Difference between revisions of "Analysis Seminar"

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'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
  
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
+
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).
  
If you wish to invite a speaker please contact  Brian at street(at)math
+
Zoom links will be sent to those who have signed up for the Analysis Seminar List.  If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. 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.
  
===[[Previous Analysis seminars]]===
+
If you'd like to suggest speakers for the spring  semester please contact David and Andreas.
  
 
= Analysis Seminar Schedule =
 
= Analysis Seminar Schedule =
Line 16: Line 15:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 10
+
|September 21, VV B139
| José Madrid
+
| Dóminique Kemp
| UCLA
+
| UW-Madison
|[[#José Madrid On the regularity of maximal operators on Sobolev Spaces ]]
+
|[[#Dóminique Kemp Decoupling by way of approximation ]]
| Andreas, David
+
|
 +
|-
 +
|September 28, VV B139
 +
| Jack Burkart
 +
| UW-Madison
 +
|[[#Jack Burkart  |  Transcendental Julia Sets with Fractional Packing Dimension ]]
 +
|
 +
|-
 +
|October 5, Online
 +
| Giuseppe Negro
 +
| University of Birmingham
 +
|[[#Giuseppe Negro  |  Stability of sharp Fourier restriction to spheres ]]
 +
|
 +
|-
 +
|October 12, VV B139
 +
|Rajula Srivastava
 +
|UW Madison
 +
|[[#Rajula Srivastava  |  Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]]
 +
|  
 
|-
 
|-
|Sept 13 (Friday, B139)
+
|October 19, Online
| Yakun Xi
+
|Itamar Oliveira
| University of  Rochester
+
|Cornell University  
|[[#Yakun Xi Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
+
|[[#Itamar Oliveira A new approach to the Fourier extension problem for the paraboloid ]]
| Shaoming
+
|  
 
|-
 
|-
|Sept 17
+
|October 26, VV B139
| Joris Roos
+
| Changkeun Oh
 
| UW Madison
 
| UW Madison
|[[#Joris Roos L^p improving estimates for maximal spherical averages ]]
+
|[[#Changkeun Oh Decoupling inequalities for quadratic forms and beyond ]]
| Brian
+
|  
 
|-
 
|-
|Sept 20 (2:25 PM Friday, Room B139 VV)
+
|October 29, Colloquium, Online
| Xiaojun Huang
+
| Alexandru Ionescu
| Rutgers University–New Brunswick
+
| Princeton University
|[[#linktoabstract A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
+
|[[#Alexandru Ionescu Polynomial averages and pointwise ergodic theorems on nilpotent groups]]
| Xianghong
 
 
|-
 
|-
|Oct 1
+
|November 2, VV B139
| Xiaocheng Li
+
| Liding Yao
 
| UW Madison
 
| UW Madison
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
+
|[[#Liding Yao |   An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains ]]
| Simon
+
|  
|-
 
|Oct 8
 
| Jeff Galkowski
 
| Northeastern University
 
|[[#Jeff Galkowski  |  Concentration and Growth of Laplace Eigenfunctions ]]
 
| Betsy
 
 
|-
 
|-
|Oct 15
+
|November 9, VV B139
| David Beltran
+
| Lingxiao Zhang
 
| UW Madison
 
| UW Madison
|[[#David Beltran Regularity of the centered fractional maximal function ]]
+
|[[#Lingxiao Zhang Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition ]]
| Brian
+
|  
 
|-
 
|-
|Oct 22
+
|November 12, Colloquium, Online
| Laurent Stolovitch
+
| Kasso Okoudjou
| University of Côte d'Azur
+
| Tufts University
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
+
|[[#Kasso Okoudjou |   An exploration in analysis on fractals ]]
| Xianghong
 
 
|-
 
|-
|<b>Wednesday Oct 23 in B129</b>
+
|November 16, VV B139
|Dominique Kemp
+
| Rahul Parhi
|Indiana University
+
| UW Madison (EE)
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
+
|[[#Rahul Parhi  |   On BV Spaces, Splines, and Neural Networks ]]
|Betsy
+
| Betsy
 
