirac equations<br />
<br />
Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15489Analysis Seminar2018-05-01T15:17:54Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1 '''at 3:30pm'''<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 2 in B239 at 4pm'''<br />
| Keith Rush<br />
| senior data scientist with the Milwaukee Brewers<br />
| [[#Keith Rush | Guerilla warfare: ruling the data jungle]]<br />
|-<br />
| '''May 7''' in '''B223'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
===Ebru Toprak===<br />
<br />
Title: Dispersive estimates for massive Dirac equations<br />
<br />
Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15488Analysis Seminar2018-05-01T15:17:18Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 2 in B239 at 4pm'''<br />
| Keith Rush<br />
| senior data scientist with the Milwaukee Brewers<br />
| [[#Keith Rush | Guerilla warfare: ruling the data jungle]]<br />
|-<br />
| '''May 7''' in '''B223'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
===Ebru Toprak===<br />
<br />
Title: Dispersive estimates for massive Dirac equations<br />
<br />
Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15469Analysis Seminar2018-04-25T19:20:54Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 7''' in '''B223'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
===Ebru Toprak===<br />
<br />
Title: Dispersive estimates for massive Dirac equations<br />
<br />
Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15468Analysis Seminar2018-04-25T19:19:44Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 7''' in '''B223'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15431Analysis Seminar2018-04-18T19:12:42Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 7''' in '''B223'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15401Analysis Seminar2018-04-13T02:44:21Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
===Xianghong Gong===<br />
<br />
Title: Smooth equivalence of deformations of domains in complex euclidean spaces<br />
<br />
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15400Analysis Seminar2018-04-13T02:43:13Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15381Analysis Seminar2018-04-09T20:29:25Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=15380Colloquia/Fall182018-04-09T20:28:48Z<p>Stovall: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16 (Room: 911)<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 5 (Thursday, Room: 911)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[#April 5 John Baez (UC Riverside)| Monoidal categories of networks ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)<br />
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13 (911 Van Vleck)<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[#April 13, Jill Pipher, Brown University| Mathematical ideas in cryptography ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 16 (Monday)<br />
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)<br />
|[[#Berkesch| Free complexes on smooth toric varieties ]]<br />
| Erman, Sam<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
| May 1 (Tuesday, 4:30pm)<br />
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University Chicago and Imperial College London) Distinguished lecture<br />
|[[# TBA| TBA ]]<br />
| Lu Wang<br />
|<br />
|-<br />
| May 2 (Wednesday, 3pm)<br />
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University of Chicago and Imperial College London) Distinguished lecture<br />
|[[# TBA| TBA ]]<br />
| Lu Wang<br />
|<br />
|-<br />
| May 4<br />
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)<br />
|[[# TBA| TBA ]]<br />
| Ellenberg<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
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|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
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|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 2 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
===March 16 Anne Gelb (Dartmouth)===<br />
<br />
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity<br />
<br />
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.<br />
<br />
<br />
<br />
<br />
===April 5 John Baez (UC Riverside)===<br />
<br />
Title: Monoidal categories of networks<br />
<br />
Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.<br />
<br />
<br />
<br />
<br />
<br />
===April 6 Edray Goins (Purdue)===<br />
<br />
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups<br />
<br />
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math> A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math> Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math> Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math><br />
<br />
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math> Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math> We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus. <br />
<br />
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.<br />
<br />
===April 13, Jill Pipher, Brown University===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===April 16 Christine Berkesch Zamaere (Minnesota)===<br />
Title: Free complexes on smooth toric varieties<br />
<br />
Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=15373Colloquia/Fall182018-04-08T22:30:28Z<p>Stovall: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16 (Room: 911)<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 5 (Thursday, Room: 911)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[#April 5 John Baez (UC Riverside)| Monoidal categories of networks ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)<br />
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[#April 13, Jill Pipher, Brown University| Mathematical ideas in cryptography ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 16 (Monday)<br />
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)<br />
|[[#Berkesch| Free complexes on smooth toric varieties ]]<br />
| Erman, Sam<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
| May 4<br />
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)<br />
|[[# TBA| TBA ]]<br />
| Ellenberg<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 2 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
===March 16 Anne Gelb (Dartmouth)===<br />
<br />
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity<br />
<br />
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.<br />
