Difference between revisions of "Analysis Seminar"

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'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
  
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
+
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
 +
It will be online at least for the Fall semester, with details to be announced in September.
  
If you wish to invite a speaker please contact Brian at street(at)math
+
If you'd like to suggest  speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).
  
===[[Previous Analysis seminars]]===
 
  
= Analysis Seminar Schedule =
+
 
 +
=[[Previous_Analysis_seminars]]=
 +
 
 +
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
 +
 
 +
= Current Analysis Seminar Schedule =
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 16: Line 19:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 10
+
|September 22
| José Madrid
+
|Alexei Poltoratski
| UCLA
+
|UW Madison
|[[#José Madrid  |  On the regularity of maximal operators on Sobolev Spaces ]]
+
|[[#linktoabstract  |  Title ]]
| Andreas, David
 
|-
 
|Sept 13 (Friday, B139)
 
| Yakun Xi
 
| University of  Rochester
 
|[[#Yakun Xi  |  Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
 
| Shaoming
 
|-
 
|Sept 17
 
| Joris Roos
 
| UW Madison
 
|[[#Joris Roos  |  L^p improving estimates for maximal spherical averages ]]
 
| Brian
 
|-
 
|Sept 20 (2:25 PM Friday, Room B139 VV)
 
| Xiaojun Huang
 
| Rutgers University–New Brunswick
 
|[[#linktoabstract  |  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
 
| Xianghong
 
|-
 
|Oct 1
 
| Xiaocheng Li
 
| UW Madison
 
|[[#Xiaocheng Li  |  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
 
| Simon
 
|-
 
|Oct 8
 
| Jeff Galkowski
 
| Northeastern University
 
|[[#Jeff Galkowski  |  Concentration and Growth of Laplace Eigenfunctions ]]
 
| Betsy
 
|-
 
|Oct 15
 
| David Beltran
 
| UW Madison
 
|[[#David Beltran  |  Regularity of the centered fractional maximal function ]]
 
| Brian
 
|-
 
|Oct 22
 
| Laurent Stolovitch
 
| University of Côte d'Azur
 
|[[#Laurent Stolovitch  | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
 
| Xianghong
 
|-
 
|<b>Wednesday Oct 23 in B129</b>
 
|Dominique Kemp
 
|Indiana University
 
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
 
|Betsy
 
|-
 
|Oct 29
 
| Bingyang Hu
 
| UW Madison
 
|[[#Bingyang Hu  |    Sparse bounds of singular Radon transforms]]
 
| Street
 
|-
 
|Nov 5
 
| Kevin O'Neill
 
| UC Davis
 
|[[#Kevin O'Neill  |  A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
 
| Betsy
 
|-
 
|Nov 12
 
| Francesco di Plinio
 
| Washington University in St. Louis
 
|[[#Francesco di Plinio  |  Maximal directional integrals along algebraic and lacunary sets]]
 
| Shaoming
 
|-
 
|Nov 13 (Wednesday)
 
| Xiaochun Li
 
| UIUC
 
|[[#Xiaochun Li  |  Roth's type theorems on progressions]]
 
| Brian, Shaoming
 
|-
 
|Nov 19
 
| Joao Ramos
 
| University of Bonn
 
|[[#Joao Ramos  |  Fourier uncertainty principles, interpolation and uniqueness sets ]]
 
| Joris, Shaoming
 
|-
 
|Jan 21
 
| No Seminar
 
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Jan 28
+
|September 29
| Person
+
|Polona Durcik
| Institution
+
| Chapman University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
|-
 
|Friday, Jan 31, 3 pm, B119
 
| Lillian Pierce
 
| Duke University
 
|[[#Lillian Pierce  |  On Bourgain’s counterexample for the Schrödinger maximal function ]]
 
| Andreas
 
 
|-
 
|-
|Feb 4
+
|September 30
| Ruixiang Zhang
+
|Joris Roos
| UW Madison
+
|University of Massachusetts - Lowell
|[[#Ruixiang Zhang  |  Local smoothing for the wave equation in 2+1 dimensions ]]
 
