Difference between revisions of "Analysis Seminar"

From UW-Math Wiki
Jump to: navigation, search
(Abstracts)
(Current Analysis Seminar Schedule)
 
(190 intermediate revisions by 8 users not shown)
Line 1: Line 1:
'''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 for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate 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.  If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
  
===[[Previous Analysis seminars]]===
+
If you'd like to suggest speakers for the spring semester please contact David and Andreas.
  
= 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 21:
 
!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
+
|Joris Roos
| University of Rochester
+
|University of Massachusetts - Lowell
|[[#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
+
|Polona Durcik
| UW Madison
+
|Chapman University
|[[#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 ]]
+
|[[#Andrew Zimmer Complex analytic problems on domains with good intrinsic geometry ]]
| Xianghong
+
|  
 
|-
 
|-
|Oct 1
+
|October 13
| Xiaocheng Li
+
|Hong Wang
| UW Madison
+
|Princeton/IAS
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
+
|[[#Hong Wang |   Improved decoupling for the parabola ]]
| Simon
+
|  
 
|-
 
|-
|Oct 8
+
|October 20
| Jeff Galkowski
+
|Kevin Luli
| Northeastern University
+
|UC Davis
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
+
|[[#Kevin Luli Smooth Nonnegative Interpolation ]]
| Betsy
+
|  
 
|-
 
|-
|Oct 15
+
|October 21, 4.00 p.m.
| David Beltran
+
|Niclas Technau
| UW Madison
+
|UW Madison
|[[#David Beltran Regularity of the centered fractional maximal function ]]
+
|[[#Niclas Technau Number theoretic applications of oscillatory integrals ]]
| Brian
+
|  
 
|-
 
|-
|Oct 22
+
|October 27
| Laurent Stolovitch
+
|Terence Harris
| University of Côte d'Azur
+
| Cornell University
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
+
|[[#Terence Harris |   Low dimensional pinned distance sets via spherical averages ]]
| Xianghong
+
|  
 
|-
 
|-
|<b>Wednesday Oct 23 in B129</b>
+
|Monday, November 2, 4 p.m.
|Dominique Kemp
+
|Yuval Wigderson
|Indiana University
+
|Stanford  University
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
+
|[[#Yuval Wigderson  |   New perspectives on the uncertainty principle ]]
|Betsy
+
|  
 
|-
 
|-
|Oct 29
+
|November 10, 10 a.m.
| Bingyang Hu
+
|Óscar Domínguez
| UW Madison
+
| Universidad Complutense de Madrid
|[[#Bingyang Hu |   Sparse bounds of singular Radon transforms]]
+
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]
| Street
+
|  
 
|-
 
|-
|Nov 5
+
|November 17
| Kevin O'Neill
+
|Tamas Titkos
| UC Davis
+
|BBS U of Applied Sciences and Renyi Institute
|[[#Kevin O'Neill |  A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
+
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]
| Betsy
+
|  
 
|-
 
|-
|Nov 12
+
|November 24
| Francesco di Plinio
+
|Shukun Wu
| Washington University in St. Louis
+
|University of Illinois (Urbana-Champaign)
|[[#Francesco di Plinio |   Maximal directional integrals along algebraic and lacunary sets]]
+
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]]  
| Shaoming
+
|  
 
|-
 
|-
|Nov 13 (Wednesday)
+
|December 1
| Xiaochun Li
+
| Jonathan Hickman
| UIUC
+
| The University of Edinburgh
|[[#Xiaochun Li |   Roth's type theorems on progressions]]
+
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]
| Brian, Shaoming
+
|  
 
|-
 
|-
|Nov 19
+
|February 2, 7:00 p.m.
| Joao Ramos
+
|Hanlong Fang
| University of Bonn
+
|UW Madison
|[[#Joao Ramos |  Fourier uncertainty principles, interpolation and uniqueness sets ]]
+
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]
| Joris, Shaoming
+
|  
 
|-
 
|-
|Nov 26
+
|February 9
| No Seminar
+
|Bingyang Hu
 +
|Purdue University
 +
|[[#Bingyang Hu  |  Some structure theorems on general doubling measures ]]
 
|  
 
|  
 +
|-
 +
|February 16
 +
|Krystal Taylor
 +
|The Ohio State University
 +
|[[#Krystal Taylor  |  Quantifications of the Besicovitch Projection theorem in a nonlinear setting  ]]
 