<br />
<br />
<br />
<br />
===April 5 John Baez (UC Riverside)===<br />
<br />
Title: Monoidal categories of networks<br />
<br />
Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.<br />
<br />
<br />
<br />
<br />
<br />
===April 6 Edray Goins (Purdue)===<br />
<br />
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups<br />
<br />
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math> A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math> Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math> Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math><br />
<br />
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math> Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math> We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus. <br />
<br />
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.<br />
<br />
===April 13, Jill Pipher, Brown University===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===April 16 Christine Berkesch Zamaere (Minnesota)===<br />
Title: Free complexes on smooth toric varieties<br />
<br />
Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=15372Colloquia/Fall182018-04-08T22:29:57Z<p>Stovall: /* Spring Abstracts */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16 (Room: 911)<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 5 (Thursday, Room: 911)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[#April 5 John Baez (UC Riverside)| Monoidal categories of networks ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)<br />
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[#Jill Pipher| Mathematical ideas in cryptography ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 16 (Monday)<br />
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)<br />
|[[#Berkesch| Free complexes on smooth toric varieties ]]<br />
| Erman, Sam<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
| May 4<br />
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)<br />
|[[# TBA| TBA ]]<br />
| Ellenberg<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 2 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
===March 16 Anne Gelb (Dartmouth)===<br />
<br />
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity<br />
<br />
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.<br />
<br />
<br />
<br />
<br />
===April 5 John Baez (UC Riverside)===<br />
<br />
Title: Monoidal categories of networks<br />
<br />
Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.<br />
<br />
<br />
<br />
<br />
<br />
===April 6 Edray Goins (Purdue)===<br />
<br />
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups<br />
<br />
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math> A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math> Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math> Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math><br />
<br />
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math> Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math> We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus. <br />
<br />
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.<br />
<br />
===April 13, Jill Pipher, Brown University===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===April 16 Christine Berkesch Zamaere (Minnesota)===<br />
Title: Free complexes on smooth toric varieties<br />
<br />
Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=15371Colloquia/Fall182018-04-08T22:28:51Z<p>Stovall: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16 (Room: 911)<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 5 (Thursday, Room: 911)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[#April 5 John Baez (UC Riverside)| Monoidal categories of networks ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)<br />
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[#Jill Pipher| Mathematical ideas in cryptography ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 16 (Monday)<br />
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)<br />
|[[#Berkesch| Free complexes on smooth toric varieties ]]<br />
| Erman, Sam<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
| May 4<br />
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)<br />
|[[# TBA| TBA ]]<br />
| Ellenberg<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 2 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
===March 16 Anne Gelb (Dartmouth)===<br />
<br />
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity<br />
<br />
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.<br />
<br />
<br />
<br />
<br />
===April 5 John Baez (UC Riverside)===<br />
<br />
Title: Monoidal categories of networks<br />
<br />
Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.<br />
<br />
<br />
<br />
<br />
<br />
===April 6 Edray Goins (Purdue)===<br />
<br />
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups<br />
<br />
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math> A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math> Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math> Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math><br />
<br />
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math> Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math> We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus. <br />
<br />
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.<br />
<br />
===April 16 Christine Berkesch Zamaere (Minnesota)===<br />
Title: Free complexes on smooth toric varieties<br />
<br />
Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15366Analysis Seminar2018-04-06T21:24:05Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas, Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15365Analysis Seminar2018-04-06T21:21:29Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| [[#linkofabstract | TBA]]<br />
| Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15364Analysis Seminar2018-04-06T21:20:56Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| '''May 15'''<br />
| Gennady Uraltsev<br />
| Cornell<br />
| Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15363Analysis Seminar2018-04-06T21:20:00Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
| '''May 7'''<br />
| Ebru Toprak<br />
| UIUC<br />
| [[#linkofabstract | TBA]]<br />
