Difference between revisions of "Analysis Seminar"

From UW-Math Wiki
Jump to: navigation, search
(Analysis Seminar Schedule)
(Oleg Safronov)
 
(317 intermediate revisions by 10 users not shown)
Line 1: Line 1:
'''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 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 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'''
 
| Kyle Hambrook
 
| San Jose State University
 
|[[#Kyle Hambrook |   Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
 
| Andreas
 
|-
 
|Nov 13
 
| Laurent Stolovitch
 
| Université de Nice - Sophia Antipolis
 
|[[#Laurent Stolovitch  |  Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
 
|Xianghong
 
 
|-
 
|-
|Nov 20
+
|November 17
| Grad Student Seminar
+
|Tamas Titkos
|  
+
|BBS U of Applied Sciences and Renyi Institute
|[[#linktoabstract |    ]]
+
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]
 
|  
 
|  
 
|-
 
|-
|Nov 27
+
|November 24
| No Seminar
+
|Shukun Wu
|  
+
|University of Illinois (Urbana-Champaign)
|[[#linktoabstract |   Title ]]
+
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]]  
 
|  
 
|  
 
|-
 
|-
|Dec 4
+
|December 1
| No Seminar
+
| Jonathan Hickman
|  
+
| The University of Edinburgh
|[[#linktoabstract |   Title ]]
+
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]
 
|  
 
|  
 
|-
 
|-
|Jan 22
+
|February 2, 7:00 p.m.
| Brian Cook
+
|Hanlong Fang
| Kent
+
|UW Madison
|[[#Brian Cook |  Equidistribution results for integral points on affine homogenous algebraic varieties ]]
+
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]
| Street
 
|-
 
|Jan 29
 
| No Seminar
 
 
|  
 
|  
|[[#linktoabstract  |  Title ]]
 
|
 
|-
 
|Feb 5, '''B239'''
 
| Alexei Poltoratski
 
| Texas A&M
 
|[[#linktoabstract  |  Title ]]
 
| Denisov
 
|-
 
|'''Friday, Feb 8'''
 
| Aaron Naber
 
| Northwestern University
 
|[[#linktoabstract  |  Title ]]
 
| See colloquium website for location
 
 
|-
 
|-
|Feb 12
+
|February 9
| Shaoming Guo
+
|Bingyang Hu
| UW Madison
+
|Purdue University
|[[# Shaoming Guo | Polynomial Roth theorems in Salem sets  ]]
+
|[[#Bingyang Hu  | Some structure theorems on general doubling measures ]]
 
|  
 
|  
 
|-
 
|-
|'''Wed, Feb 13, B239'''
+
|February 16
| Dean Baskin
+
|Krystal Taylor
| TAMU
+
|The Ohio State University
|[[# Dean Baskin Radiation fields for wave equations ]]
+
|[[#Krystal Taylor Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]
| Colloquium
+
|
 
|-
 
|-
|'''Friday, Feb 15'''
+
|February 23
| Lillian Pierce
+
|Dominique Maldague
| Duke
+
|MIT
|[[# Lillian Pierce |   Short character sums ]]
+
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]
| Colloquium
+
|
 
|-
 
|-
|'''Monday,  Feb 18, 3:30 p.m, B239.'''
+
|March 2
| Daniel Tataru
+
|Diogo Oliveira e Silva
| UC Berkeley
+
|University of Birmingham
|[[#linktoabstract Title ]]
+
|[[#Diogo Oliveira e Silva Global maximizers for spherical restriction ]]
 
|-
 
|Feb 19
 
| PDE seminar in B139
 
|
 
|
 
 
|
 
|
 
|-
 
|-
|Feb 26
+
|March 9
| No Seminar
+
|Oleg Safronov
|
+
|University of North Carolina Charlotte
 +
|[[#Oleg Safronov  | Relations between discrete and continuous spectra of differential operators ]]
 
|
 
|
 
|-
 
|-
|Mar 5
+
|March 16
| Loredana Lanzani
+
|Ziming Shi
| Syracuse University
+
|Rutgers University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
+
|
 
