Difference between revisions of "Analysis Seminar"

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The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
+
The 2021-2022 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).
+
Some of the talks will be in person (room Van Vleck B139) and some will be online. 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.
+
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. If you are from an institution different than UW-Madison, please send 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.
+
If you'd like to suggest speakers for the spring semester please contact David and Andreas.
  
 
+
= Analysis Seminar Schedule =
 
 
=[[Previous_Analysis_seminars]]=
 
 
 
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
 
 
 
= Current Analysis Seminar Schedule =
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 21: Line 15:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 22
+
|September 21, VV B139
|Alexei Poltoratski
+
| Dóminique Kemp
|UW Madison
+
| UW-Madison
|[[#Alexei Poltoratski Dirac inner functions ]]
+
|[[#Dóminique Kemp Decoupling by way of approximation ]]
 
|  
 
|  
 
|-
 
|-
|September 29
+
|September 28, VV B139
|Joris Roos
+
| Jack Burkart
|University of Massachusetts - Lowell
+
| UW-Madison
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]
+
|[[#Jack Burkart |   Transcendental Julia Sets with Fractional Packing Dimension ]]
 
|  
 
|  
 
|-
 
|-
|Wednesday September 30, 4 p.m.
+
|October 5, Online
|Polona Durcik
+
| Giuseppe Negro
|Chapman University
+
| University of Birmingham
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]
+
|[[#Giuseppe Negro |   Stability of sharp Fourier restriction to spheres ]]
 
|  
 
|  
 
|-
 
|-
|October 6
+
|October 12, VV B139
|Andrew Zimmer
+
|Rajula Srivastava
 
|UW Madison
 
|UW Madison
|[[#Andrew Zimmer Complex analytic problems on domains with good intrinsic geometry ]]
+
|[[#Rajula Srivastava Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]]
 
|  
 
|  
 
|-
 
|-
|October 13
+
|October 19, Online
|Hong Wang
+
|Itamar Oliveira
|Princeton/IAS
+
|Cornell University
|[[#Hong Wang Improved decoupling for the parabola ]]
+
|[[#Itamar Oliveira A new approach to the Fourier extension problem for the paraboloid ]]
 
|  
 
|  
 
|-
 
|-
|October 20
+
|October 26, VV B139
|Kevin Luli
+
| Changkeun Oh
|UC Davis
+
| UW Madison
|[[#Kevin Luli Smooth Nonnegative Interpolation ]]
+
|[[#Changkeun Oh Decoupling inequalities for quadratic forms and beyond ]]
 
|  
 
|  
 
|-
 
|-
|October 21, 4.00 p.m.
+
|October 29, Colloquium
|Niclas Technau
+
| Alexandru Ionescu
|UW Madison
+
| Princeton University
|[[#Niclas Technau Number theoretic applications of oscillatory integrals ]]
+
|[[#Alexandru Ionescu  |  Polynomial averages and pointwise ergodic theorems on nilpotent groups]]
 +
|-
 +
|November 2, VV B139
 +
| Liding Yao
 +
| UW Madison
 +
|[[#Liding Yao An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains ]]
 
|  
 
|  
 
|-
 
|-
|October 27
+
|November 9, VV B139
|Terence Harris
+
| Lingxiao Zhang
| Cornell University
+
| UW Madison
|[[#Terence Harris Low dimensional pinned distance sets via spherical averages ]]
+
|[[#Lingxiao Zhang Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition ]]
 
|  
 
|  
 
|-
 
|-
|Monday, November 2, 4 p.m.
+
|November 12, Colloquium
|Yuval Wigderson
+
| Kasso Okoudjou
|Stanford  University
+
| Tufts University
|[[#Yuval Wigderson New perspectives on the uncertainty principle ]]
+
|[[#Kasso Okoudjou An exploration in analysis on fractals ]]
 +
|-
 +
|November 16, VV B139
 +
| Rahul Parhi
 +
| UW Madison (EE)
 +
|[[#Rahul Parhi  |    On BV Spaces, Splines, and Neural Networks ]]
 
|  
 
|  
 
|-
 
|-
|November 10, 10 a.m.
+
|November 30, VV B139
|Óscar Domínguez
+
| Alexei Poltoratski
| Universidad Complutense de Madrid
+
| UW Madison
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]
+
|[[#Alexei Poltoratski | Pointwise convergence for the scattering data and non-linear Fourier transform. ]]
 