|Betsy<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15361Analysis Seminar2018-04-06T16:05:52Z<p>Stovall: /* Lenka Slavíková */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15360Analysis Seminar2018-04-06T16:05:04Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
===Lenka Slavíková===<br />
<br />
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria<br />
<br />
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the </math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15359Analysis Seminar2018-04-06T16:01:43Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15358Analysis Seminar2018-04-06T15:59:16Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
===Jill Pipher===<br />
<br />
Title: Mathematical ideas in cryptography<br />
<br />
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,<br />
including homomorphic encryption.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15357Analysis Seminar2018-04-06T15:58:22Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#Jill Pipher | Mathematical ideas in cryptography]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15356Analysis Seminar2018-04-06T15:57:12Z<p>Stovall: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#linkofabstract | Title]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
===Martina Neuman===<br />
<br />
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces<br />
<br />
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15355Analysis Seminar2018-04-06T15:56:18Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neuman<br />
| UC Berkeley<br />
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#linkofabstract | Title]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15329Analysis Seminar2018-04-03T15:14:24Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neumann<br />
| UC Berkeley<br />
| [[#linkofabstract | Title]]<br />
| Betsy<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#linkofabstract | Title]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| tentatively reserved <br />
| <br />
| [[#linkofabstract | Title]]<br />
| Betsy<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15328Analysis Seminar2018-04-03T15:13:45Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| UW<br />
| [[#linkofabstract | Two endpoint bounds via inverse problems]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| Martina Neumann<br />
| UC Berkeley<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#linkofabstract | Title]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovallhttps://www.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=15212Analysis Seminar2018-03-05T19:13:26Z<p>Stovall: /* 2017-2018 Analysis Seminar Schedule */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Betsy at stovall(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= 2017-2018 Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8 in B239 (Colloquium)<br />
| Tess Anderson<br />
| UW Madison<br />
|[[#linktoabstract | A Spherical Maximal Function along the Primes]]<br />
|Tonghai<br />
|-<br />
|September 19<br />
| Brian Street<br />
| UW Madison<br />
|[[#Brian Street | Convenient Coordinates ]]<br />
| Betsy<br />
|-<br />
|September 26<br />
| Hiroyoshi Mitake<br />
| Hiroshima University<br />
|[[#Hiroyoshi Mitake | Derivation of multi-layered interface system and its application ]]<br />
| Hung<br />
|-<br />
|October 3<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | A polynomial Roth theorem on the real line ]]<br />
| Betsy<br />
|-<br />
|October 10<br />
| Michael Greenblatt<br />
| UI Chicago<br />
|[[#Michael Greenblatt | Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]<br />
| Andreas<br />
|-<br />
|October 17<br />
| David Beltran<br />
| Basque Center of Applied Mathematics<br />
|[[#David Beltran | Fefferman-Stein inequalities ]]<br />
| Andreas<br />
|-<br />
|Wednesday, October 18, 4:00 p.m. in B131<br />
|Jonathan Hickman<br />
|University of Chicago<br />
|[[#Jonathan Hickman | Factorising X^n ]]<br />
|Andreas<br />
|-<br />
|October 24<br />
| Xiaochun Li<br />
| UIUC<br />
|[[#Xiaochun Li | Recent progress on the pointwise convergence problems of Schroedinger equations ]]<br />
| Betsy<br />
|-<br />
|Thursday, October 26, 4:30 p.m. in B139<br />
| Fedor Nazarov<br />
| Kent State University<br />
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp ]]<br />
| Sergey, Andreas<br />
|-<br />
|Friday, October 27, 4:00 p.m. in B239<br />
| Stefanie Petermichl<br />
| University of Toulouse<br />
|[[#Stefanie Petermichl | Higher order Journé commutators ]]<br />
| Betsy, Andreas<br />
|-<br />
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)<br />
| Shaoming Guo<br />
| Indiana University<br />
|[[#Shaoming Guo | Parsell-Vinogradov systems in higher dimensions ]]<br />
| Andreas<br />
|-<br />
|November 14<br />
| Naser Talebizadeh Sardari<br />
| UW Madison<br />
|[[#Naser Talebizadeh Sardari | Quadratic forms and the semiclassical eigenfunction hypothesis ]]<br />
| Betsy<br />
|-<br />
|November 28<br />
| Xianghong Chen<br />
| UW Milwaukee<br />
|[[#Xianghong Chen | Some transfer operators on the circle with trigonometric weights ]]<br />
| Betsy<br />
|-<br />
|Monday, December 4, 4:00, B139<br />
| Bartosz Langowski and Tomasz Szarek<br />
| Institute of Mathematics, Polish Academy of Sciences<br />
|[[#Bartosz Langowski and Tomasz Szarek | Discrete Harmonic Analysis in the Non-Commutative Setting ]]<br />
| Betsy<br />
|-<br />
|Wednesday, December 13, 4:00, B239 (Colloquium)<br />
|Bobby Wilson <br />
|MIT<br />
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]<br />
| Andreas<br />
|-<br />
| Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar)<br />
| Andreas Seeger<br />
| UW<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]] <br />
|<br />
|-<br />
|February 6<br />
| Dong Dong<br />
| UIUC<br />
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]<br />
|Betsy<br />
|-<br />
|February 13<br />
| Sergey Denisov<br />
| UW Madison<br />
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]<br />
| <br />
|-<br />
|February 20<br />
| Ruixiang Zhang <br />
| IAS (Princeton)<br />
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]<br />
| Betsy, Jordan, Andreas<br />
|-<br />
|February 27<br />
|Detlef Müller <br />
|University of Kiel<br />
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]<br />
|Betsy, Andreas<br />
|-<br />
|Wednesday, March 7, 4:00 p.m.<br />
| Winfried Sickel <br />
|Friedrich-Schiller-Universität Jena<br />