|-
 
|-
|Mar 12
+
|March 23
| Trevor Leslie
+
|Xiumin Du
| UW Madison
+
|Northwestern University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 
|
 
|
 
|-
 
|-
|Mar 19
+
|March 30, 10:00  a.m.
|Spring Break!!!
+
|Etienne Le Masson
|  
+
|Cergy Paris University
|
+
|[[#linktoabstract  |  Title ]]
 
|
 
|
 
|-
 
|-
|Mar 26
+
|April 6
| Person
+
|Theresa Anderson
| Institution
+
|Purdue University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|
 
|-
 
|-
|Apr 2
+
|April 13
| Stefan Steinerberger
+
|Nathan Wagner
| Yale
+
|Washington University  St. Louis
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Shaoming, Andreas
+
|
 
|-
 
|-
 
+
|April 20
|Apr 9
+
|Jongchon Kim
| Franc Forstnerič
+
| University of British Columbia
| Unversity of Ljubljana
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong, Andreas
+
|
 
|-
 
|-
|Apr 16
+
|April 27
| Andrew Zimmer
+
|Yumeng Ou
| Louisiana State University
+
|University of Pennsylvania
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
+
|
 
|-
 
|-
|Apr 23
+
|May 4
| Person
+
|
| Institution
+
|
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Apr 30
 
| Person
 
| Institution
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Simon Marshall===
+
===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.
  
''Integrals of eigenfunctions on hyperbolic manifolds''
+
===Polona Durcik and Joris Roos===
  
Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity.  I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
+
Title: 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
 +
 
 +
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 $p\geq 1$.
  
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.  
+
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).
  
===Polona Durcik===
+
[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.
  
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
+
[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.
  
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.
+
[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.
  
===Song-Ying Li===
+
[5] C. Villani, \emph{Optimal Transport: Old and New,}
 +
(Grundlehren der mathematischen Wissenschaften)
 +
Springer, 2009.
  
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
+
===Shukun Wu===
  
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
+
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
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.
 
  
 +
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.
  
===Hanlong Fan===
 
  
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
+
===Jonathan Hickman===
  
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: Sobolev improving for averages over space curves
  
===Kyle Hambrook===
+
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.
  
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
+
===Hanlong Fang===
  
I will discuss my recent work on some problems concerning
+
Title: Canonical blow-ups of Grassmann manifolds
Fourier decay and Fourier restriction for fractal measures on curves.
 
  
===Laurent Stolovitch===
+
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.
  
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
+
===Bingyang Hu===
  
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: 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.
  
===Brian Cook===
+
===Krystal Taylor===
  
''Equidistribution results for integral points on affine homogenous algebraic varieties''
+
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting
  
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.
+
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.
  
===Shaoming Guo===
+
===Dominique Maldague===
''Polynomial Roth theorems in Salem sets''
 
  
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.
+
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===
  
===Dean Baskin===
+
Title: Global maximizers for spherical restriction
  
''Radiation fields for wave equations''
+
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.
  
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.
+
===Oleg Safronov===
  
 +
Title: Relations between discrete and continuous spectra of differential operators
  
===Lillian Pierce===
+
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.
  
''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:
 +
 
 +
Abstract:
 +
 
 +
===Name===
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
===Name===
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
===Name===
 +
 
 +
Title:
 +
 
 +
Abstract:
  
 
=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 20:04, 26 February 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 Title
March 23 Xiumin Du Northwestern University Title
March 30, 10:00 a.m. Etienne Le Masson Cergy Paris University Title
April 6 Theresa Anderson Purdue University Title
April 13 Nathan Wagner Washington University St. Louis Title
April 20 Jongchon Kim University of British Columbia Title
April 27 Yumeng Ou University of Pennsylvania 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.

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.

Name

Title:

Abstract:

Name

Title:

Abstract:

Name

Title:

Abstract:

Name

Title:

Abstract:

Extras

Blank Analysis Seminar Template


Graduate Student Seminar:

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