|  
 
|  
 
|-
 
|-
|November 17
+
|December 7, Online
|Tamas Titkos
+
| John Green
|BBS U of Applied Sciences and Renyi Institute
+
| The University of Edinburgh
|[[#Tamas Titkos Isometries of Wasserstein spaces ]]
+
|[[#John Green  Estimates for oscillatory integrals via sublevel set estimates ]]
 
|  
 
|  
 
|-
 
|-
|November 24
+
|December 14, VV B139
|Shukun Wu
+
| Tao Mei
|University of Illinois (Urbana-Champaign)
+
| Baylor University
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]]  
+
|[[#Tao Mei |   Fourier Multipliers on free groups ]]
 
|  
 
|  
 
|-
 
|-
|December 1
+
|Winter break
| Jonathan Hickman
+
|
| The University of Edinburgh
+
|
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]
+
|-
 +
|February 8, VV B139
 +
|Alex Nagel
 +
| UW Madison
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|February 15, Online
 +
| Sebastian Bechtel
 +
| Institut de Mathématiques de Bordeaux
 +
|[[#linktoabstract |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 2, 7:00 p.m.
+
|February 22, VV B139
|Hanlong Fang
+
|Betsy Stovall
 
|UW Madison
 
|UW Madison
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]
+
|[[#linktoabstract  |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 9
+
|March 1, Online
|Bingyang Hu
+
| Po Lam Yung
|Purdue University
+
| Australian National University  
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]
+
|[[#linktoabstract |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 16
+
|March 8, VV B139
|Krystal Taylor
+
| Brian Street
|The Ohio State University
+
| UW Madison
|[[#Krystal Taylor Quantifications of the Besicovitch Projection theorem in a nonlinear setting  ]]
+
|[[#linktoabstract Title ]]
|
+
|  
 
|-
 
|-
|February 23
+
|March 15: No Seminar
|Dominique Maldague
+
|  
|MIT
+
|  
|[[#Dominique Maldague  |  A new proof of decoupling for the parabola ]]
 
 
|
 
|
 +
|
 
|-
 
|-
|March 2
+
|March 22
|Diogo Oliveira e Silva
+
| Laurent Stolovitch
|University of Birmingham
+
| University of Cote d'Azur
|[[#Diogo Oliveira e Silva Global maximizers for spherical restriction ]]
+
|[[#linktoabstract Title ]]
|
+
|  
 
|-
 
|-
|March 9
+
|March 29
|Oleg Safronov
+
| TBA
|University of North Carolina Charlotte
+
| Institution
|[[#linktoabstract  | Relations between discrete and continuous spectra of differential operators ]]
+
|[[#linktoabstract  |   Title ]]
|
+
|  
 
|-
 
|-
|March 16
+
|April 5, Online
|Ziming Shi
+
|Malabika Pramanik
|Rutgers University
+
|University of British Columbia
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|March 23
+
|April 12
|Xiumin Du
+
| Hongki Jung
|Northwestern University
+
| IU Bloomington
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|March 30, 10:00  a.m.
+
|April 15, Colloquium
|Etienne Le Masson
+
| Bernhard Lamel
|Cergy Paris University
+
| Texas A&M University at Qatar
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|April 19:  No seminar
 +
|
 +
|
 
|
 
|
 +
|
 
|-
 
|-
|April 6
+
|April 22, Colloquium
|Theresa Anderson
+
|Detlef Müller
|Purdue University
+
|University of Kiel
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|April 25, 4:00 p.m., Distinguished Lecture Series
 +
|Larry Guth
 +
|MIT
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
 
 
|-
 
|-
|April 13
+
|April 26, 4:00 p.m., Distinguished Lecture Series
|Nathan Wagner
+
|Larry Guth
|Washington University  St. Louis
+
|MIT
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
 
 
|-
 
|-
|April 20
+
|April 27, 4:00 p.m., Distinguished Lecture Series
|Jongchon Kim
+
|Larry Guth
| University of British Columbia
+
|MIT
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|April 27
+
|May 3
|Yumeng Ou
+
| Jingjing Huang
|University of Pennsylvania
+
| University of Nevada, Reno
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|
 
|
 
|
|-
 
|May 4
 
 
|
 
|
 
|
 
|
 +
|-
 +
|Talks in the Fall semester 2022:
 +
|-
 +
|September 20,  PDE and Analysis Seminar
 +
|Andrej Zlatoš
 +
|UCSD
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 +
| Hung Tran
 +
|-
 +
 +
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Alexei Poltoratski===
+
===Dóminique Kemp===
  
Title: Dirac inner functions
+
Decoupling by way of approximation
  
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.
+
Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$.
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===
+
===Jack Burkart===
  
Title: A triangular Hilbert transform with curvature, I & II.
+
Transcendental Julia Sets with Fractional Packing Dimension
  
Abstract: The triangular Hilbert is a two-dimensional bilinear singular
+
If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set.
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===
+
In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.
  