|-
 
|-
|Oct 29
+
|November 30, VV B139
| Bingyang Hu
+
| Alexei Poltoratski
 
| UW Madison
 
| UW Madison
|[[#Bingyang Hu |   Sparse bounds of singular Radon transforms]]
+
|[[#Alexei Poltoratski | Pointwise convergence for the scattering data and non-linear Fourier transform. ]]
| Brian
+
|  
 
|-
 
|-
|Nov 5
+
|December 7, Online
| Kevin O'Neill
+
| John Green
| UC Davis
+
| The University of Edinburgh
|[[#Kevin O'Neill |   A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
+
|[[#John Green | Estimates for oscillatory integrals via sublevel set estimates ]]
| Betsy
+
|  
 
|-
 
|-
|Nov 12
+
|December 14, VV B139
| Francesco di Plinio
+
| Tao Mei
| Washington University in St. Louis
+
| Baylor University
|[[#Francesco di Plinio Maximal directional integrals along algebraic and lacunary sets]]
+
|[[#Tao Mei Fourier Multipliers on free groups ]]
 
| Shaoming
 
| Shaoming
 
|-
 
|-
|Nov 13 (Wednesday)
+
|Winter break
| Xiaochun Li
+
|
| UIUC
+
|
|[[#Xiaochun Li  |  Roth's type theorems on progressions]]
 
| Brian, Shaoming
 
 
|-
 
|-
|Nov 19
+
|February 8, VV B139
| Joao Ramos
+
|Alexander  Nagel
| University of Bonn
+
| UW Madison
|[[#Joao Ramos Fourier uncertainty principles, interpolation and uniqueness sets ]]
+
|[[#Alex Nagel Global estimates for a class of kernels and multipliers with multiple homogeneities]]
| Joris, Shaoming
+
|
 +
|-
 +
|February 15, Online
 +
| Sebastian Bechtel
 +
| Institut de Mathématiques de Bordeaux
 +
|[[#Sebastian Bechtel  | Square roots of elliptic systems on open sets]]
 +
|  
 
|-
 
|-
|Jan 21
+
|Friday,  February 18,  Colloquium, VVB239
| No Seminar
+
| Andreas Seeger
 +
| UW Madison
 +
|[[#Andreas Seeger | Spherical maximal functions and fractal dimensions of dilation sets]]
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Friday, Jan 31, 4 pm, B239, Colloquium
+
|February 22, VV B139
| Lillian Pierce
+
|Tongou Yang
| Duke University
+
|University of British Comlumbia
|[[#Lillian Pierce |   On Bourgain’s counterexample for the Schrödinger maximal function ]]
+
|[[#linktoabstract  |  Restricted projections along $C^2$ curves on the sphere ]]
| Andreas, Simon
+
| Shaoming
 +
|-
 +
|Monday, February 28, 4:30 p.m., Online
 +
| Po Lam Yung
 +
| Australian National University  
 +
|[[#Po Lam Yung | Revisiting an old argument for Vinogradov's Mean Value Theorem ]]
 +
|  
 
|-
 
|-
|Feb 4
+
|March 8, VV B139
| Ruixiang Zhang
+
| Brian Street
 
| UW Madison
 
| UW Madison
|[[#Ruixiang Zhang |   Local smoothing for the wave equation in 2+1 dimensions ]]
+
|[[#Brian Street | Maximal Subellipticity ]]
| Andreas
+
|  
 
|-
 
|-
|Feb 11
+
|March 15: No Seminar
| Zane Li
+
|  
| Indiana University
+
|  
|[[#Zane Li  |  A bilinear proof of decoupling for the moment curve ]]
+
|
| Betsy
+
|  
 
|-
 
|-
|Feb 18
+
|March 22
| Sergey Denisov
+
| Laurent Stolovitch
| UW Madison
+
| University of Cote d'Azur
|[[#linktoabstract  |  De Branges canonical systems with finite logarithmic integral ]]
+
|[[#linktoabstract  |  Classification of reversible parabolic diffeomorphisms of
| Brian
+
$(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional
 +
hyperbolic type ]]
 +
| Xianghong
 