| [[#Winfried Sickel | On the regularity of compositions of functions]]<br />
|Andreas<br />
|-<br />
|March 13<br />
|<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|March 20<br />
| Betsy Stovall<br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 3<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 10<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|Friday, April 13, 4:00 p.m. (Colloquium)<br />
|Jill Pipher<br />
|Brown<br />
| [[#linkofabstract | Title]]<br />
|WIMAW<br />
|-<br />
|April 17<br />
| <br />
| <br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|April 24<br />
| Lenka Slavíková<br />
| University of Missouri<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
|May 1<br />
| Xianghong Gong<br />
| UW<br />
| [[#linkofabstract | Title]]<br />
|<br />
|-<br />
|May 15<br />
|Gennady Uraltsev<br />
|Cornell University<br />
| [[#linkofabstract | TBA]]<br />
|Betsy, Andreas<br />
|-<br />
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]<br />
|<br />
|<br />
|<br />
|Betsy, Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Brian Street===<br />
<br />
Title: Convenient Coordinates<br />
<br />
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Title: Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can <br />
be formally derived by the singular limit of the weakly coupled system of <br />
the Allen-Cahn equation. By using the level set approach, this system can be <br />
written as a quasi-monotone degenerate parabolic system. <br />
We give results of the well-posedness of viscosity solutions, and study the <br />
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
===Joris Roos===<br />
<br />
Title: A polynomial Roth theorem on the real line<br />
<br />
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.<br />
<br />
===Michael Greenblatt===<br />
<br />
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces<br />
<br />
Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.<br />
<br />
===David Beltran===<br />
<br />
Title: Fefferman Stein Inequalities<br />
<br />
Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Factorising X^n.<br />
<br />
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.<br />
<br />
===Xiaochun Li ===<br />
<br />
Title: Recent progress on the pointwise convergence problems of Schrodinger equations<br />
<br />
Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.<br />
<br />
===Fedor Nazarov=== <br />
<br />
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden<br />
conjecture is sharp.<br />
<br />
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for<br />
the norm of the Hilbert transform on the line as an operator from $L^1(w)$<br />
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work<br />
with Andrei Lerner and Sheldy Ombrosi.<br />
<br />
===Stefanie Petermichl===<br />
Title: Higher order Journé commutators<br />
<br />
Abstract: We consider questions that stem from operator theory via Hankel and<br />
Toeplitz forms and target (weak) factorisation of Hardy spaces. In<br />
more basic terms, let us consider a function on the unit circle in its<br />
Fourier representation. Let P_+ denote the projection onto<br />
non-negative and P_- onto negative frequencies. Let b denote<br />
multiplication by the symbol function b. It is a classical theorem by<br />
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and<br />
only if b is in an appropriate space of functions of bounded mean<br />
oscillation. The necessity makes use of a classical factorisation<br />
theorem of complex function theory on the disk. This type of question<br />
can be reformulated in terms of commutators [b,H]=bH-Hb with the<br />
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such<br />
as in the real variable setting, in the multi-parameter setting or<br />
other, these classifications can be very difficult.<br />
<br />
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and<br />
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of<br />
spaces of bounded mean oscillation via L^p boundedness of commutators.<br />
We present here an endpoint to this theory, bringing all such<br />
characterisation results under one roof.<br />
<br />
The tools used go deep into modern advances in dyadic harmonic<br />
analysis, while preserving the Ansatz from classical operator theory.<br />
<br />
===Shaoming Guo ===<br />
Title: Parsell-Vinogradov systems in higher dimensions<br />
<br />
Abstract: <br />
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.<br />
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.<br />
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.<br />
<br />
===Naser Talebizadeh Sardari===<br />
<br />
Title: Quadratic forms and the semiclassical eigenfunction hypothesis<br />
<br />
Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.<br />
<br />
===Xianghong Chen===<br />
<br />
Title: Some transfer operators on the circle with trigonometric weights<br />
<br />
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer. <br />
<br />
===Bobby Wilson===<br />
<br />
Title: Projections in Banach Spaces and Harmonic Analysis<br />
<br />
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Dong Dong===<br />
<br />
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory<br />
<br />
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.<br />
<br />
===Sergey Denisov===<br />
<br />
Title: Spectral Szegő theorem on the real line<br />
<br />
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.<br />
<br />
===Ruixiang Zhang===<br />
<br />
Title: The (Euclidean) Fractal Uncertainty Principle<br />
<br />
Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).<br />
<br />
===Detlef Müller===<br />
<br />
Title: On Fourier restriction for a non-quadratic hyperbolic surface<br />
<br />
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.<br />
<br />
===Winfried Sickel===<br />
<br />
Title: On the regularity of compositions of functions<br />
<br />
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math><br />
we associate the composition operator<br />
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>. <br />
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.<br />
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.<br />
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math><br />
such that the composition operator maps such a space <math>E</math> into itself.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Stovall