Title:  Complex analytic problems on domains with good intrinsic geometry
+
===Giuseppe Negro===
  
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).
+
Stability of sharp Fourier restriction to spheres
  
===Hong Wang===
+
In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$, which is exactly the classical Stein-Tomas estimate.
  
Title: Improved decoupling for the parabola
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
Abstract: In 2014, Bourgain and Demeter proved the  $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. 
+
===Rajula Srivastava===
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===
+
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
Title: Smooth Nonnegative Interpolation
+
We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.
  
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.
+
===Itamar Oliveira===
  
===Niclas Technau===
+
A new approach to the Fourier extension problem for the paraboloid
  
Title: Number theoretic applications of oscillatory integrals
+
An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. One can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $ g $  of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}) $. We will present this theorem as a proof of concept of a more general framework and set of techniques that can also address multilinear versions of this problem and get similar results. This is joint work with Camil Muscalu.
  
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.
+
===Changkeun Oh===
  
===Terence Harris===
+
Decoupling inequalities for quadratic forms and beyond
  
Title: Low dimensional pinned distance sets via spherical averages
+
In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.
  
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.
+
===Alexandru Ionescu===
  
===Yuval Wigderson===
+
Polynomial averages and pointwise ergodic theorems on nilpotent groups
  
Title: New perspectives on the uncertainty principle
+
I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.
  
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.
+
===Liding Yao===
  
===Oscar Dominguez===
+
An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
  
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
+
Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.
  
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.
+
===Lingxiao Zhang===
  
===Tamas Titkos===
+
Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
  
Title: Isometries of Wasserstein spaces
+
We study operators of the form
 +
$Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt$
 +
where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a `multi-parameter singular kernel' with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the single-parameter case when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.
  
Abstract: Due to its nice theoretical properties and an astonishing number of
+
===Kasso Okoudjou===
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
+
An exploration in analysis on fractals
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
+
Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.
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
+
===Rahul Parhi===
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
+
On BV Spaces, Splines, and Neural Networks
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
+
Many problems in science and engineering can be phrased as the problem
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373
+
of reconstructing a function from a finite number of possibly noisy
(2020), 5855--5883.
+
measurements. The reconstruction problem is inherently ill-posed when
 +
the allowable functions belong to an infinite set. Classical techniques
 +
to solve this problem assume, a priori, that the underlying function has
 +
some kind of regularity, typically Sobolev, Besov, or BV regularity. The
 +
field of applied harmonic analysis is interested in studying efficient
 +
decompositions and representations for functions with certain
 +
regularity. Common representation systems are based on splines and
 +
wavelets. These are well understood mathematically and have been
 +
successfully applied in a variety of signal processing and statistical
 +
tasks. Neural networks are another type of representation system that is
 +
useful in practice, but poorly understood mathematically.
  
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
+
In this talk, I will discuss my research which aims to rectify this
Wasserstein spaces: The Hilbertian case}, submitted manuscript.
+
issue by understanding the regularity properties of neural networks in a
 +
similar vein to classical methods based on splines and wavelets. In
 +
particular, we will show that neural networks are optimal solutions to
 +
variational problems over BV-type function spaces defined via the Radon
 +
transform. These spaces are non-reflexive Banach spaces, generally
 +
distinct from classical spaces studied in analysis. However, in the
 +
univariate setting, neural networks reduce to splines and these function
 +
spaces reduce to classical univariate BV spaces. If time permits, I will
 +
also discuss approximation properties of these spaces, showing that they
 +
are, in some sense, "small" compared to classical multivariate spaces
 +
such as Sobolev or Besov spaces.
  
[5] C. Villani, \emph{Optimal Transport: Old and New,}
+
This is joint work with Robert Nowak.
(Grundlehren der mathematischen Wissenschaften)
 
Springer, 2009.
 
  
===Shukun Wu===
+
===Alexei Poltoratski===
  
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
+
Title: Pointwise convergence for the scattering data and non-linear Fourier transform.
  