|-
 
|-
|Feb 25
+
|March 29, VV B139
| Michel Alexis
+
|Betsy Stovall
| UW Madison
+
|UW Madison
|[[#Michel Alexis The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
+
|[[#Betsy Stovall On extremizing sequences for adjoint Fourier restriction to the sphere ]]
| Sergey
+
|  
 
|-
 
|-
|Friday, Feb 28 (Colloquium)
+
|April 5, Online
| Brett Wick
+
|Malabika Pramanik
| Washington University - St. Louis
+
|University of British Columbia
|[[#MBrett Wick  The Corona Theorem]]
+
|[[#Malabika Pramanik Dimensionality and Patterns with Curvature]]
| Andreas
+
|  
 
|-
 
|-
|Mar 3
+
|April 12, VV B139
| William Green
+
| Hongki Jung
| Rose-Hulman Institute of Technology
+
| IU Bloomington
|[[#William Green |   Dispersive estimates for the Dirac equation ]]
+
|[[#Hongki Jung | A small cap decoupling for the twisted cubic ]]
| Betsy
+
| Shaoming
 
|-
 
|-
|Mar 10
+
|Friday, April 15, Colloquium, VV B239
| Yifei Pan
+
| Bernhard Lamel
| Purdue University Fort Wayne
+
| Texas A&M University at Qatar
|[[#linktoabstract | On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n]]
+
|[[#Bernhard Lamel |   Convergence and Divergence of Formal Power Series Maps ]]
 
| Xianghong
 
| Xianghong
 
|-
 
|-
|Mar 17
+
|April 19, Online
| Spring Break!
+
| Carmelo Puliatti
|
+
| Euskal Herriko Unibertsitatea
 +
|[[#Carmelo Puliatti  |  Gradients of single layer potentials for elliptic operators 
 +
with coefficients of Dini mean oscillation-type ]]
 +
| David
 
|
 
|
 
|  
 
|  
 
|-
 
|-
|Mar 24
+
|April 25-26-27, Distinguished Lecture Series
| Oscar Dominguez
+
|Larry Guth
| Universidad Complutense de Madrid
+
|MIT
|[[#linktoabstract |   Title ]]
+
|[[#Larry Guth | Reflections on decoupling and Vinogradov's mean value problem. ]]
| Andreas
+
|-
 +
|April 25, 4:00 p.m., Lecture I, VV B239
 +
|
 +
|
 +
|[[#linktoabstract  |  Introduction to decoupling and Vinogradov's mean value problem ]]
 
|-
 
|-
|Mar 31
+
|April 26, 4:00 p.m., Lecture II, Chamberlin 2241
| Brian Street
+
|
| University of Wisconsin-Madison
+
|
|[[#linktoabstract  |   Title ]]
+
|[[#linktoabstract  | Features of the proof of decoupling  ]]
| Local
 
 
|-
 
|-
|Apr 7
+
|April 27, 4:00 p.m., Lecture III, VV B239
| Hong Wang
+
|
| Institution
+
|
|[[#linktoabstract  |   Title ]]
+
|[[#linktoabstract  |   Open problems ]]
| Street
+
|  
 +
|
 
|-
 
|-
|<b>Monday, Apr 13</b>
+
|
|Yumeng Ou
+
|
|CUNY, Baruch College
+
|
|[[#linktoabstract  |  TBA ]]
+
|
|Zhang
 
 
|-
 
|-
|Apr 14
+
|Talks in the Fall semester 2022:
| Tamás Titkos
 
| BBS University of Applied Sciences & Rényi Institute
 
|[[#linktoabstract  |  Distance preserving maps on spaces of probability measures ]]
 
| Street
 
 
|-
 
|-
|Apr 21
+
|September 20,  PDE and Analysis Seminar
| Diogo Oliveira e Silva
+
|Andrej Zlatoš
| University of Birmingham
+
|UCSD
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
| Hung Tran
|-
 
|Apr 28
 
| No Seminar
 
 
|-
 
|-
|May 5
+
|Friday, September 23, 4:00 p.m., Colloquium
|Jonathan Hickman
+
|Pablo Shmerkin
|University of Edinburgh
+
|University of British Columbia
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Andreas
+
|Shaoming and Andreas
 
|-
 
|-
 +
|September 24-25, RTG workshop in Harmonic Analysis
 +
|
 +
|
 +
|
 +
|Shaoming and Andreas
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===José Madrid===
+
===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===
  
