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 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  Betsy at stovall(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.
  
= 2017-2018 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)
 
|-
 
|-
|September 8 in B239 (Colloquium)
+
|September 22
| Tess Anderson
+
|Alexei Poltoratski
| UW Madison
+
|UW Madison
|[[#linktoabstract A Spherical Maximal Function along the Primes]]
+
|[[#Alexei Poltoratski Dirac inner functions ]]
|Tonghai
+
|  
 
|-
 
|-
|September 19
+
|September 29
| Brian Street
+
|Joris Roos
| UW Madison
+
|University of Massachusetts - Lowell
|[[#Brian Street |   Convenient Coordinates ]]
+
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]
| Betsy
+
|  
 
|-
 
|-
|September 26
+
|Wednesday September 30, 4 p.m.
| Hiroyoshi Mitake
+
|Polona Durcik
| Hiroshima University
+
|Chapman University
|[[#Hiroyoshi Mitake |   Derivation of multi-layered interface system and its application ]]
+
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]
| Hung
+
|  
 
|-
 
|-
|October 3
+
|October 6
| Joris Roos
+
|Andrew Zimmer
| UW Madison
+
|UW Madison
|[[#Joris Roos A polynomial Roth theorem on the real line ]]
+
|[[#Andrew Zimmer Complex analytic problems on domains with good intrinsic geometry ]]
| Betsy
+
|  
 
|-
 
|-
|October 10
+
|October 13
| Michael Greenblatt
+
|Hong Wang
| UI Chicago
+
|Princeton/IAS
|[[#Michael Greenblatt Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
+
|[[#Hong Wang Improved decoupling for the parabola ]]
| Andreas
+
|  
 
|-
 
|-
|October 17
+
|October 20
| David Beltran
+
|Kevin Luli
| Basque Center of Applied Mathematics
+
|UC Davis
|[[#David Beltran Fefferman-Stein inequalities ]]
+
|[[#Kevin Luli Smooth Nonnegative Interpolation ]]
| Andreas
+
|  
 
|-
 
|-
|Wednesday, October 18, 4:00 p.m. in B131
+
|October 21, 4.00 p.m.
|Jonathan Hickman
+
|Niclas Technau
|University of Chicago
+
|UW Madison
|[[#Jonathan Hickman | Factorising X^n  ]]
+
|[[#Niclas Technau |   Number theoretic applications of oscillatory integrals ]]
|Andreas
+
|  
 
|-
 
|-
|October 24
+
|October 27
| Xiaochun Li
+
|Terence Harris
| UIUC
+
| Cornell University
|[[#Xiaochun Li Recent progress on the pointwise convergence problems of Schroedinger equations ]]
+
|[[#Terence Harris Low dimensional pinned distance sets via spherical averages ]]
| Betsy
+
|  
 
|-
 
|-
|Thursday, October 26, 4:30 p.m. in B139
+
|Monday, November 2, 4 p.m.
| Fedor Nazarov
+
|Yuval Wigderson
| Kent State University
+
|Stanford  University
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp  ]]
+
|[[#Yuval Wigderson |   New perspectives on the uncertainty principle ]]
| Sergey, Andreas
+
|  
 
|-
 
|-
|Friday, October 27, 4:00 p.m. in B239
+
|November 10, 10 a.m.  
| Stefanie Petermichl
+
|Óscar Domínguez
| University of Toulouse
+
| Universidad Complutense de Madrid
|[[#Stefanie Petermichl | Higher order Journé commutators  ]]
+
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]
| Betsy, Andreas
+
|  
 
|-
 
|-
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
+
|November 17
| Shaoming Guo
+
|Tamas Titkos
| Indiana University
+
|BBS U of Applied Sciences and Renyi Institute
|[[#Shaoming Guo |  Parsell-Vinogradov systems in higher dimensions ]]
+
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]
| Andreas
+
|  
 