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.  
+
Abstract: This talk is about applications of complex and harmonic analysis in spectral and scattering theory
 +
for differential operators. The scattering transform for the Dirac system of differential equations
 +
can be viewed as the non-linear version of the classical Fourier transform. This connection raises many natural
 +
problems on extensions of classical results of Fourier analysis to non-linear settings. In this talk
 +
I will discuss one of such problems, an extension of Carleson's theorem on pointwise convergence of Fourier series to the non-linear case.
  
  
===Jonathan Hickman===
 
  
Title: Sobolev improving for averages over space curves
+
===John Green===
  
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.
+
Estimates for oscillatory integrals via sublevel set estimates.
  
===Hanlong Fang===
+
In many situations, oscillatory integral estimates are known to imply sublevel set estimates in a stable manner. Reversing this implication is much more difficult, but understanding when this is true is helpful for understanding scalar oscillatory integral estimates. We shall motivate a line of investigation in which we seek to reverse the implication in the presence of a qualitative structural assumption. After considering some one-dimensional results, we turn to the setting of convex functions in higher dimensions.
  
Title: Canonical blow-ups of Grassmann manifolds
+
===Tao Mei===
  
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.  
+
Fourier Multipliers on free groups.
  
===Bingyang Hu===
+
In this introductory talk,  I will try to explain what is the noncommutative Lp spaces associated with the free groups, and what are the to be answered questions on  the corresponding Fourier multiplier operators.  At the end, I will explain a recent work on an analogue of Mikhlin’s Lp Fourier multiplier theory on free groups (joint with Eric Ricard and Quanhua Xu).
  
Title: Some structure theorems on general doubling measures.
+
=[[Previous_Analysis_seminars]]=
  
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.
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
===Krystal Taylor===
+
=Extras=
  
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.
 
 
===Name===
 
 
Title:
 
 
Abstract:
 
 
===Name===
 
 
Title:
 
 
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===Name===
 
 
Title:
 
 
Abstract:
 
 
===Name===
 
 
Title:
 
 
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===Name===
 
 
Title:
 
 
Abstract:
 
 
===Name===
 
 
Title:
 
 
Abstract:
 
 
=Extras=
 
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
  

Latest revision as of 17:39, 25 January 2022

The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger. Some of the talks will be in person (room Van Vleck B139) and some will be online. 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. If you are from an institution different than UW-Madison, please send 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.

Analysis Seminar Schedule

date speaker institution title host(s)
September 21, VV B139 Dóminique Kemp UW-Madison Decoupling by way of approximation
September 28, VV B139 Jack Burkart UW-Madison Transcendental Julia Sets with Fractional Packing Dimension
October 5, Online Giuseppe Negro University of Birmingham Stability of sharp Fourier restriction to spheres
October 12, VV B139 Rajula Srivastava UW Madison Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
October 19, Online Itamar Oliveira Cornell University A new approach to the Fourier extension problem for the paraboloid
October 26, VV B139 Changkeun Oh UW Madison Decoupling inequalities for quadratic forms and beyond
October 29, Colloquium Alexandru Ionescu Princeton University Polynomial averages and pointwise ergodic theorems on nilpotent groups
November 2, VV B139 Liding Yao UW Madison An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
November 9, VV B139 Lingxiao Zhang UW Madison Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
November 12, Colloquium Kasso Okoudjou Tufts University An exploration in analysis on fractals
November 16, VV B139 Rahul Parhi UW Madison (EE) On BV Spaces, Splines, and Neural Networks
November 30, VV B139 Alexei Poltoratski UW Madison Pointwise convergence for the scattering data and non-linear Fourier transform.
December 7, Online John Green The University of Edinburgh Estimates for oscillatory integrals via sublevel set estimates
December 14, VV B139 Tao Mei Baylor University Fourier Multipliers on free groups
Winter break
February 8, VV B139 Alex Nagel UW Madison Title
February 15, Online Sebastian Bechtel Institut de Mathématiques de Bordeaux Title
February 22, VV B139 Betsy Stovall UW Madison Title
March 1, Online Po Lam Yung Australian National University Title
March 8, VV B139 Brian Street UW Madison Title
March 15: No Seminar
March 22 Laurent Stolovitch University of Cote d'Azur Title
March 29 TBA Institution Title
April 5, Online Malabika Pramanik University of British Columbia Title
April 12 Hongki Jung IU Bloomington Title
April 15, Colloquium Bernhard Lamel Texas A&M University at Qatar Title
April 19: No seminar
April 22, Colloquium Detlef Müller University of Kiel Title
April 25, 4:00 p.m., Distinguished Lecture Series Larry Guth MIT Title
April 26, 4:00 p.m., Distinguished Lecture Series Larry Guth MIT Title
April 27, 4:00 p.m., Distinguished Lecture Series Larry Guth MIT Title
May 3 Jingjing Huang University of Nevada, Reno Title
Talks in the Fall semester 2022:
September 20, PDE and Analysis Seminar Andrej Zlatoš UCSD Title Hung Tran

Abstracts

Dóminique Kemp

Decoupling by way of approximation

Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$.