Title: On the regularity of maximal operators on Sobolev Spaces
+
Stability of sharp Fourier restriction to spheres
  
Abstract:  In this talk, we will discuss the regularity properties (boundedness and
+
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.  
continuity) of the classical and fractional maximal
 
operators when these act on the Sobolev space W^{1,p}(\R^n). We will
 
focus on the endpoint case p=1. We will talk about
 
some recent results and current open problems.
 
  
===Yakun Xi===
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
+
===Rajula Srivastava===
  
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.
+
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
===Joris Roos===
+
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.
  
Title: L^p improving estimates for maximal spherical averages
+
===Itamar Oliveira===
  
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
+
A new approach to the Fourier extension problem for the paraboloid
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.
 
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.
 
  
 +
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===
  
===Joao Ramos===
+
Decoupling inequalities for quadratic forms and beyond
  
Title: Fourier uncertainty principles, interpolation and uniqueness sets
+
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: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that
+
===Alexandru Ionescu===
  
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
+
Polynomial averages and pointwise ergodic theorems on nilpotent groups
  
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.  
+
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.
  
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$
+
===Liding Yao===
  
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.
+
An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
  
===Xiaojun Huang===
+
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.
  
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
+
===Lingxiao Zhang===
  
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.
+
Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
  
===Xiaocheng Li===
+
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 $K$.
  
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
+
===Kasso Okoudjou===
  
Abstract:  We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.
+
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.
  
===Xiaochun Li===
+
===Rahul Parhi===
  
Title:  Roth’s type theorems on progressions
+
On BV Spaces, Splines, and Neural Networks
  
Abstract:  The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.
+
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.
  
===Jeff Galkowski===
+
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.
  
<b>Concentration and Growth of Laplace Eigenfunctions</b>
+
This is joint work with Robert Nowak.
  
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.
+
===Alexei Poltoratski===
  
===David Beltran===
+
Title: Pointwise convergence for the scattering data and non-linear Fourier transform.
  
Title: Regularity of the centered fractional maximal function
+
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.
  
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.
+
===John Green===
  
This is joint work with José Madrid.
+
Estimates for oscillatory integrals via sublevel set estimates.
  
===Dominique Kemp===
+
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.
  
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
+
===Tao Mei===
  
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone.  Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.
+
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).
  
===Kevin O'Neill===
+
===Alex Nagel===
  
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
+
Global estimates for a class of kernels and multipliers with multiple homogeneities
  
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.
+
In joint work with Fulvio Ricci we obtain global estimates for a class of kernels and multipliers which contain homogeneous Calderon-Zygmund operators for several different homogeneities. This is an extension of earlier work with Ricci, Stein, and Wainger on the local theory.
  
===Francesco di Plinio===
+
===Sebastian Bechtel===
  
<b>Maximal directional integrals along algebraic and lacunary sets </b>
+
Square roots of elliptic systems on open sets
  
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of  Nagel-Stein-Wainger.
+
In my talk, we will consider elliptic systems in divergence form with measurable and elliptic complex coefficients on possibly unbounded open sets which are subject to mixed boundary conditions. First, I will present and discuss minimal geometric conditions under which Kato’s square root problem can be solved. In particular, I will present an argument that allows to work on a set that is not supposed to satisfy the interior thickness condition.
 +
Afterwards, we will investigate the question for which integrability parameters p the square root isomorphism $W^{1,2} \to L^2$ extrapolates to an isomorphism $W^{1,p} \to L^p$. We focus on the case $p>2$. I will introduce a critical number that describes the range in which $L$ (compatibly) acts as an isomorphism $W^{1,p} \to W^{-1,p}$. We will then see that this critical number also yields an optimal range in which the square root extrapolates to a $p$-isomorphism, even in the case of mixed boundary conditions.
 +
 
