Fall 2021 and Spring 2022 Analysis Seminars: Difference between revisions

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'''Analysis Seminar
'''Fall 2019 and Spring 2020 Analysis Seminar Series
'''
'''


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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|Sept 11
|Sept 10
| Simon Marshall
| José Madrid
| Madison
| UCLA
|[[#Simon Marshall | Integrals of eigenfunctions on hyperbolic manifolds ]]
|[[#José Madrid |   On the regularity of maximal operators on Sobolev Spaces ]]
|  
| Andreas, David
|-
|-
|'''Wednesday, Sept 12'''
|Sept 13 (Friday, B139)
| Gunther Uhlmann 
| Yakun Xi
| University of Washington
| University of Rochester
| Distinguished Lecture Series
|[[#Yakun Xi  |  Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
| See colloquium website for location
| Shaoming
|-
|-
|'''Friday, Sept 14'''
|Sept 17
| Gunther Uhlmann 
| Joris Roos
| University of Washington
| UW Madison
| Distinguished Lecture Series
|[[#Joris Roos  |  L^p improving estimates for maximal spherical averages ]]
| See colloquium website for location
| Brian
|-
|-
|Sept 18
|Sept 20 (2:25 PM Friday, Room B139 VV)
| Grad Student Seminar
| Xiaojun Huang
|  
| Rutgers University–New Brunswick
|
|[[#linktoabstract  |  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
|
| Xianghong
|-
|-
|Sept 25
|Oct 1
| Grad Student Seminar
| Xiaocheng Li
|
| UW Madison
|
|[[#Xiaocheng Li  |  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
|
| Simon
|-
|-
|Oct 2
|Oct 8
| Person
| Jeff Galkowski
| Institution
| Northeastern University
|[[#linktoabstract Title ]]
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
| Sponsor
| Betsy
|-
|-
|Oct 9
|Oct 15
| Hong Wang
| David Beltran
| MIT
|[[#Hong Wang  |  About Falconer distance problem in the plane ]]
| Ruixiang
|-
|Oct 16
| Polona Durcik
| Caltech
|[[#Polona Durcik  |  Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
| Joris
|-
|Oct 23
| Song-Ying Li
| UC Irvine
|[[#Song-Ying Li  |  Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
| Xianghong
|-
|Oct 30
|Grad student seminar
|
|
|
|-
|Nov 6
| Hanlong Fang
| UW Madison
| UW Madison
|[[#HanlongFang A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
|[[#David Beltran Regularity of the centered fractional maximal function ]]
| Brian
| Brian
|-
|-
||'''Monday, Nov. 12'''
|Oct 22
| Kyle Hambrook
| Laurent Stolovitch
| San Jose State University
| University of Côte d'Azur
|[[#Kyle Hambrook |   Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
| Andreas
| Xianghong
|-
|-
|Nov 13
|<b>Wednesday Oct 23 in B129</b>
| Laurent Stolovitch
|Dominique Kemp
| Université de Nice - Sophia Antipolis
|Indiana University
|[[#Laurent Stolovitch  |   Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
|Xianghong
|Betsy
|-
|-
|Nov 20
|Oct 29
| Grad Student Seminar
| Bingyang Hu
|  
| UW Madison
|[[#linktoabstract |   Title ]]
|[[#Bingyang Hu |   Sparse bounds of singular Radon transforms]]
|  
| Street
|-
|-
|Nov 27
|Nov 5
| Person
| Kevin O'Neill
| Institution
| UC Davis
|[[#linktoabstract Title ]]
|[[#Kevin O'Neill A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
| Sponsor
| Betsy
|-
|-
|Dec 4
|Nov 12
| Person
| Francesco di Plinio
| Institution
| Washington University in St. Louis
|[[#linktoabstract Title ]]
|[[#Francesco di Plinio Maximal directional integrals along algebraic and lacunary sets]]
| Sponsor
| Shaoming
|-
|-
|Jan 22
|Nov 13 (Wednesday)
| Brian Cook
| Xiaochun Li
| Kent
| UIUC
|[[#linktoabstract Title ]]
|[[#Xiaochun Li Roth's type theorems on progressions]]
| Street
| Brian, Shaoming
|-
|-
|Jan 29
|Nov 19
| Trevor Leslie
| Joao Ramos
| UW Madison
| University of Bonn
|[[#linktoabstract Title ]]
|[[#Joao Ramos Fourier uncertainty principles, interpolation and uniqueness sets ]]
|  
| Joris, Shaoming
|-
|-
|Feb 5
|Jan 21
| No seminar
| No Seminar
|  
|  
|
|
|
|
|-
|-
|'''Friday, Feb 8'''
|Friday, Jan 31, 4 pm, B239, Colloquium
| Aaron Naber
| Lillian Pierce
| Northwestern University
| Duke University
|[[#linktoabstract Title ]]
|[[#Lillian Pierce On Bourgain’s counterexample for the Schrödinger maximal function ]]
| See colloquium website for location
| Andreas, Simon
|-
|Feb 4
| Ruixiang Zhang
| UW Madison
|[[#Ruixiang Zhang  |  Local smoothing for the wave equation in 2+1 dimensions ]]
| Andreas
|-
|-
|Feb 12
|Feb 11
| No seminar
| Zane Li
|
| Indiana University
|
|[[#Zane Li  |  A bilinear proof of decoupling for the moment curve ]]
|
| Betsy
|-
|-
|'''Friday, Feb 15'''
|Feb 18
| Charles Smart
| Sergey Denisov
| University of Chicago
| UW Madison
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  De Branges canonical systems with finite logarithmic integral ]]
| See colloquium website for information
| Street
|-
|-
|Feb 19
|Feb 25
| Person
| Michel Alexis
| Institution
| Local
|[[#linktoabstract Title ]]
|[[#Michel Alexis The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
| Sponsor
| Denisov
|-
|-
|Feb 26
|Mar 3
| Person
| William Green
| Institution
| Rose-Hulman Institute of Technology
|[[#linktoabstract Title ]]
|[[#William Green Dispersive estimates for the Dirac equation ]]
| Sponsor
| Betsy
|-
|-
|Mar 5
|Mar 10
| Person
| Yifei Pan
| Institution
| Indiana University-Purdue University Fort Wayne
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  Title ]]
| Sponsor
| Xianghong
|-
|-
|Mar 12
|Mar 17
| No Seminar
| Spring Break!
|
|
|[[#linktoabstract  |  Title ]]
|
|
|-
|Mar 19
|Spring Break!!!
|  
|  
|
|
|-
|-
|Apr 2
|Mar 24
| Person
| Oscar Dominguez
| Institution
| Universidad Complutense de Madrid
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  Title ]]
| Sponsor
| Andreas
|-
|-
 
