Difference between revisions of "Analysis Seminar"

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'''Analysis Seminar
 
'''
 
  
The seminar will  meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
+
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
 +
It will be online at least for the Fall semester, with details to be announced in September.
 +
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).
  
If you wish to invite a speaker please contact Betsy at stovall(at)math
+
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230
  
===[[Previous Analysis seminars]]===
+
If you'd like to suggest  speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).
  
= 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 22:
 
!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
| Bartosz Langowski and Tomasz Szarek
+
|Jongchon Kim
| Institute of Mathematics, Polish Academy of Sciences
+
| UBC
|[[#Bartosz Langowski and Tomasz Szarek Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|[[#linktoabstract Title ]]
| 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 ]]
+
|[[#linktoabstract  |   Title ]]
| Andreas
+
|  
 
|-
 
|-
|January 30
+
|February 16
|  
+
|Krystal Taylor
|  
+
|The Ohio State University
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
 
|
 
|
 
|-
 
|-
|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]]
+
|[[#linktoabstract  |   Title ]]
|Betsy
+
|
 
|-
 
|-
|February 13
+
|March 2
| TBA
+
|Diogo Oliveira e Silva
|  
+
|University of Birmingham
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
 
|
 
|
 
|-
 
|-
|February 20
+
|March 9
| TBA
+
|
|  
+
|
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
 
|
 
|
 
|-
 
|-
|February 27
+
|March 16
|Detlef Müller (tent.)
+
|Ziming Shi
|University of Kiel
+
|Rutgers University
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|Betsy, Andreas
+
|
 
|-
 
|-
|Wednesday, March 7, 4:00 p.m.
+
|March 23
| Winfried Sickel
+
|
|Friedrich-Schiller-Universität Jena
+
|
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|Andreas
+
|
 
|-
 
|-
|March 13
+
|March 30
| TBA
+
|
|  
+
|
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
 
|
 
|
 
|-
 
|-
|March 20
+
|April 6
|  
+
|
|  
+
|
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
 
|
 
|
 
|-
 
|-
|April 3
+
|April 13
|  
+
|
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
|-
+
|[[#linktoabstract  |   Title ]]
|April 10
 
|
 
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
 
|-
 
|-
|April 17
+
|April 20
|
 
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
|-
 
|April 24
 
|
 
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
|-
+
|[[#linktoabstract  |   Title ]]
|May 1
 
|
 
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
 
|-
 
|-
| May 16-18, Workshop in Fourier Analysis
+
|April 27
 
|
 
|
 
|
 
|
 +
|[[#linktoabstract  |  Title ]]
 
|
 
|
|Betsy, Andreas
 
 
|-
 
|-
 +
|May 4
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 
|}
 
|}
  
 
=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: Fefferman Stein Inequalities
+
Title: Smooth Nonnegative Interpolation
  
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.
+
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.  
  
===Jonathan Hickman===
+
===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
  
Title: Factorising X^n.
+
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}.
  
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.
+
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 $p\geq 1$.
  
===Xiaochun Li ===
+
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).
  
Title: Recent progress on the pointwise convergence problems of Schrodinger equations
+
[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.
  
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.
+
[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.
  
===Fedor Nazarov=== 
+
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of
 +
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373
 +
(2020), 5855--5883.
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
+
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
conjecture is sharp.
+
Wasserstein spaces: The Hilbertian case}, submitted manuscript.
  
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
+
[5] C. Villani, \emph{Optimal Transport: Old and New,}
the norm of the Hilbert transform on the line as an operator from $L^1(w)$
+
(Grundlehren der mathematischen Wissenschaften)
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work
+
Springer, 2009.
with Andrei Lerner and Sheldy Ombrosi.
 
  
===Stefanie Petermichl===
+
===Shukun Wu===
Title: Higher order Journé commutators
 
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
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: 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.  
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
 
analysis, while preserving the Ansatz from classical operator theory.
 
