Difference between revisions of "Analysis Seminar"

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'''Analysis Seminar
 
'''
 
  
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.
 +
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, 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: Line 22:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 11
+
|September 22
| Simon Marshall
+
|Alexei Poltoratski
| UW Madison
+
|UW Madison
|[[#Simon Marshall | Integrals of eigenfunctions on hyperbolic manifolds ]]
+
|[[#Alexei Poltoratski |   Dirac inner functions ]]
 
|  
 
|  
 
|-
 
|-
|'''Wednesday, Sept 12'''
+
|September 29
| Gunther Uhlmann 
+
|Joris Roos
| University of Washington
+
|University of Massachusetts - Lowell
| Distinguished Lecture Series
+
|[[#Polona Durcik and Joris Rooslinktoabstract  |  A triangular Hilbert transform with curvature, I ]]
| See colloquium website for location
+
|  
 
|-
 
|-
|'''Friday, Sept 14'''
+
|Wednesday September 30, 4 p.m.
| Gunther Uhlmann 
+
|Polona Durcik
| University of Washington
+
|Chapman University
| Distinguished Lecture Series
+
|[[#Polona Durcik and Joris Roos  |  A triangular Hilbert transform with curvature, II ]]
| See colloquium website for location
+
|  
 
|-
 
|-
|Sept 18
+
|October 6
| Grad Student Seminar
+
|Andrew Zimmer
 +
|UW Madison
 +
|[[#Andrew Zimmer  |  Complex analytic problems on domains with good intrinsic geometry ]]
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Sept 25
+
|October 13
| Grad Student Seminar
+
|Hong Wang
|
+
|Princeton/IAS
|
+
|[[#Hong Wang  |  Improved decoupling for the parabola ]]
|
+
|  
 
|-
 
|-
|Oct 9
+
|October 20
| Hong Wang
+
|Kevin Luli
| MIT
+
|UC Davis
|[[#Hong Wang About Falconer distance problem in the plane ]]
+
|[[#Kevin Luli Smooth Nonnegative Interpolation ]]
| Ruixiang
+
|  
 
|-
 
|-
|Oct 16
+
|October 21, 4.00 p.m.
| Polona Durcik
+
|Niclas Technau
| Caltech
+
|UW Madison
|[[#Polona Durcik Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
+
|[[#Niclas Technau Number theoretic applications of oscillatory integrals ]]
| Joris
+
|  
 
|-
 
|-
|Oct 23
+
|October 27
| Song-Ying Li
+
|Terence Harris
| UC Irvine
+
| Cornell University
|[[#Song-Ying Li Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
+
|[[#Terence Harris Low dimensional pinned distance sets via spherical averages ]]
| Xianghong
+
|  
 
|-
 
|-
|Oct 30
+
|Monday, November 2, 4 p.m.
|Grad student seminar
+
|Yuval Wigderson
|
+
|Stanford  University
|
+
|[[#Yuval Wigderson  |  New perspectives on the uncertainty principle ]]
|
+
|  
 
|-
 
|-
|Nov 6
+
|November 10, 10 a.m.
| Hanlong Fang
+
|Óscar Domínguez
| UW Madison
+
| Universidad Complutense de Madrid
|[[#Hanlong Fang |   A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
+
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]
| Brian
+
|  
 
|-
 
|-
||'''Monday, Nov. 12, B139'''
+
|November 17
| Kyle Hambrook
+
|Tamas Titkos
| San Jose State University
+
|BBS U of Applied Sciences and Renyi Institute
|[[#Kyle Hambrook  |  Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
+
|[[#Tamas Titkos Isometries of Wasserstein spaces ]]
| Andreas
 
|-
 
|Nov 13
 
| Laurent Stolovitch
 
| Université de Nice - Sophia Antipolis
 
|[[#Laurent Stolovitch |  Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
 
|Xianghong
 
|-
 
|Nov 20
 
| Grad Student Seminar
 
|
 
|[[#linktoabstract  |    ]]
 