 +
===Tongou Yang===
 +
 
 +
Restricted projections along $C^2$ curves on the sphere
 +
 
 +
Given a $C^2$ closed curve $\gamma(\theta)$ lying on the sphere
 +
$\mathbb S^2$ and a Borel set $A\subseteq \mathbb R^3$. Consider the
 +
projections $P_\theta(A)$ of $A$ into straight lines in the directions
 +
$\gamma(\theta)$. We prove that if $\gamma$ satisfies the torsion
 +
condition: $\det(\gamma,\gamma',\gamma")(\theta)\neq 0$ for any $\theta$,
 +
then for almost every $\theta$, the Hausdorff dimension of $P_\theta(A)$ is
 +
equal to $\min\{1,\dim_H(A)\}$. This solves a conjecture of Fässler and
 +
Orponen. One key feature of our argument is a result of Marcus-Tardos in
 +
topological graph theory. This is a joint work with Malabika Pramanik, Orit Raz and Josh Zahl.
 +
 
 +
===Po Lam Yung===
 +
 
 +
Revisiting an old argument for Vinogradov's Mean Value Theorem
 +
 
 +
We will examine an old argument for the Vinogradov's Mean Value Theorem due to Karatsuba, and interpret it in the language of Fourier decoupling. This is ongoing work in progress with Brian Cook, Kevin Hughes, Zane Kun Li, Akshat Mudgal and Olivier Robert.
 +
 
 +
===Brian Street===
 +
 
 +
Maximal Subellipticity
 +
 
 +
The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general linear and fully nonlinear elliptic PDEs. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced: now known as maximal subellipticity or maximal hypoellipticity.  In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry.  The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis.
  
 
===Laurent Stolovitch===
 
===Laurent Stolovitch===
  
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
+
Classification of reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional hyperbolic type
  
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.
+
The aim of this joint work with Martin Klimes is twofold:
  
===Bingyang Hu===
+
First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants.
  
<b>Sparse bounds of singular Radon transforms</b>
+
Related to it, we solve the problem of both formal and analytic classification of germs of real analytic surfaces in $\mathbb{C}^2$ with non-degenerate CR singularities of exceptional hyperbolic type, under the assumption that the surface is holomorphically flat, i.e. that it can be locally holomorphically embedded in a real hypersurface of $\mathbb{C}^2$.
  
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.
+
===Betsy Stovall===
  
===Lillian Pierce===
+
On extremizing sequences for adjoint Fourier restriction to the sphere
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
 
  
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices.
+
In this talk we will provide a soft answer to the question, "What properties must a function $f$ obeying $\|Ef\|_q \geq C \|f\|_p$ have?," where $E$ denotes the spherical extension operator.  We will use our answer (called a linear profile decomposition) to establish new results about the existence of extremizers (functions obeying $\|Ef\|_q = \|E\|\|f\|_p$) for $E$. This is joint work with Taryn C. Flock.
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.
 
  
===Ruixiang Zhang===
+
===Malabika Pramanik===
  
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
+
https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf
  
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate  gains a fractional  derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.
+
===Hongki Jung===
  
===Zane Li===
+
A small cap decoupling for the twisted cubic
  
<b> A bilinear proof of decoupling for the moment curve</b>
+
===Bernhard Lamel===
  
We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.
+
Convergence and Divergence of Formal Power Series Maps
  
 +
Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.
  
===Sergey Denisov===
+
===Carmelo Puliatti===
  
<b> De Branges canonical systems with finite logarithmic integral </b>
+
Gradients of single layer potentials for elliptic operators 
 +
with coefficients of Dini mean oscillation-type
  