|
 
|
|
 
 
|-
 
|-
|Dec 3
+
|February 23
| Person
+
|Dominique Maldague
| Institution
+
|MIT
|[[#linktoabstract |   Title ]]
+
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]
| Sponsor
+
|
 
|-
 
|-
|Dec 10
+
|March 2
| No Seminar
+
|Diogo Oliveira e Silva
|  
+
|University of Birmingham
 +
|[[#Diogo Oliveira e Silva  |  Global maximizers for spherical restriction ]]
 
|
 
|
 +
|-
 +
|March 9
 +
|Oleg Safronov
 +
|University of North Carolina Charlotte
 +
|[[#Oleg Safronov  | Relations between discrete and continuous spectra of differential operators ]]
 
|
 
|
 
|-
 
|-
|Jan 21
+
|March 16
| No Seminar
+
|Ziming Shi
|  
+
|Rutgers University
 +
|[[#Ziming Shi  | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary  ]]
 
|
 
|
 +
|-
 +
|March 23
 +
|Xiumin Du
 +
|Northwestern University
 +
|[[#Xiumin Du  |  Falconer's distance set problem ]]
 
|
 
|
 
|-
 
|-
|Jan 28
+
|March 30, 10:00  a.m.
| Person
+
|Etienne Le Masson
| Institution
+
|Cergy Paris University
|[[#linktoabstract |   Title ]]
+
|[[#Etienne Le Masson | Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces ]]
| Sponsor
+
|
 
|-
 
|-
|Friday, Jan 31, 3 pm, B119
+
|April 6
| Lillian Pierce
+
|Theresa Anderson
| Duke University
+
|Purdue University
|[[#Lillian Pierce On Bourgain’s counterexample for the Schrödinger maximal function ]]
+
|[[#Theresa Anderson Dyadic analysis (virtually) meets number theory ]]
| Andreas
+
|
 
|-
 
|-
|Feb 4
+
|April 13
| Person
+
|Nathan Wagner
| Institution
+
|Washington University  St. Louis
|[[#linktoabstract |   Title ]]
+
|[[#Nathan Wagner | Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness ]]
| Sponsor
+
|
 
|-
 
|-
|Feb 11
+
|April 20
| Zane Li
+
|David Beltran
| Indiana University
 
|[[#linktoabstract  |  Title ]]
 
| Betsy
 
|-
 
|Feb 18
 
| Sergey Denisov
 
 
| UW Madison
 
| UW Madison
|[[#linktoabstract Title ]]
+
|[[#David Beltran Sobolev improving for averages over curves in $\mathbb{R}^4$]]
| Street
 
|-
 
|Feb 25
 
|  Speaker
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Host
 
|-
 
|Mar 3
 
| William Green
 
| Rose-Hulman Institute of Technology
 
|[[#linktoabstract  |  Title ]]
 
| Betsy
 
|-
 
|Mar 10
 
| Yifei Pan
 
| Indiana University-Purdue University Fort Wayne
 
|[[#linktoabstract  |  Title ]]
 
| Xianghong
 
|-
 
|Mar 17
 
| Spring Break!
 
|
 
 
|
 
|
|
 
|-
 
|Mar 24
 
| Oscar Dominguez
 
| Universidad Complutense de Madrid
 
|[[#linktoabstract  |  Title ]]
 
| Andreas
 
|-
 
|Mar 31
 
| Brian Street
 
| University of Wisconsin-Madison
 
|[[#linktoabstract  |  Title ]]
 
| Local
 
|-
 
|Apr 7
 
| Hong Wang
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Street
 
 
|-
 
|-
|<b>Monday, Apr 13</b>
+
|April 27
 
|Yumeng Ou
 
|Yumeng Ou
|CUNY, Baruch College
+
|University of Pennsylvania
|[[#linktoabstract  |  TBA ]]
+
|[[#Yumeng Ou On the multiparameter distance problem]]
|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 ]]
 
| Betsy
 
|-
 
|Apr 28
 
| No Seminar
 
|-
 
|May 5
 
|Jonathan Hickman
 
|University of Edinburgh
 
|[[#linktoabstract  |  Title ]]
 
| Andreas
 
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===José Madrid===
+
===Alexei Poltoratski===
 +
 
 +
Title: Dirac inner functions
 +
 
 +
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.
 +
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential
 +
operators and the non-linear Fourier transform.
 +
 