|Mar 31
|Apr 9
| Brian Street
| Franc Forstnerič
| University of Wisconsin-Madison
| Unversity of Ljubljana
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  Title ]]
| Xianghong, Andreas
| Local
|-
|-
|Apr 16
|Apr 7
| Person
| Hong Wang
| Institution
| Institution
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  Title ]]
| Sponsor
| Street
|-
|<b>Monday, Apr 13</b>
|Yumeng Ou
|CUNY, Baruch College
|[[#linktoabstract  |  TBA ]]
|Zhang
|-
|Apr 14
| Tamás Titkos
| BBS University of Applied Sciences & Rényi Institute
|[[#linktoabstract  |  Distance preserving maps on spaces of probability measures ]]
| Street
|-
|-
|Apr 23
|Apr 21
| Person
| Diogo Oliveira e Silva
| Institution
| University of Birmingham
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  Title ]]
| Sponsor
| Betsy
|-
|-
|Apr 30
|Apr 28
| Person
| No Seminar
| Institution
|-
|May 5
|Jonathan Hickman
|University of Edinburgh
|[[#linktoabstract  |  Title ]]
|[[#linktoabstract  |  Title ]]
| Sponsor
| Andreas
|-
|-
|}
|}


=Abstracts=
=Abstracts=
===Simon Marshall===
===José Madrid===
 
Title: On the regularity of maximal operators on Sobolev Spaces
 
Abstract:  In this talk, we will discuss the regularity properties (boundedness and
continuity) of the classical and fractional maximal
operators when these act on the Sobolev space W^{1,p}(\R^n). We will
focus on the endpoint case p=1. We will talk about
some recent results and current open problems.
 
===Yakun Xi===
 
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
 
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.
 