  
===Shaoming Guo ===
+
===Jonathan Hickman===
Title: Parsell-Vinogradov systems in higher dimensions
 
  
Abstract:  
+
Title: Sobolev improving for averages over space curves
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===
+
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.
  
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
+
===Name===
  
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.
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===Xianghong Chen===
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Title:  Some transfer operators on the circle with trigonometric weights
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===Name===
  
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.
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===Bobby Wilson===
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Title: Projections in Banach Spaces and Harmonic Analysis
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===Name===
  
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.
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===Dong Dong===
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Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
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===Name===
  
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.
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Graduate Student Seminar:
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https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html

Latest revision as of 15:52, 3 December 2020

The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. It will be online at least for the Fall semester, with details to be announced in September. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).

Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230

If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).


Previous_Analysis_seminars

https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars

Current Analysis Seminar Schedule

date speaker institution title host(s)
September 22 Alexei Poltoratski UW Madison Dirac inner functions
September 29 Joris Roos University of Massachusetts - Lowell A triangular Hilbert transform with curvature, I
Wednesday September 30, 4 p.m. Polona Durcik Chapman University A triangular Hilbert transform with curvature, II
October 6 Andrew Zimmer UW Madison Complex analytic problems on domains with good intrinsic geometry
October 13 Hong Wang Princeton/IAS Improved decoupling for the parabola
October 20 Kevin Luli UC Davis Smooth Nonnegative Interpolation
October 21, 4.00 p.m. Niclas Technau UW Madison Number theoretic applications of oscillatory integrals
October 27 Terence Harris Cornell University Low dimensional pinned distance sets via spherical averages
Monday, November 2, 4 p.m. Yuval Wigderson Stanford University New perspectives on the uncertainty principle
November 10, 10 a.m. Óscar Domínguez Universidad Complutense de Madrid New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
November 17 Tamas Titkos BBS U of Applied Sciences and Renyi Institute Isometries of Wasserstein spaces
November 24 Shukun Wu University of Illinois (Urbana-Champaign) On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
December 1 Jonathan Hickman The University of Edinburgh Sobolev improving for averages over space curves
February 2 Jongchon Kim UBC Title
February 9 Bingyang Hu Purdue University Title
February 16 Krystal Taylor The Ohio State University Title
February 23 Dominique Maldague MIT Title
March 2 Diogo Oliveira e Silva University of Birmingham Title
March 9 Title
March 16 Ziming Shi Rutgers University Title
March 23 Title
March 30 Title
April 6 Title
April 13 Title
April 20 Title
April 27 Title
May 4 Title

Abstracts

Alexei Poltoratski

Title: Dirac inner functions

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

Polona Durcik and Joris Roos

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

Abstract: The triangular Hilbert is a two-dimensional bilinear singular originating in time-frequency analysis. No Lp bounds are currently known for this operator. In these two talks we discuss a recent joint work with Michael Christ on a variant of the triangular Hilbert transform involving curvature. This object is closely related to the bilinear Hilbert transform with curvature and a maximally modulated singular integral of Stein-Wainger type. As an application we also discuss a quantitative nonlinear Roth type theorem on patterns in the Euclidean plane. The second talk will focus on the proof of a key ingredient, a certain regularity estimate for a local operator.

Andrew Zimmer

Title: Complex analytic problems on domains with good intrinsic geometry

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

Hong Wang

Title: Improved decoupling for the parabola

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

Kevin Luli

Title: Smooth Nonnegative Interpolation

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

Niclas Technau

Title: Number theoretic applications of oscillatory integrals

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

Terence Harris

Title: Low dimensional pinned distance sets via spherical averages

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

Yuval Wigderson

Title: New perspectives on the uncertainty principle

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

Oscar Dominguez

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

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

Tamas Titkos

Title: Isometries of Wasserstein spaces

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

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

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

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

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

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

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

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

Shukun Wu

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

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


Jonathan Hickman

Title: Sobolev improving for averages over space curves

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

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Graduate Student Seminar:

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