 +
===Polona Durcik and Joris Roos===
 +
 
 +
Title: A triangular Hilbert transform with curvature, I & II.
 +
 
 +
Abstract: The triangular Hilbert is a two-dimensional bilinear singular
 +
originating in time-frequency 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 Stein-Wainger
 +
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.
 +
 
 +
===Andrew Zimmer===
 +
 
 +
Title:  Complex analytic problems on domains with good intrinsic geometry
 +
 
 +
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).
 +
 
 +
===Hong Wang===
  
Title: On the regularity of maximal operators on Sobolev Spaces
+
Title: Improved decoupling for the parabola
  
Abstract: In this talk, we will discuss the regularity properties (boundedness and
+
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. 
continuity) of the classical and fractional maximal
+
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$.   This is joint work with Larry Guth and Dominique Maldague.
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===
+
===Kevin Luli===
  
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
+
Title: Smooth Nonnegative Interpolation
  
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.
+
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E  \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets.  
  
===Joris Roos===
+
===Niclas Technau===
  
Title: L^p improving estimates for maximal spherical averages
+
Title: Number theoretic applications of oscillatory integrals
  
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
+
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.
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.
 
  
 +
===Terence Harris===
  
 +
Title: Low dimensional pinned distance sets via spherical averages
  
===Joao Ramos===
+
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.
  
Title: Fourier uncertainty principles, interpolation and uniqueness sets
+
===Yuval Wigderson===
  
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
+
Title: New perspectives on the uncertainty principle
  
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$  
+
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.
  
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.
+
===Oscar Dominguez===
  
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: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
  
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: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, ), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.
  
===Xiaojun Huang===
+
===Tamas Titkos===
  
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
+
Title: Isometries of Wasserstein spaces
  
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.
+
Abstract: Due to its nice theoretical properties and an astonishing number of
 +
applications via optimal transport problems, probably the most
 +
intensively studied metric nowadays is the p-Wasserstein metric. Given
 +
a complete and separable metric space $X$ and a real number $p\geq1$,
 +
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection
 +
of Borel probability measures with finite $p$-th moment, endowed with a
 +
distance which is calculated by means of transport plans \cite{5}.
  
===Xiaocheng Li===
+
The main aim of our research project is to reveal the structure of the
 +
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although
 +
$\mathrm{Isom}(X)$ embeds naturally into
 +
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding
 +
turned out to be surjective in many cases (see e.g. [1]), these two
 +
groups are not isomorphic in general. Kloeckner in [2] described
 +
the isometry group of the quadratic Wasserstein space
 +
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$
 +
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$
 +
is extremely rich. Namely, it contains a large subgroup of wild behaving
 +
isometries that distort the shape of measures. Following this line of
 +
investigation, in \cite{3} we described
 +
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and
 +
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.
  
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
+
In this talk I will survey first some of the earlier results in the
 +
subject, and then I will present the key results of [3]. If time
 +
permits, I will also report on our most recent manuscript [4] in
 +
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)
 +
and D\'aniel Virosztek (IST Austria).
  
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.
+
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein
 +
spaces: isometric rigidity in negative curvature}, International
 +
Mathematics Research Notices, 2016 (5), 1368--1386.
  
 +
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean
 +
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di
 +
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.
  
===Xiaochun Li===
+
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of
 +
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373
 +
(2020), 5855--5883.
  
Title: Roth’s type theorems on progressions
+
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
 +
Wasserstein spaces: The Hilbertian case}, submitted manuscript.
  
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.
+
[5] C. Villani, \emph{Optimal Transport: Old and New,}
 +
(Grundlehren der mathematischen Wissenschaften)
 +
Springer, 2009.
  
===Jeff Galkowski===
+
===Shukun Wu===
  
<b>Concentration and Growth of Laplace Eigenfunctions</b>
+
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
  
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.
+
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem.  
  