===Joris Roos===
 
Title: L^p improving estimates for maximal spherical averages
 
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
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 $E$.
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.
 
 
 
===Joao Ramos===
 
Title: Fourier uncertainty principles, interpolation and uniqueness sets
 
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
 
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
 
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.
 
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 $f \equiv 0.$
 
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.
 
===Xiaojun Huang===
 
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
 
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.
 
===Xiaocheng Li===
 
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
 
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.
 
 
===Xiaochun Li===
 
Title:  Roth’s type theorems on progressions
 
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.
 
===Jeff Galkowski===
 
<b>Concentration and Growth of Laplace Eigenfunctions</b>
 
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.


''Integrals of eigenfunctions on hyperbolic manifolds''
===David Beltran===


Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity.  I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
Title: Regularity of the centered fractional maximal function


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.


===Hong Wang===
This is joint work with José Madrid.


''About Falconer distance problem in the plane''
===Dominique Kemp===


If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>


===Polona Durcik===
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.


''Singular Brascamp-Lieb inequalities and extended boxes in R^n''


Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
===Kevin O'Neill===


<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>


===Song-Ying Li===
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.


''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
===Francesco di Plinio===


In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
<b>Maximal directional integrals along algebraic and lacunary sets </b>
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
Kohn Laplacian on strictly pseudoconvex hypersurfaces.


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.


===Laurent Stolovitch===
===Laurent Stolovitch===


''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
 
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.
 
===Bingyang Hu===
 
<b>Sparse bounds of singular Radon transforms</b>
 
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.
 
===Lillian Pierce===
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
 
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.
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.”
 
===Ruixiang Zhang===
 
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
 
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.
 
===Zane Li===
 
<b> A bilinear proof of decoupling for the moment curve</b>
 
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.


We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$  are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.


===Sergey Denisov===


===Hanlong Fan===
<b> De Branges canonical systems with finite logarithmic integral </b>


''A generalization of the theorem of Weil and Kodaira on prescribing residues''
We consider measures  m on the real line for which logarithmic
integral exists and give a complete characterization of all Hamiltonians
in de Branges canonical system for which m  is the spectral measure.
This characterization involves the matrix A_2 Muckenhoupt condition on a
fixed scale. Our result provides a generalization of the classical
theorem of Szego for polynomials orthogonal on the unit circle and
complements the Krein-Wiener theorem. Based on the joint work with R.
Bessonov.


An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.


===Kyle Hambrook===
===William Green===


"Fourier Decay and Fourier Restriction for Fractal Measures on Curves"
<b> Dispersive estimates for the Dirac equation </b>


I will discuss my recent work on some problems concerning
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
Fourier decay and Fourier restriction for fractal measures on curves.
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.
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).


=Extras=
=Extras=
[[Blank Analysis Seminar Template]]
[[Blank Analysis Seminar Template]]

