Difference between revisions of "Fall 2021 and Spring 2022 Analysis Seminars"

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'''Analysis Seminar
 
'''
 
  
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  Betsy at stovall(at)math
+
Zoom links will be sent to those who have signed up for the Analysis Seminar List.  If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. 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.
  
= 2017-2018 Analysis Seminar Schedule =
+
= Analysis Seminar Schedule =
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
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!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 8 in B239 (Colloquium)
+
|September 21, VV B139
| Tess Anderson
+
| Dóminique Kemp
 +
| UW-Madison
 +
|[[#Dóminique Kemp  |  Decoupling by way of approximation ]]
 +
|
 +
|-
 +
|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 ]]
 +
|
 +
|-
 +
|October 19, Online
 +
|Itamar Oliveira
 +
|Cornell University
 +
|[[#Itamar Oliveira  |  A new approach to the Fourier extension problem for the paraboloid ]]
 +
|
 +
|-
 +
|October 26, VV B139
 +
| Changkeun Oh
 
| UW Madison
 
| UW Madison
|[[#linktoabstract A Spherical Maximal Function along the Primes]]
+
|[[#Changkeun Oh Decoupling inequalities for quadratic forms and beyond ]]
|Tonghai
+
|  
 
|-
 
|-
|September 19
+
|October 29, Colloquium, Online
| Brian Street
+
| Alexandru Ionescu
 +
| Princeton University
 +
|[[#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 ]]
 +
|
 +
|-
 +
|November 9, VV B139
 +
| Lingxiao Zhang
 
| UW Madison
 
| UW Madison
|[[#Brian Street Convenient Coordinates ]]
+
|[[#Lingxiao Zhang Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition ]]
| Betsy
+
|  
 
|-
 
|-
|September 26
+
|November 12, Colloquium, Online
| Hiroyoshi Mitake
+
| Kasso Okoudjou
| Hiroshima University
+
| Tufts University
|[[#Hiroyoshi Mitake Derivation of multi-layered interface system and its application ]]
+
|[[#Kasso Okoudjou An exploration in analysis on fractals ]]
| Hung
 
 
|-
 
|-
|October 3
+
|November 16, VV B139
| Joris Roos
+
| Rahul Parhi
| UW Madison
+
| UW Madison (EE)
|[[#Joris Roos |   A polynomial Roth theorem on the real line ]]
+
|[[#Rahul Parhi |   On BV Spaces, Splines, and Neural Networks ]]
 
| Betsy
 
| Betsy
 
|-
 
|-
|October 10
+
|November 30, VV B139
| Michael Greenblatt
+
| Alexei Poltoratski
| UI Chicago
+
| UW Madison
|[[#Michael Greenblatt |   Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
+
|[[#Alexei Poltoratski | Pointwise convergence for the scattering data and non-linear Fourier transform. ]]
| Andreas
+
|  
 
|-
 
|-
|October 17
+
|December 7, Online
| David Beltran
+
| John Green
| Basque Center of Applied Mathematics
+
| The University of Edinburgh
|[[#David Beltran |   Fefferman-Stein inequalities ]]
+
|[[#John Green | Estimates for oscillatory integrals via sublevel set estimates ]]
| Andreas
+
|  
 
|-
 
|-
|Wednesday, October 18, 4:00 p.m.  in B131
+
|December 14, VV B139
|Jonathan Hickman
+
| Tao Mei
|University of Chicago
+
| Baylor University
|[[#Jonathan Hickman | Factorising X^n  ]]
+
|[[#Tao Mei |   Fourier Multipliers on free groups ]]
|Andreas
+
| Shaoming
 
|-
 
|-
|October 24
+
|Winter break
| Xiaochun Li
+
|
| UIUC
+
|
|[[#Xiaochun Li  |  Recent progress on the pointwise convergence problems of Schroedinger equations ]]
 
| Betsy
 
 
|-
 
|-
|Thursday, October 26, 4:30 p.m. in B139
+
|February 8, VV B139
| Fedor Nazarov
+
|Alexander  Nagel
| Kent State University
+
| UW Madison
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp  ]]
+
|[[#Alex Nagel |   Global estimates for a class of kernels and multipliers with multiple homogeneities]]
| Sergey, Andreas
+
|  
 
|-
 
|-
|Friday, October 27, 4:00 p.m.  in B239
+
|February 15, Online
| Stefanie Petermichl
+
| Sebastian Bechtel
| University of Toulouse
+
| Institut de Mathématiques de Bordeaux
|[[#Stefanie Petermichl | Higher order Journé commutators  ]]
+
|[[#Sebastian Bechtel | Square roots of elliptic systems on open sets]]
| Betsy, Andreas
+
|  
 
|-
 
|-
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
+
|Friday, February 18, Colloquium, VVB239
| Shaoming Guo
+
| Andreas Seeger
| Indiana University
 
|[[#Shaoming Guo  |  Parsell-Vinogradov systems in higher dimensions ]]
 
| Andreas
 
|-
 
|November 14
 
| Naser Talebizadeh Sardari
 
 
| UW Madison
 
| UW Madison
|[[#Naser Talebizadeh Sardari  |   Quadratic forms and the semiclassical eigenfunction hypothesis ]]
+
|[[#Andreas Seeger | Spherical maximal functions and fractal dimensions of dilation sets]]
| Betsy
+
|  
 