===David Beltran===
 
  
Title: Regularity of the centered fractional maximal function
+
===Jonathan Hickman===
 +
 
 +
Title: Sobolev improving for averages over space curves
 +
 
 +
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity.  Joint with D. Beltran, S. Guo and A. Seeger.
 +
 
 +
===Hanlong Fang===
 +
 
 +
Title: Canonical blow-ups of Grassmann manifolds
 +
 
 +
Abstract:  We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification.
 +
 
 +
===Bingyang Hu===
 +
 
 +
Title: Some structure theorems on general doubling measures.
 +
 
 +
Abstract: In this talk, we will first  several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely,  we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.
 +
 
 +
===Krystal Taylor===
 +
 
 +
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting
 +
 
 +
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections.
 +
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points.  In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.
 +
 
 +
===Dominique Maldague===
 +
 
 +
Title: A new proof of decoupling for the parabola
 +
 
 +
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.
 +
 
 +
===Diogo Oliveira e Silva===
 +
 
 +
Title: Global maximizers for spherical restriction
 +
 
 +
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.
 +
 
 +
===Oleg Safronov===
 +
 
 +
Title: Relations between discrete and continuous spectra of differential operators
 +
 
 +
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation:  one needs to consider two operators one of which is obtained  from the other
 +
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.
 +
 
 +
===Ziming Shi===
  
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.
+
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary
  
This is joint work with José Madrid.
+
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$  and $s>1/p$.
 +
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work  of X. Gong.
 +
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate.
 +
Joint work with Liding Yao.
  
===Dominique Kemp===
+
===Xiumin Du===
  
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
+
Title: Falconer's distance set problem
  
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.
+
Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.
  
 +
===Etienne Le Masson===
  
===Kevin O'Neill===
+
Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces
  
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
+
Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods.
 +
Joint work with Tuomas Sahlsten.
  
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.
+
===Theresa Anderson===
  
===Francesco di Plinio===
+
Title: Dyadic analysis (virtually) meets number theory
  
<b>Maximal directional integrals along algebraic and lacunary sets </b>
+
Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.
  
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.
+
===Nathan Wagner===
  
===Laurent Stolovitch===
+
Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness
  
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
+
Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.
  
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.
+
===David Beltran===
  
===Bingyang Hu===
+
Title: Sobolev improving for averages over curves in $\mathbb{R}^4$
  
<b>Sparse bounds of singular Radon transforms</b>
+
Abstract: Given a smooth non-degenerate space curve (that is, a smooth curve whose n-1 curvature functions are non-vanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$-Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the non-trivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.
  
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.
+
===Yumeng Ou===
  
===Lillian Pierce===
+
Title: On the multiparameter distance problem
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
 
  
n 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.  
+
Abstract: In this talk, we will describe some recent progress on the Falconer distance problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{|x-y|: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent work on this problem, which also includes some new results on the multiparameter radial projection theory of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.
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.
 
  
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
 +
 +
 +
Graduate Student Seminar:
 +
 +
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html

Latest revision as of 08:47, 21 April 2021

The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).

Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.

If you'd like to suggest speakers for the spring semester please contact David and Andreas.


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 Dirac inner functions
September 29 Joris Roos University of Massachusetts - Lowell A triangular Hilbert transform with curvature, I
Wednesday September 30, 4 p.m. Polona Durcik Chapman University A triangular Hilbert transform with curvature, II
October 6 Andrew Zimmer UW Madison Complex analytic problems on domains with good intrinsic geometry
October 13 Hong Wang Princeton/IAS Improved decoupling for the parabola
October 20 Kevin Luli UC Davis Smooth Nonnegative Interpolation
October 21, 4.00 p.m. Niclas Technau UW Madison Number theoretic applications of oscillatory integrals
October 27 Terence Harris Cornell University Low dimensional pinned distance sets via spherical averages
Monday, November 2, 4 p.m. Yuval Wigderson Stanford University New perspectives on the uncertainty principle
November 10, 10 a.m. Óscar Domínguez Universidad Complutense de Madrid New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
November 17 Tamas Titkos BBS U of Applied Sciences and Renyi Institute Isometries of Wasserstein spaces
November 24 Shukun Wu University of Illinois (Urbana-Champaign) On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
December 1 Jonathan Hickman The University of Edinburgh Sobolev improving for averages over space curves
February 2, 7:00 p.m. Hanlong Fang UW Madison Canonical blow-ups of Grassmann manifolds
February 9 Bingyang Hu Purdue University Some structure theorems on general doubling measures
February 16 Krystal Taylor The Ohio State University Quantifications of the Besicovitch Projection theorem in a nonlinear setting
February 23 Dominique Maldague MIT A new proof of decoupling for the parabola
March 2 Diogo Oliveira e Silva University of Birmingham Global maximizers for spherical restriction
March 9 Oleg Safronov University of North Carolina Charlotte Relations between discrete and continuous spectra of differential operators
March 16 Ziming Shi Rutgers University Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary
March 23 Xiumin Du Northwestern University Falconer's distance set problem
March 30, 10:00 a.m. Etienne Le Masson Cergy Paris University Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces
April 6 Theresa Anderson Purdue University Dyadic analysis (virtually) meets number theory
April 13 Nathan Wagner Washington University St. Louis Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness
April 20 David Beltran UW Madison Sobolev improving for averages over curves in $\mathbb{R}^4$
April 27 Yumeng Ou University of Pennsylvania On the multiparameter distance problem

Abstracts

Alexei Poltoratski

Title: Dirac inner functions

Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations. We will discuss connections between problems in complex function theory, spectral and scattering problems for differential operators and the non-linear Fourier transform.