 
|  
 
|  
 
|-
 
|-
|Nov 27
+
|November 24
| No Seminar
+
|Shukun Wu
|  
+
|University of Illinois (Urbana-Champaign)
|[[#linktoabstract |   ]]
+
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]]  
 
|  
 
|  
 
|-
 
|-
|Dec 4
+
|December 1
| No Seminar
+
| Jonathan Hickman
|[[#linktoabstract |   ]]
+
| The University of Edinburgh
 +
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]
 
|  
 
|  
 
|-
 
|-
|Jan 22
+
|February 2
| Brian Cook
+
|Jongchon Kim
| Kent
+
| UBC
|[[#Brian Cook Equidistribution results for integral points on affine homogenous algebraic varieties ]]
+
|[[#linktoabstract Title ]]
| Street
 
|-
 
|Jan 29
 
| No Seminar
 
 
|  
 
|  
|[[#linktoabstract  |    ]]
 
|
 
 
|-
 
|-
|Feb 5, '''B239'''
+
|February 9
| Alexei Poltoratski
+
|Bingyang Hu
| Texas A&M
+
|Purdue University
|[[#Alexei Poltoratski  |  Completeness of exponentials: Beurling-Malliavin and type problems ]]
+
|[[#linktoabstract  |  Title ]]
| Denisov
 
|-
 
|'''Friday, Feb 8'''
 
| Aaron Naber
 
| Northwestern University
 
|[[#linktoabstract  |  A structure theory for spaces with lower Ricci curvature bounds ]]
 
| See colloquium website for location
 
|-
 
|Feb 12
 
| Shaoming Guo
 
| UW Madison
 
|[[# Shaoming Guo | Polynomial Roth theorems in Salem sets  ]]
 
 
|  
 
|  
 
|-
 
|-
|'''Wed, Feb 13, B239'''
+
|February 16
| Dean Baskin
+
|Krystal Taylor
| TAMU
+
|The Ohio State University
|[[# Dean Baskin Radiation fields for wave  equations ]]
+
|[[#linktoabstract Title ]]
| Colloquium
+
|
 
|-
 
|-
|'''Friday, Feb 15'''
+
|February 23
| Lillian Pierce
+
|Dominique Maldague
| Duke
+
|MIT
|[[# Lillian Pierce Short character sums ]]
+
|[[#linktoabstract Title ]]
| Colloquium
+
|
 
|-
 
|-
|'''Monday,  Feb 18, 3:30 p.m, B239.'''
+
|March 2
| Daniel Tataru
+
|Diogo Oliveira e Silva
| UC Berkeley
+
|University of Birmingham
|[[#lDaniel Tataru A Morawetz inequality for water waves ]]
+
|[[#linktoabstract Title ]]
| PDE Seminar
+
|
|-
 
|Feb 19
 
| Wenjia Jing
 
|Tsinghua University
 
|Periodic  homogenization of Dirichlet problems in perforated domains: a unified proof
 
| PDE Seminar
 
 
|-
 
|-
|Feb 26
+
|March 9
| No Seminar
 
 
|
 
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 
|
 
|
 
|-
 
|-
|Mar 5
+
|March 16
| Loredana Lanzani
+
|Ziming Shi
| Syracuse University
+
|Rutgers University
|[[# Loredana Lanzani On regularity and irregularity of the Cauchy-Szegő projection in several complex variables ]]
+
|[[#linktoabstract Title ]]
| Xianghong
+
|
 
|-
 
|-
|Mar 12
+
|March 23
| Trevor Leslie
+
|
| UW Madison
+
|
|[[# Trevor Leslie Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time ]]
+
|[[#linktoabstract Title ]]
 
|
 
|
 
|-
 
|-
|Mar 19
+
|March 30
|Spring Break!!!
 