| Andreas
 
|-
 
|Feb 11
 
| Zane Li
 
| Indiana University
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
|  
 
|-
 
|-
|Feb 18
+
|October 6
| Sergey Denisov
+
|Andrew Zimmer
| UW Madison
+
|UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|Feb 25
+
|October 13
| Speaker
+
|Hong Wang
| Institution
+
|Princeton/IAS
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Host
+
|  
 
|-
 
|-
|Mar 3
+
|October 20
| William Green
+
|Kevin Luli
| Rose-Hulman Institute of Technology
+
|UC Davis
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
|  
 
|-
 
|-
|Mar 10
+
|October 27
| Yifei Pan
+
|Terence Harris
| Indiana University-Purdue University Fort Wayne
+
| Cornell University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
+
|  
 
|-
 
|-
|Mar 17
+
|November 3
| Spring Break!
+
|No seminar
|
+
|  
 
|
 
|
 
|  
 
|  
 
|-
 
|-
|Mar 24
+
|November 10
| Oscar Dominguez
+
|Óscar Domínguez
 
| Universidad Complutense de Madrid
 
| Universidad Complutense de Madrid
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Andreas
+
|  
 
|-
 
|-
|Mar 31
+
|November 17
| Brian Street
+
|Tamas Titkos
| University of Wisconsin-Madison
+
|BBS U of Applied Sciences and Renyi Institute
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Local
+
|  
 
|-
 
|-
|Apr 7
+
|November 24
| Hong Wang
+
| No seminar
| Institution
+
|  
|[[#linktoabstract  |  Title ]]
+
|
| Street
+
|  
 
|-
 
|-
|<b>Monday, Apr 13</b>
+
|December 1
|Yumeng Ou
+
| Jonathan Hickman
|CUNY, Baruch College
+
| The University of Edinburgh
|[[#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 21
 
| Diogo Oliveira e Silva
 
| University of Birmingham
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
|  
 
|-
 
|-
|Apr 28
+
|December 8
| No Seminar
+
|David Beltran
|-
+
| UW Madison
|May 5
 
|Jonathan Hickman
 
|University of Edinburgh
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Andreas
+
|  
 
|-
 
|-
 +
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===José Madrid===
+
===Name===
 
 
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.
 
 
 
===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===
 
 
 
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
 
  
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.
+
Title
  
 +
Abstract
  
===Kevin O'Neill===
 
  
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
+
===Name===
  
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.
+
Title
  
===Francesco di Plinio===
+
Abstract
  
<b>Maximal directional integrals along algebraic and lacunary sets </b>
 
  
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.
+
===Name===
  
===Laurent Stolovitch===
+
Title
  
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
+
Abstract
  
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===
+
===Name===
  
<b>Sparse bounds of singular Radon transforms</b>
+
Title
  
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.
+
Abstract
  
===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.
+
===Name===
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 optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, in particular to number theorists and analysts.
 
  
===Ruixiang Zhang===
+
Title
  
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
+
Abstract
  
Sogge's local smoothing conjecture for the wave equation predicts that the solution to this equation gets smoother  when averaged over time. 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.
 
  
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]

Latest revision as of 17:35, 12 August 2020

The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. It will be online at least for the Fall semester, with details to be announced in September.

If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).


Previous_Analysis_seminars

https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars

Current Analysis Seminar Schedule

date speaker institution title host(s)
September 22 Alexei Poltoratski UW Madison Title
September 29 Polona Durcik Chapman University Title
September 30 Joris Roos University of Massachusetts - Lowell Title
October 6 Andrew Zimmer UW Madison Title
October 13 Hong Wang Princeton/IAS Title
October 20 Kevin Luli UC Davis Title
October 27 Terence Harris Cornell University Title
November 3 No seminar
November 10 Óscar Domínguez Universidad Complutense de Madrid Title
November 17 Tamas Titkos BBS U of Applied Sciences and Renyi Institute Title
November 24 No seminar
December 1 Jonathan Hickman The University of Edinburgh Title
December 8 David Beltran UW Madison Title

Abstracts

Name

Title

Abstract


Name

Title

Abstract


Name

Title

Abstract


Name

Title

Abstract


Name

Title

Abstract


Extras

Blank Analysis Seminar Template