We consider measures m on the real line for which logarithmic
+
We consider a uniformly elliptic operator $L_A$ in divergence form  
integral exists and give a complete characterization of all Hamiltonians
+
associated with a matrix A with real, bounded, and possibly 
in de Branges canonical system for which m is the spectral measure.
+
non-symmetric coefficients. If a proper $L^1$-mean oscillation of the 
This characterization involves the matrix A_2 Muckenhoupt condition on a
+
coefficients of A satisfies suitable Dini-type assumptions, we prove 
fixed scale. Our result provides a generalization of the classical
+
the following: if \mu is a compactly supported Radon measure in   
theorem of Szego for polynomials orthogonal on the unit circle and
+
$\mathbb{R}^{n+1}, n >= 2$,  the $L^2(\mu)$-operator norm of the gradient of the 
complements the Krein-Wiener theorem. Based on the joint work with R.
+
single layer potential $T_\mu$ associated with $L_A$ is comparable to the
Bessonov.
+
$L^2$-norm of the n-dimensional Riesz transform $R_\mu$, modulo an 
 +
additive constant.
 +
This makes possible to obtain direct generalizations of some deep 
 +
geometric results, initially proved for the Riesz transform, which 
 +
were recently extended to $T_\mu$ under a H\"older continuity assumption 
 +
on the coefficients of the matrix $A$.
  
 +
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and 
 +
Xavier Tolsa.
  
===Michel Alexis===
+
===Larry Guth===
  
<b>The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight</b>
+
Series title: Reflections on decoupling and Vinogradov's mean value problem.
  
Let $\{\varphi_n\}_{n=0}^{\infty}$ be the sequence of degree $n$ polynomials on $\mathbb{T}$, orthonormal with respect to a positive weight $w$. Steklov conjectured whenever $w \geq \delta> 0$ a.e.\ then $\{\varphi_n\}$ are uniformly bounded in $L^{\infty}$. While false, this conjecture brings us to ask the following: under what regularity conditions on $w$ are $\{\varphi_n\}$ uniformly bounded in $L^p (w)$ for some $p > 2$?
+
Series abstract: Decoupling is a recent development in Fourier analysis that has solved several longstanding problems.  The goal of the lectures is to describe this development to a general mathematical audience. We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory.  This problem is about the number of solutions of a certain system of diophantine equations. It was raised in the 1930s and resolved in the last decade. We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof.  The method of decoupling came as a big surprise to me, and I think to other people working in the field. The main idea in the proof of decoupling is to combine estimates from many different scales.  We will describe this process and reflect on why it is helpful.
 
We discuss some answers to this question using the contraction principle and operator estimates for the Hilbert transform, in particular recent joint work with Alexander Aptakarev and Sergey Denisov for when $w$ is a Muckenhoupt weight.
 
  
===William Green===
 
  
<b> Dispersive estimates for the Dirac equation </b>
+
Lecture 1: Introduction to decoupling and Vinogradov's mean value problem.
 +
Abstract: In this lecture, we introduce Vinogradov's problem and give an overview of the proof.
  
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations
+
Lecture 2: Features of the proof of decoupling.
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.
+
Abstract: In this lecture, we look more closely at some features of the proof of decoupling.  The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scalesThe second feature we examine is called the wave packet decompositionThis structure has roots in quantum physics and in information theory.
   
 
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations.  We will survey recent work on time-decay estimates for the solution operatorSpecifically the mapping properties of the solution operator between L^p spacesAs in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution.  We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay.  The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).
 
  
===Yifei Pan===
+
Lecture 3: Open problems.
 +
Abstract: In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*.  In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems.  Hopefully this will give a sense of some of the issues and difficulties involved in these problems.
  
<b>On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n</b>
+
=[[Previous_Analysis_seminars]]=
  
In this talk, I will present a proof of the following Sobolev space property of logarithmic modulus of holomorphic functions in C^n. If f is a holomorphic function on the unit ball B(0,1) in C^n vanishing at the origin (i.e., f(0) = 0) but it is not identically zero, then log |f| ∈ W^{1,p}(B(0, r)) for any p < 2, but log |f| is not in W^{1,2}(B(0, r)) (r < 1). As you may see, this result is rather simple to prove in the complex plane due to the discreteness of zeros of holomorphic functions. In higher dimensions, we are going to apply Hironaka’s resolution of singularity and then Harvey- Polking removable singularity method to prove the existence of weak derivatives of log |f(z)|. This is part of a joint project with Ziming Shi at Madison.
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
 