Polona Durcik and Joris Roos

Title: A triangular Hilbert transform with curvature, I & II.

Abstract: The triangular Hilbert is a two-dimensional bilinear singular originating in time-frequency 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 Stein-Wainger 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.

Andrew Zimmer

Title: Complex analytic problems on domains with good intrinsic geometry

Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).

Hong Wang

Title: Improved decoupling for the parabola

Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.

Kevin Luli

Title: Smooth Nonnegative Interpolation

Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets.

Niclas Technau

Title: Number theoretic applications of oscillatory integrals

Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.

Terence Harris

Title: Low dimensional pinned distance sets via spherical averages

Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.

Yuval Wigderson

Title: New perspectives on the uncertainty principle

Abstract: The phrase ``uncertainty principle refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.

Oscar Dominguez

Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions

Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.

Tamas Titkos

Title: Isometries of Wasserstein spaces

Abstract: Due to its nice theoretical properties and an astonishing number of applications via optimal transport problems, probably the most intensively studied metric nowadays is the p-Wasserstein metric. Given a complete and separable metric space $X$ and a real number $p\geq1$, one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection of Borel probability measures with finite $p$-th moment, endowed with a distance which is calculated by means of transport plans \cite{5}.

The main aim of our research project is to reveal the structure of the isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although $\mathrm{Isom}(X)$ embeds naturally into $\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding turned out to be surjective in many cases (see e.g. [1]), these two groups are not isomorphic in general. Kloeckner in [2] described the isometry group of the quadratic Wasserstein space $\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$ is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$ is extremely rich. Namely, it contains a large subgroup of wild behaving isometries that distort the shape of measures. Following this line of investigation, in \cite{3} we described $\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and $\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.

In this talk I will survey first some of the earlier results in the subject, and then I will present the key results of [3]. If time permits, I will also report on our most recent manuscript [4] in which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading) and D\'aniel Virosztek (IST Austria).

[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein spaces: isometric rigidity in negative curvature}, International Mathematics Research Notices, 2016 (5), 1368--1386.

[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.

[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373 (2020), 5855--5883.

[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of Wasserstein spaces: The Hilbertian case}, submitted manuscript.

[5] C. Villani, \emph{Optimal Transport: Old and New,} (Grundlehren der mathematischen Wissenschaften) Springer, 2009.

Shukun Wu

Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator

Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem.


Jonathan Hickman

Title: Sobolev improving for averages over space curves

Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.

Hanlong Fang

Title: Canonical blow-ups of Grassmann manifolds

Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification.

Bingyang Hu

Title: Some structure theorems on general doubling measures.

Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.

Krystal Taylor

Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting

Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.

Dominique Maldague

Title: A new proof of decoupling for the parabola

Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.

Diogo Oliveira e Silva

Title: Global maximizers for spherical restriction

Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.

Oleg Safronov

Title: Relations between discrete and continuous spectra of differential operators

Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.

Ziming Shi

Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary

Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. Joint work with Liding Yao.

Xiumin Du

Title: Falconer's distance set problem

Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.

Etienne Le Masson

Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces

Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods. Joint work with Tuomas Sahlsten.

Theresa Anderson

Title: Dyadic analysis (virtually) meets number theory

Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.

Nathan Wagner

Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness

Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.

David Beltran

Title: Sobolev improving for averages over curves in $\mathbb{R}^4$

Abstract: Given a smooth non-degenerate space curve (that is, a smooth curve whose n-1 curvature functions are non-vanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$-Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the non-trivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

Yumeng Ou

Title: On the multiparameter distance problem

Abstract: In this talk, we will describe some recent progress on the Falconer distance problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{|x-y|: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent work on this problem, which also includes some new results on the multiparameter radial projection theory of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.

Extras

Blank Analysis Seminar Template


Graduate Student Seminar:

https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html