Revision as of 20:47, 14 February 2020

Fall 2019 and Spring 2020 Analysis Seminar Series

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

If you wish to invite a speaker please contact Brian at street(at)math

Previous Analysis seminars

Analysis Seminar Schedule

date speaker institution title host(s)
Sept 10 José Madrid UCLA On the regularity of maximal operators on Sobolev Spaces Andreas, David
Sept 13 (Friday, B139) Yakun Xi University of Rochester Distance sets on Riemannian surfaces and microlocal decoupling inequalities Shaoming
Sept 17 Joris Roos UW Madison L^p improving estimates for maximal spherical averages Brian
Sept 20 (2:25 PM Friday, Room B139 VV) Xiaojun Huang Rutgers University–New Brunswick A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries Xianghong
Oct 1 Xiaocheng Li UW Madison An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ Simon
Oct 8 Jeff Galkowski Northeastern University Concentration and Growth of Laplace Eigenfunctions Betsy
Oct 15 David Beltran UW Madison Regularity of the centered fractional maximal function Brian
Oct 22 Laurent Stolovitch University of Côte d'Azur Linearization of neighborhoods of embeddings of complex compact manifolds Xianghong
Wednesday Oct 23 in B129 Dominique Kemp Indiana University Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature Betsy
Oct 29 Bingyang Hu UW Madison Sparse bounds of singular Radon transforms Street
Nov 5 Kevin O'Neill UC Davis A Quantitative Stability Theorem for Convolution on the Heisenberg Group Betsy
Nov 12 Francesco di Plinio Washington University in St. Louis Maximal directional integrals along algebraic and lacunary sets Shaoming
Nov 13 (Wednesday) Xiaochun Li UIUC Roth's type theorems on progressions Brian, Shaoming
Nov 19 Joao Ramos University of Bonn Fourier uncertainty principles, interpolation and uniqueness sets Joris, Shaoming
Jan 21 No Seminar
Friday, Jan 31, 4 pm, B239, Colloquium Lillian Pierce Duke University On Bourgain’s counterexample for the Schrödinger maximal function Andreas, Simon
Feb 4 Ruixiang Zhang UW Madison Local smoothing for the wave equation in 2+1 dimensions Andreas
Feb 11 Zane Li Indiana University A bilinear proof of decoupling for the moment curve Betsy
Feb 18 Sergey Denisov UW Madison De Branges canonical systems with finite logarithmic integral Street
Feb 25 Michel Alexis Local The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight Denisov
Mar 3 William Green Rose-Hulman Institute of Technology Dispersive estimates for the Dirac equation Betsy
Mar 10 Yifei Pan Indiana University-Purdue University Fort Wayne Title Xianghong
Mar 17 Spring Break!
Mar 24 Oscar Dominguez Universidad Complutense de Madrid Title Andreas
Mar 31 Brian Street University of Wisconsin-Madison Title Local
Apr 7 Hong Wang Institution Title Street
Monday, Apr 13 Yumeng Ou CUNY, Baruch College TBA Zhang
Apr 14 Tamás Titkos BBS University of Applied Sciences & Rényi Institute Distance preserving maps on spaces of probability measures Street
Apr 21 Diogo Oliveira e Silva University of Birmingham Title Betsy
Apr 28 No Seminar
May 5 Jonathan Hickman University of Edinburgh Title Andreas

Abstracts

José Madrid

Title: On the regularity of maximal operators on Sobolev Spaces

Abstract: In this talk, we will discuss the regularity properties (boundedness and continuity) of the classical and fractional maximal operators when these act on the Sobolev space W^{1,p}(\R^n). We will focus on the endpoint case p=1. We will talk about some recent results and current open problems.

Yakun Xi

Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities

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.

Joris Roos

Title: L^p improving estimates for maximal spherical averages

Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$. 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 $E$. Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.


Joao Ramos

Title: Fourier uncertainty principles, interpolation and uniqueness sets

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

$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$

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.

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 $f \equiv 0.$

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.

Xiaojun Huang

Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries

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.

Xiaocheng Li

Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$

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.


Xiaochun Li

Title: Roth’s type theorems on progressions

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.

Jeff Galkowski

Concentration and Growth of Laplace Eigenfunctions

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.

David Beltran

Title: Regularity of the centered fractional maximal function

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.

This is joint work with José Madrid.

Dominique Kemp

Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature

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.


Kevin O'Neill

A Quantitative Stability Theorem for Convolution on the Heisenberg Group

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.

Francesco di Plinio

Maximal directional integrals along algebraic and lacunary sets

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.

Laurent Stolovitch

Linearization of neighborhoods of embeddings of complex compact manifolds

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.

Bingyang Hu

Sparse bounds of singular Radon transforms

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.

Lillian Pierce

On Bourgain’s counterexample for the Schrödinger maximal function

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. 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.”

Ruixiang Zhang

Local smoothing for the wave equation in 2+1 dimensions

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.

Zane Li

A bilinear proof of decoupling for the moment curve

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.


Sergey Denisov

De Branges canonical systems with finite logarithmic integral

We consider measures m on the real line for which logarithmic integral exists and give a complete characterization of all Hamiltonians in de Branges canonical system for which m is the spectral measure. This characterization involves the matrix A_2 Muckenhoupt condition on a fixed scale. Our result provides a generalization of the classical theorem of Szego for polynomials orthogonal on the unit circle and complements the Krein-Wiener theorem. Based on the joint work with R. Bessonov.


William Green

Dispersive estimates for the Dirac equation

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 That can be interpreted as a sort of square root of a system of Klein-Gordon equations.

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).

Extras

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