Jack Burkart

Transcendental Julia Sets with Fractional Packing Dimension

If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set.

In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.

Giuseppe Negro

Stability of sharp Fourier restriction to spheres

In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$, which is exactly the classical Stein-Tomas estimate.

Joint work with E.Carneiro and D.Oliveira e Silva.

Rajula Srivastava

Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups

We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.

Itamar Oliveira

A new approach to the Fourier extension problem for the paraboloid

An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. One can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $ g $ of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}) $. We will present this theorem as a proof of concept of a more general framework and set of techniques that can also address multilinear versions of this problem and get similar results. This is joint work with Camil Muscalu.

Changkeun Oh

Decoupling inequalities for quadratic forms and beyond

In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.

Alexandru Ionescu

Polynomial averages and pointwise ergodic theorems on nilpotent groups

I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.

Liding Yao

An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains

Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.

Lingxiao Zhang

Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition

We study operators of the form $Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt$ where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a `multi-parameter singular kernel' with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the single-parameter case when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.

Kasso Okoudjou

An exploration in analysis on fractals

Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.

Rahul Parhi

On BV Spaces, Splines, and Neural Networks

Many problems in science and engineering can be phrased as the problem of reconstructing a function from a finite number of possibly noisy measurements. The reconstruction problem is inherently ill-posed when the allowable functions belong to an infinite set. Classical techniques to solve this problem assume, a priori, that the underlying function has some kind of regularity, typically Sobolev, Besov, or BV regularity. The field of applied harmonic analysis is interested in studying efficient decompositions and representations for functions with certain regularity. Common representation systems are based on splines and wavelets. These are well understood mathematically and have been successfully applied in a variety of signal processing and statistical tasks. Neural networks are another type of representation system that is useful in practice, but poorly understood mathematically.

In this talk, I will discuss my research which aims to rectify this issue by understanding the regularity properties of neural networks in a similar vein to classical methods based on splines and wavelets. In particular, we will show that neural networks are optimal solutions to variational problems over BV-type function spaces defined via the Radon transform. These spaces are non-reflexive Banach spaces, generally distinct from classical spaces studied in analysis. However, in the univariate setting, neural networks reduce to splines and these function spaces reduce to classical univariate BV spaces. If time permits, I will also discuss approximation properties of these spaces, showing that they are, in some sense, "small" compared to classical multivariate spaces such as Sobolev or Besov spaces.

This is joint work with Robert Nowak.

Alexei Poltoratski

Title: Pointwise convergence for the scattering data and non-linear Fourier transform.

Abstract: This talk is about applications of complex and harmonic analysis in spectral and scattering theory for differential operators. The scattering transform for the Dirac system of differential equations can be viewed as the non-linear version of the classical Fourier transform. This connection raises many natural problems on extensions of classical results of Fourier analysis to non-linear settings. In this talk I will discuss one of such problems, an extension of Carleson's theorem on pointwise convergence of Fourier series to the non-linear case.


John Green

Estimates for oscillatory integrals via sublevel set estimates.

In many situations, oscillatory integral estimates are known to imply sublevel set estimates in a stable manner. Reversing this implication is much more difficult, but understanding when this is true is helpful for understanding scalar oscillatory integral estimates. We shall motivate a line of investigation in which we seek to reverse the implication in the presence of a qualitative structural assumption. After considering some one-dimensional results, we turn to the setting of convex functions in higher dimensions.

Tao Mei

Fourier Multipliers on free groups.

In this introductory talk, I will try to explain what is the noncommutative Lp spaces associated with the free groups, and what are the to be answered questions on the corresponding Fourier multiplier operators. At the end, I will explain a recent work on an analogue of Mikhlin’s Lp Fourier multiplier theory on free groups (joint with Eric Ricard and Quanhua Xu).

Previous_Analysis_seminars

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

Extras

Blank Analysis Seminar Template


Graduate Student Seminar:

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