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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 20202021 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. 
 +  The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier to accomodate speakers). 
   
−  If you wish to invite a speaker please contact Brian at street(at)math
 +  Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230 
   
−  ===[[Previous Analysis seminars]]===
 +  If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math). 
   
−  = 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: 
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 !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 ]]  +  [[#Alexei Poltoratski  Dirac inner functions ]] 
−   Andreas, David  +   
     
−  Sept 13 (Friday, B139)  +  September 29 
−   Yakun Xi  +  Polona Durcik 
−   University of Rochester  +   Chapman University 
−  [[#Yakun Xi  Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]  +  [[#Polona Durcik and Joris Rooslinktoabstract  A triangular Hilbert transform with curvature, I ]] 
−   Shaoming  +   
     
−  Sept 17  +  Wednesday September 30, 4 p.m. 
−   Joris Roos  +  Joris Roos 
−   UW Madison  +  University of Massachusetts  Lowell 
−  [[#Joris Roos  L^p improving estimates for maximal spherical averages ]]  +  [[#Polona Durcik and Joris Roos  A triangular Hilbert transform with curvature, II ]] 
−   Brian  +   
     
−  Sept 20 (2:25 PM Friday, Room B139 VV)  +  October 6 
−   Xiaojun Huang  +  Andrew Zimmer 
−   Rutgers University–New Brunswick
 +  UW Madison 
−  [[#linktoabstract  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
 +  [[#linktoabstract  Title ]] 
−   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  +  October 13 
−   Person  +  Hong Wang 
−   Institution  +  Princeton/IAS 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Sponsor  +   
     
−  Friday, Jan 31, 4 pm, B239, Colloquium  +  October 20 
−   Lillian Pierce  +  Kevin Luli 
−   Duke University  +  UC Davis 
−  [[#Lillian Pierce  On Bourgain’s counterexample for the Schrödinger maximal function ]]
 
−   Andreas
 
−  
 
−  Feb 4
 
−   Ruixiang Zhang
 
−   UW Madison
 
−  [[#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 27 
−   Sergey Denisov  +  Terence Harris 
−   UW Madison  +   Cornell University 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Street  +   
     
−  Feb 25  +  Monday, November 2, 4 p.m. 
−   Speaker  +  Yuval Wigderson 
−   Institution  +  Stanford University 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Host  +   
     
−  Mar 3  +  November 10 
−   William Green  +  Óscar Domínguez 
−   RoseHulman Institute of Technology  +   Universidad Complutense de Madrid 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Betsy  +   
     
−  Mar 10  +  November 17 
−   Yifei Pan  +  Tamas Titkos 
−   Indiana UniversityPurdue University Fort Wayne  +  BBS U of Applied Sciences and Renyi Institute 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Xianghong  +   
     
−  Mar 17  +  November 24 
−   Spring Break!  +  Shukun Wu 
−    +  University of Illinois (UrbanaChampaign) 
−    +  [[#linktoabstract  Title ]] 
     
     
−  Mar 24  +  December 1 
−   Oscar Dominguez  +   Jonathan Hickman 
−   Universidad Complutense de Madrid  +   The University of Edinburgh 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Andreas  +   
     
−  Mar 31  +  December 8 
−   Brian Street  +  TBA 
−   University of WisconsinMadison  +   
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Local  +   
     
−  Apr 7  +  February 2 
−   Hong Wang  +  Jongchon Kim 
−   Institution  +   UBC 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Street  +   
     
−  <b>Monday, Apr 13</b>  +  February 9 
−  Yumeng Ou  +  Bingyang Hu 
−  CUNY, Baruch College
 +  Purdue University 
−  [[#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
 
−   No Seminar
 
     
−  May 5  +  February 16 
−  Jonathan Hickman  +  David Beltran 
−  University of Edinburgh  +  UW  Madison 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Andreas
 
−  
 
 }   } 
   
 =Abstracts=   =Abstracts= 
−  ===José Madrid===  +  ===Alexei Poltoratski=== 
   
−  Title: On the regularity of maximal operators on Sobolev Spaces  +  Title: Dirac inner functions 
   
−  Abstract: In this talk, we will discuss the regularity properties (boundedness and  +  Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations. 
−  continuity) of the classical and fractional maximal
 +  We will discuss connections between problems in complex function theory, spectral and scattering problems for differential 
−  operators when these act on the Sobolev space W^{1,p}(\R^n). We will
 +  operators and the nonlinear Fourier transform. 
−  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
 +  ===Polona Durcik and Joris Roos=== 
   
−  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 GuthIosevichOuWang 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 CarlesonSj\”olin condition. This is joint work with Iosevich and Liu.
 +  Title: A triangular Hilbert transform with curvature 
   
−  ===Joris Roos===
 +  Abstract: The triangular Hilbert is a twodimensional bilinear singular 
 +  originating in timefrequency analysis. No Lp bounds are currently 
 +  known for this operator. 
 +  In these two talks we discuss a recent joint work with Michael Christ 
 +  on a variant of the triangular Hilbert transform involving curvature. 
 +  This object is closely related to the bilinear Hilbert transform with 
 +  curvature and a maximally modulated singular integral of SteinWainger 
 +  type. As an application we also discuss a quantitative nonlinear Roth 
 +  type theorem on patterns in the Euclidean plane. 
 +  The second talk will focus on the proof of a key ingredient, a certain 
 +  regularity estimate for a local operator. 
   
−  Title: L^p improving estimates for maximal spherical averages
 +  ===Name=== 
   
−  Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
 +  Title 
−  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.
 
   
 +  Abstract 
   
   
−  ===Joao Ramos===  +  ===Name=== 
   
−  Title: Fourier uncertainty principles, interpolation and uniqueness sets  +  Title 
   
−  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  +  Abstract 
   
−  $$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(xn))}{\pi(x n)}. $$
 
   
−  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.
 +  ===Name=== 
   
−  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 $f \equiv 0.$
 +  Title 
   
−  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.
 +  Abstract 
   
−  ===Xiaojun Huang===  +  =Extras= 
−   +  [[Blank Analysis Seminar Template]] 
−  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\”osTuran, 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 HardyLittlewood maximal function. A key step in our analysis is a relation between the centered and noncentered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the noncentered 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 ndimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zerocurvature twodimensional surfaces without umbilical points (the socalled 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===
 
−   
−  <b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
 
−   
−  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 nearmaximizers. Specifically, any nearmaximizing 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===
 
   
−  <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 twodimensional and $n$dimensional coverings combining seemingly contrasting ideas of ParcetRogers and of NagelSteinWainger.
 +  Graduate Student Seminar: 
   
−  ===Laurent Stolovitch===
 +  https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html 
−   
−  <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 setup 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 spacetime estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n1). 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=
 
−  [[Blank Analysis Seminar Template]]
 