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=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 00:07, 28 April 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, Online 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, Online 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 Betsy
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 Shaoming
Winter break
February 8, VV B139 Alexander Nagel UW Madison Global estimates for a class of kernels and multipliers with multiple homogeneities
February 15, Online Sebastian Bechtel Institut de Mathématiques de Bordeaux Square roots of elliptic systems on open sets
Friday, February 18, Colloquium, VVB239 Andreas Seeger UW Madison Spherical maximal functions and fractal dimensions of dilation sets
February 22, VV B139 Tongou Yang University of British Comlumbia Restricted projections along $C^2$ curves on the sphere Shaoming
Monday, February 28, 4:30 p.m., Online Po Lam Yung Australian National University Revisiting an old argument for Vinogradov's Mean Value Theorem
March 8, VV B139 Brian Street UW Madison Maximal Subellipticity
March 15: No Seminar
March 22 Laurent Stolovitch University of Cote d'Azur Classification of reversible parabolic diffeomorphisms of

$(\mathbb{C}^2,0)$ and of flat CR-singularities of exceptional hyperbolic type

Xianghong
March 29, VV B139 Betsy Stovall UW Madison On extremizing sequences for adjoint Fourier restriction to the sphere
April 5, Online Malabika Pramanik University of British Columbia Dimensionality and Patterns with Curvature
April 12, VV B139 Hongki Jung IU Bloomington A small cap decoupling for the twisted cubic Shaoming
Friday, April 15, Colloquium, VV B239 Bernhard Lamel Texas A&M University at Qatar Convergence and Divergence of Formal Power Series Maps Xianghong
April 19, Online Carmelo Puliatti Euskal Herriko Unibertsitatea Gradients of single layer potentials for elliptic operators

with coefficients of Dini mean oscillation-type

David
April 25-26-27, Distinguished Lecture Series Larry Guth MIT Reflections on decoupling and Vinogradov's mean value problem.
April 25, 4:00 p.m., Lecture I, VV B239 Introduction to decoupling and Vinogradov's mean value problem
April 26, 4:00 p.m., Lecture II, Chamberlin 2241 Features of the proof of decoupling
April 27, 4:00 p.m., Lecture III, VV B239 Open problems
Talks in the Fall semester 2022:
September 20, PDE and Analysis Seminar Andrej Zlatoš UCSD Title Hung Tran
Friday, September 23, 4:00 p.m., Colloquium Pablo Shmerkin University of British Columbia Title Shaoming and Andreas
September 24-25, RTG workshop in Harmonic Analysis Shaoming and Andreas

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).

Alex Nagel

Global estimates for a class of kernels and multipliers with multiple homogeneities

In joint work with Fulvio Ricci we obtain global estimates for a class of kernels and multipliers which contain homogeneous Calderon-Zygmund operators for several different homogeneities. This is an extension of earlier work with Ricci, Stein, and Wainger on the local theory.

Sebastian Bechtel

Square roots of elliptic systems on open sets

In my talk, we will consider elliptic systems in divergence form with measurable and elliptic complex coefficients on possibly unbounded open sets which are subject to mixed boundary conditions. First, I will present and discuss minimal geometric conditions under which Kato’s square root problem can be solved. In particular, I will present an argument that allows to work on a set that is not supposed to satisfy the interior thickness condition. Afterwards, we will investigate the question for which integrability parameters p the square root isomorphism $W^{1,2} \to L^2$ extrapolates to an isomorphism $W^{1,p} \to L^p$. We focus on the case $p>2$. I will introduce a critical number that describes the range in which $L$ (compatibly) acts as an isomorphism $W^{1,p} \to W^{-1,p}$. We will then see that this critical number also yields an optimal range in which the square root extrapolates to a $p$-isomorphism, even in the case of mixed boundary conditions.

Tongou Yang

Restricted projections along $C^2$ curves on the sphere

Given a $C^2$ closed curve $\gamma(\theta)$ lying on the sphere $\mathbb S^2$ and a Borel set $A\subseteq \mathbb R^3$. Consider the projections $P_\theta(A)$ of $A$ into straight lines in the directions $\gamma(\theta)$. We prove that if $\gamma$ satisfies the torsion condition: $\det(\gamma,\gamma',\gamma")(\theta)\neq 0$ for any $\theta$, then for almost every $\theta$, the Hausdorff dimension of $P_\theta(A)$ is equal to $\min\{1,\dim_H(A)\}$. This solves a conjecture of Fässler and Orponen. One key feature of our argument is a result of Marcus-Tardos in topological graph theory. This is a joint work with Malabika Pramanik, Orit Raz and Josh Zahl.