|
 
 
|
 
|
 
|
 
|
|-
 
|Mar 26
 
| No seminar
 
|
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|  
+
|
 
|-
 
|-
|Apr 2
+
|April 6
| Stefan Steinerberger
+
|
| Yale
+
|
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Shaoming, Andreas
+
|
 
|-
 
|-
 
+
|April 13
|Apr 9
+
|
| Franc Forstnerič
+
|
| Unversity of Ljubljana
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong, Andreas
+
|
 
|-
 
|-
|Apr 16
+
|April 20
| Andrew Zimmer
+
|
| Louisiana State University
+
|
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
+
|
 
|-
 
|-
|Apr 23
+
|April 27
| Person
+
|
| Institution
+
|
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|
 
|-
 
|-
|Apr 30
+
|May 4
| Reserved
+
|
| Institution
+
|
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Shaoming
 
|-
 
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Simon Marshall===
+
===Alexei Poltoratski===
 +
 
 +
Title: Dirac inner functions
  
''Integrals of eigenfunctions on hyperbolic manifolds''
+
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.
  
Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity.  I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
+
===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===
 
===Hong Wang===
  
''About Falconer distance problem in the plane''
+
Title: Improved decoupling for the parabola
  
If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.  
+
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.
  
===Polona Durcik===
+
===Kevin Luli===
  
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
+
Title: Smooth Nonnegative Interpolation
  
Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
+
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===
  
===Song-Ying Li===
+
Title: Number theoretic applications of oscillatory integrals
  
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
+
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.
  
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
+
===Terence Harris===
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
 
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
 
Kohn Laplacian on strictly pseudoconvex hypersurfaces.
 
  
 +
Title: Low dimensional pinned distance sets via spherical averages
  
===Hanlong Fan===
+
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.
  
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
+
===Yuval Wigderson===
  
An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.
+
Title: New perspectives on the uncertainty principle
  
===Kyle Hambrook===
+
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.
  
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
+
===Oscar Dominguez===
  
I will discuss my recent work on some problems concerning
+
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
Fourier decay and Fourier restriction for fractal measures on curves.
 
  
===Laurent Stolovitch===
+
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.
  
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
+
===Tamas Titkos===
  
We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$  are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.
+
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}.
  
===Brian Cook===
+
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 $p\geq 1$.
  
''Equidistribution results for integral points on affine homogenous algebraic varieties''
+
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).
  
Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.
+
[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.
  
===Alexei Poltoratski===
+
[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.
  
''Completeness of exponentials: Beurling-Malliavin and type problems''
+
[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.
  
This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.
+
[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.
  
===Shaoming Guo===
+
===Shukun Wu===
  
''Polynomial Roth theorems in Salem sets''
+
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
  
Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.  
+
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
  
===Dean Baskin===
+
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.
  
''Radiation fields for wave equations''
+
===Name===
  
Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
+
Title
  
===Lillian Pierce===
+
Abstract:
  
''Short character sums''
+
===Name===
  
A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
+
Title
  
===Loredana Lanzani===
+
Abstract:
  
''On regularity and irregularity of the Cauchy-Szegő projection in several complex variables''
+
===Name===
  
This talk is a survey of my latest, and now final, collaboration with Eli Stein.
+
Title
  
It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.)  We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.
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===Trevor Leslie===
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''Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time''
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In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity.  Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1.  In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time.  When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the ''concentration dimension'' of a certain measure associated to the solution.  The work described is joint with Roman Shvydkoy (UIC).
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Graduate Student Seminar:
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https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html

Latest revision as of 15:52, 3 December 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. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).

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

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 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 Jongchon Kim UBC Title
February 9 Bingyang Hu Purdue University Title
February 16 Krystal Taylor The Ohio State University Title
February 23 Dominique Maldague MIT Title
March 2 Diogo Oliveira e Silva University of Birmingham Title
March 9 Title
March 16 Ziming Shi Rutgers University Title
March 23 Title
March 30 Title
April 6 Title
April 13 Title
April 20 Title
April 27 Title
May 4 Title

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.

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Graduate Student Seminar:

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