|-
 
|-
|November 14
+
|November 24
| Naser Talebizadeh Sardari
+
|Shukun Wu
| UW Madison
+
|University of Illinois (Urbana-Champaign)
|[[#Naser Talebizadeh Sardari |   Quadratic forms and the semiclassical eigenfunction hypothesis ]]
+
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]]  
| Betsy
+
|  
 
|-
 
|-
|November 28
+
|December 1
| Xianghong Chen
+
| Jonathan Hickman
| UW Milwaukee
+
| The University of Edinburgh
|[[#Xianghong Chen |   Some transfer operators on the circle with trigonometric weights ]]
+
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]
| Betsy
+
|  
 
|-
 
|-
|Monday, December 4, 4:00, B139
+
|February 2, 7:00 p.m.
| Bartosz Langowski and Tomasz Szarek
+
|Hanlong Fang
| Institute of Mathematics, Polish Academy of Sciences
+
|UW Madison
|[[#Bartosz Langowski and Tomasz Szarek |  Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]
| Betsy
+
|  
 
|-
 
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
+
|February 9
|Bobby Wilson
+
|Bingyang Hu
|MIT
+
|Purdue University
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
+
|[[#Bingyang Hu  | Some structure theorems on general doubling measures ]]
| Andreas
+
|  
 
|-
 
|-
| Monday, February 5, 3:00-3:50, B341  (PDE-GA seminar)
+
|February 16
| Andreas Seeger
+
|Krystal Taylor
| UW
+
|The Ohio State University
|[[#Andreas Seeger Singular integrals and a problem on mixing flows]]  
+
|[[#Krystal Taylor  |   Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]
 
|
 
|
 
|-
 
|-
|February 6
+
|February 23
| Dong Dong
+
|Dominique Maldague
| UIUC
+
|MIT
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
+
|[[#Dominique Maldague  | A new proof of decoupling for the parabola ]]
|Betsy
+
|
 
|-
 
|-
|February 13
+
|March 2
| Sergey  Denisov
+
|Diogo Oliveira e Silva
| UW Madison
+
|University of Birmingham
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]
+
|[[#Diogo Oliveira e Silva  |   Global maximizers for spherical restriction ]]
|  
+
|
 
|-
 
|-
|February 20
+
|March 9
| Ruixiang Zhang
+
|Oleg Safronov
| IAS (Princeton)
+
|University of North Carolina Charlotte
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]
+
|[[#Oleg Safronov  | Relations between discrete and continuous spectra of differential operators ]]
| Betsy, Jordan, Andreas
+
|
 
|-
 
|-
|February 27
+
|March 16
|Detlef Müller
+
|Ziming Shi
|University of Kiel
+
|Rutgers University
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]
+
|[[#Ziming Shi  | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary  ]]
|Betsy, Andreas
+
|
 
|-
 
|-
|Wednesday, March 7, 4:00 p.m.
+
|March 23
| Winfried Sickel
+
|Xiumin Du
|Friedrich-Schiller-Universität Jena
+
|Northwestern University
| [[#Winfried Sickel | On the regularity of compositions of functions]]
+
|[[#Xiumin Du  |   Falconer's distance set problem ]]
|Andreas
+
|
 
|-
 
|-
|March 20
+
|March 30, 10:00  a.m.
| Betsy Stovall
+
|Etienne Le Masson
| UW
+
|Cergy Paris University
| [[#linkofabstract | Two endpoint bounds via inverse problems]]
+
|[[#Etienne Le Masson  | Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces ]]
 
|
 
|
 
|-
 
|-
|April 10
+
|April 6
| Martina Neuman
+
|Theresa Anderson
| UC Berkeley
+
|Purdue University
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]
+
|[[#Theresa Anderson  |   Dyadic analysis (virtually) meets number theory ]]
| Betsy
+
|
|-
 
|Friday, April 13, 4:00 p.m. (Colloquium)
 