Po Lam Yung

Revisiting an old argument for Vinogradov's Mean Value Theorem

We will examine an old argument for the Vinogradov's Mean Value Theorem due to Karatsuba, and interpret it in the language of Fourier decoupling. This is ongoing work in progress with Brian Cook, Kevin Hughes, Zane Kun Li, Akshat Mudgal and Olivier Robert.

Brian Street

Maximal Subellipticity

The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general linear and fully nonlinear elliptic PDEs. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced: now known as maximal subellipticity or maximal hypoellipticity. In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry. The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis.

Laurent Stolovitch

Classification of reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ and of flat CR-singularities of exceptional hyperbolic type

The aim of this joint work with Martin Klimes is twofold:

First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants.

Related to it, we solve the problem of both formal and analytic classification of germs of real analytic surfaces in $\mathbb{C}^2$ with non-degenerate CR singularities of exceptional hyperbolic type, under the assumption that the surface is holomorphically flat, i.e. that it can be locally holomorphically embedded in a real hypersurface of $\mathbb{C}^2$.

Betsy Stovall

On extremizing sequences for adjoint Fourier restriction to the sphere

In this talk we will provide a soft answer to the question, "What properties must a function $f$ obeying $\|Ef\|_q \geq C \|f\|_p$ have?," where $E$ denotes the spherical extension operator. We will use our answer (called a linear profile decomposition) to establish new results about the existence of extremizers (functions obeying $\|Ef\|_q = \|E\|\|f\|_p$) for $E$. This is joint work with Taryn C. Flock.

Malabika Pramanik

https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf

Hongki Jung

A small cap decoupling for the twisted cubic

Bernhard Lamel

Convergence and Divergence of Formal Power Series Maps

Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.

Carmelo Puliatti

Gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

We consider a uniformly elliptic operator $L_A$ in divergence form associated with a matrix A with real, bounded, and possibly non-symmetric coefficients. If a proper $L^1$-mean oscillation of the coefficients of A satisfies suitable Dini-type assumptions, we prove the following: if \mu is a compactly supported Radon measure in $\mathbb{R}^{n+1}, n >= 2$, the $L^2(\mu)$-operator norm of the gradient of the single layer potential $T_\mu$ associated with $L_A$ is comparable to the $L^2$-norm of the n-dimensional Riesz transform $R_\mu$, modulo an additive constant. This makes possible to obtain direct generalizations of some deep geometric results, initially proved for the Riesz transform, which were recently extended to $T_\mu$ under a H\"older continuity assumption on the coefficients of the matrix $A$.

This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Larry Guth

Series title: Reflections on decoupling and Vinogradov's mean value problem.

Series abstract: Decoupling is a recent development in Fourier analysis that has solved several longstanding problems. The goal of the lectures is to describe this development to a general mathematical audience. We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory. This problem is about the number of solutions of a certain system of diophantine equations. It was raised in the 1930s and resolved in the last decade. We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof. The method of decoupling came as a big surprise to me, and I think to other people working in the field. The main idea in the proof of decoupling is to combine estimates from many different scales. We will describe this process and reflect on why it is helpful.


Lecture 1: Introduction to decoupling and Vinogradov's mean value problem. Abstract: In this lecture, we introduce Vinogradov's problem and give an overview of the proof.

Lecture 2: Features of the proof of decoupling. Abstract: In this lecture, we look more closely at some features of the proof of decoupling. The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scales. The second feature we examine is called the wave packet decomposition. This structure has roots in quantum physics and in information theory.

Lecture 3: Open problems. Abstract: In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*. In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems. Hopefully this will give a sense of some of the issues and difficulties involved in these problems.

Previous_Analysis_seminars

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

Extras

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Graduate Student Seminar:

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