|Jill Pipher
 
|Brown
 
| [[#Jill Pipher | Mathematical ideas in cryptography]]
 
|WIMAW
 
 
|-
 
|-
|April 17
+
|April 13
|  
+
|Nathan Wagner
|  
+
|Washington University  St. Louis
| [[#linkofabstract | Title]]
+
|[[#Nathan Wagner  | Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness ]]
 
|
 
|
 
|-
 
|-
|April 24
+
|April 20
| Lenka Slavíková
+
|David Beltran
| University of Missouri
+
| UW Madison
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]
+
|[[#David Beltran  |   Sobolev improving for averages over curves in $\mathbb{R}^4$]]
|Betsy, Andreas
 
|-
 
|May 1
 
| Xianghong Gong
 
| UW
 
| [[#linkofabstract | Title]]
 
 
|
 
|
 
|-
 
|-
| '''May 7'''
+
|April 27
| Ebru Toprak
+
|Yumeng Ou
| UIUC
+
|University of Pennsylvania
| [[#linkofabstract | TBA]]
+
|[[#Yumeng Ou  | On the multiparameter distance problem]]
|Betsy
 
|-
 
| '''May 15'''
 
| Gennady Uraltsev
 
| Cornell
 
| [[#linkofabstract | TBA]]
 
| Andreas, Betsy
 
|-
 
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
 
|
 
 
|
 
|
|
 
|Betsy, Andreas
 
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Brian Street===
+
===Alexei Poltoratski===
  
Title: Convenient Coordinates
+
Title: Dirac inner functions
  
Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields.  We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic).  By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps.  When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger.  When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".
+
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.
  
===Hiroyoshi Mitake===
+
===Polona Durcik and Joris Roos===
  
Title: Derivation of multi-layered interface system and its application
+
Title: A triangular Hilbert transform with curvature, I & II.
  
Abstract:   In this talk, I will propose a multi-layered interface system which can
+
Abstract: The triangular Hilbert is a two-dimensional bilinear singular
be formally derived by the singular limit of the weakly coupled system of
+
originating in time-frequency analysis. No Lp bounds are currently
the Allen-Cahn equation. By using the level set approach, this system can be
+
known for this operator.
written as a quasi-monotone degenerate parabolic system.
+
In these two talks we discuss a recent joint work with Michael Christ
We give results of the well-posedness of viscosity solutions, and study the  
+
on a variant of the triangular Hilbert transform involving curvature.
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
+
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.
  
===Joris Roos===
+
===Andrew Zimmer===
  
Title: A polynomial Roth theorem on the real line
+
Title: Complex analytic problems on domains with good intrinsic geometry
  
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.
+
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).
  
===Michael Greenblatt===
+
===Hong Wang===
  
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces
+
Title: Improved decoupling for the parabola
  
Abstract:  A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.
+
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.
  
===David Beltran===
+
===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}.
  
Title:  Fefferman Stein Inequalities
+
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 $p\geq 1$.
  
Abstract:  Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.
+
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).
  
===Jonathan Hickman===
+
[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.
  
Title: Factorising X^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.
  
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.
+
[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.
  
===Xiaochun Li ===
+
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
 +
Wasserstein spaces: The Hilbertian case}, submitted manuscript.
  
Title: Recent progress on the pointwise convergence problems of Schrodinger equations
+
[5] C. Villani, \emph{Optimal Transport: Old and New,}
 +
(Grundlehren der mathematischen Wissenschaften)
 +
Springer, 2009.
  
Abstract:  Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3.  This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and  the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.
+
===Shukun Wu===
  
===Fedor Nazarov=== 
+
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
+
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.  
conjecture is sharp.
 
  
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
 
the norm of the Hilbert transform on the line as an operator from $L^1(w)$
 
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work
 
with Andrei Lerner and Sheldy Ombrosi.
 
  
===Stefanie Petermichl===
+
===Jonathan Hickman===
Title: Higher order Journé commutators
 
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
Title: Sobolev improving for averages over space curves
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
 
more basic terms, let us consider a function on the unit circle in its
 
Fourier representation. Let P_+ denote the projection onto
 
non-negative and P_- onto negative frequencies. Let b denote
 
multiplication by the symbol function b. It is a classical theorem by
 
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
 
only if b is in an appropriate space of functions of bounded mean
 
oscillation. The necessity makes use of a classical factorisation
 
theorem of complex function theory on the disk. This type of question
 
can be reformulated in terms of commutators [b,H]=bH-Hb with the
 
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
 
as in the real variable setting, in the multi-parameter setting or
 
other, these classifications can be very difficult.
 
  
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
+
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.
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
 
spaces of bounded mean oscillation via L^p boundedness of commutators.
 
We present here an endpoint to this theory, bringing all such
 
characterisation results under one roof.
 
  
The tools used go deep into modern advances in dyadic harmonic
+
===Hanlong Fang===
analysis, while preserving the Ansatz from classical operator theory.
 
  
===Shaoming Guo ===
+
Title: Canonical blow-ups of Grassmann manifolds
Title: Parsell-Vinogradov systems in higher dimensions
 
  
Abstract:  
+
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.  
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
 
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
 
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
 
  
===Naser Talebizadeh Sardari===
+
===Bingyang Hu===
  
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
+
Title: Some structure theorems on general doubling measures.
  
Abstract:  Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math>  in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove  a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given  eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.
+
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.
  
===Xianghong Chen===
+
===Krystal Taylor===
  
Title: Some transfer operators on the circle with trigonometric weights
+
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting
  
Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.  
+
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.
  
===Bobby Wilson===
+
===Dominique Maldague===
  
Title: Projections in Banach Spaces and Harmonic Analysis
+
Title: A new proof of decoupling for the parabola
  
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
+
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.
  
===Andreas Seeger===
+
===Diogo Oliveira e Silva===
  
Title: Singular integrals and a problem on mixing flows
+
Title: Global maximizers for spherical restriction
  
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
+
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.
  
===Dong Dong===
+
===Oleg Safronov===
  
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
+
Title: Relations between discrete and continuous spectra of differential operators
  
Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.
+
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.
  
===Sergey Denisov===
+
===Ziming Shi===
  
Title: Spectral Szegő  theorem on the real line
+
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary
  
Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.
+
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.
  
===Ruixiang Zhang===
+
===Xiumin Du===
  
Title: The (Euclidean) Fractal Uncertainty Principle
+
Title: Falconer's distance set problem
  
Abstract: On the real line, a  version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work  the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).
+
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.
  
===Detlef Müller===
+
===Etienne Le Masson===
  
Title: On Fourier restriction for a non-quadratic hyperbolic surface
+
Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces
  
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about  hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically  in the presence of a perturbation term,  and that new techniques, compared to the elliptic case, are required to handle more general 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.
  
===Winfried Sickel===
+
===Theresa Anderson===
  
Title: On the regularity of compositions of functions
+
Title: Dyadic analysis (virtually) meets number theory
  
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math>
+
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.
we associate the composition operator
 
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>.
 
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.
 
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.
 
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math>
 
such that the composition operator maps such a space <math>E</math> into itself.
 
  
===Martina Neuman===
+
===Nathan Wagner===
  
Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces
+
Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness
  
Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.
+
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.
  
===Jill Pipher===
+
===David Beltran===
  
Title: Mathematical ideas in cryptography
+
Title: Sobolev improving for averages over curves in $\mathbb{R}^4$
  
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,
+
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.
including homomorphic encryption.
 
  
===Lenka Slavíková===
+
===Yumeng Ou===
  
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria
+
Title: On the multiparameter distance problem
  
Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.
+
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=
 
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[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
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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 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

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Graduate Student Seminar:

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