|-
 
|-
|November 28
+
|February 22, VV B139
| Xianghong Chen
+
|Tongou Yang
| UW Milwaukee
+
|University of British Comlumbia
|[[#Xianghong Chen Some transfer operators on the circle with trigonometric weights ]]
+
|[[#linktoabstract Restricted projections along $C^2$ curves on the sphere ]]
| Betsy
+
| Shaoming
 
|-
 
|-
|Monday, December 4, 4:00, B139
+
|Monday, February 28, 4:30 p.m., Online
| Bartosz Langowski and Tomasz Szarek
+
| Po Lam Yung
| Institute of Mathematics, Polish Academy of Sciences
+
| Australian National University
|[[#Bartosz Langowski and Tomasz Szarek |   Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|[[#Po Lam Yung | Revisiting an old argument for Vinogradov's Mean Value Theorem ]]
| Betsy
+
|  
 
|-
 
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
+
|March 8, VV B139
|Bobby Wilson
+
| Brian Street
|MIT
+
| UW Madison
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
+
|[[#Brian Street  | Maximal Subellipticity ]]
| Andreas
+
|  
 
|-
 
|-
| Monday, February 5, 3:00-3:50, B341  (PDE-GA seminar)
+
|March 15: No Seminar
| Andreas Seeger
+
|  
| UW
+
|  
|[[#Andreas Seeger |  Singular integrals and a problem on mixing flows]]
 
 
|
 
|
 +
|
 
|-
 
|-
|February 6
+
|March 22
| Dong Dong
+
| Laurent Stolovitch
| UIUC
+
| University of Cote d'Azur
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
+
|[[#linktoabstract  |   Classification of reversible parabolic diffeomorphisms of
|Betsy
+
$(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional
 +
hyperbolic type ]]
 +
| Xianghong
 
|-
 
|-
|February 13
+
|March 29, VV B139
| Sergey  Denisov
+
|Betsy Stovall
| UW Madison
+
|UW Madison
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]
+
|[[#Betsy Stovall  |   On extremizing sequences for adjoint Fourier restriction to the sphere ]]
 
|  
 
|  
 
|-
 
|-
|February 20
+
|April 5, Online
| Ruixiang Zhang
+
|Malabika Pramanik
| IAS (Princeton)
+
|University of British Columbia
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]
+
|[[#Malabika Pramanik |   Dimensionality and Patterns with Curvature]]
| Betsy, Jordan, Andreas
+
|  
 
|-
 
|-
|February 27
+
|April 12, VV B139
|Detlef Müller
+
| Hongki Jung
|University of Kiel
+
| IU Bloomington
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]
+
|[[#Hongki Jung  | A small cap decoupling for the twisted cubic ]]
|Betsy, Andreas
+
| Shaoming
 
|-
 
|-
|Wednesday, March 7, 4:00 p.m.
+
|Friday, April 15, Colloquium, VV B239
| Winfried Sickel
+
| Bernhard Lamel
|Friedrich-Schiller-Universität Jena
+
| Texas A&M University at Qatar
| [[#Winfried Sickel | On the regularity of compositions of functions]]
+
|[[#Bernhard Lamel  |   Convergence and Divergence of Formal Power Series Maps ]]
|Andreas
+
| Xianghong
 
|-
 
|-
|March 20
+
|April 19, Online
| Betsy Stovall
+
| Carmelo Puliatti
| UW
+
| Euskal Herriko Unibertsitatea
| [[#linkofabstract | Two endpoint bounds via inverse problems]]
+
|[[#Carmelo Puliatti  |   Gradients of single layer potentials for elliptic operators 
 +
with coefficients of Dini mean oscillation-type ]]
 +
| David
 
|
 
|
 +
|
 
|-
 
|-
|April 10
+
|April 25-26-27, Distinguished Lecture Series
| Martina Neuman
+
|Larry Guth
| UC Berkeley
+
|MIT
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]
+
|[[#Larry Guth  | Reflections on decoupling and Vinogradov's mean value problem. ]]
| Betsy
+
|-
 +
|April 25, 4:00 p.m., Lecture I, VV B239
 +
|
 +
|
 +
|[[#linktoabstract  |  Introduction to decoupling and Vinogradov's mean value problem ]]
 
|-
 
|-
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)
+
|April 26, 4:00 p.m., Lecture II, Chamberlin 2241
|Jill Pipher
+
|
|Brown
+
|
| [[#Jill Pipher | Mathematical ideas in cryptography]]
+
|[[#linktoabstract  | Features of the proof of decoupling  ]]
|WIMAW
 
 
|-
 
|-
|April 17
+
|April 27, 4:00 p.m., Lecture III, VV B239
 +
|
 +
|
 +
|[[#linktoabstract  |    Open problems ]]
 
|  
 
|  
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
 
|-
 
|-
|April 24
+
|
| Lenka Slavíková
+
|
| University of Missouri
+
|
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]
+
|
|Betsy, Andreas
 
 
|-
 
|-
|May 1
+
|Talks in the Fall semester 2022:
| Xianghong Gong
 
| UW
 
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]
 
|
 
 
|-
 
|-
| '''May 7''' in '''B223'''
+
|September 20,  PDE and Analysis Seminar
| Ebru Toprak
+
|Andrej Zlatoš
| UIUC
+
|UCSD
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]
+
|[[#linktoabstract  |   Title ]]
|Betsy
+
| Hung Tran
 
|-
 
|-
| '''May 15'''
+
|Friday, September 23, 4:00 p.m., Colloquium
| Gennady Uraltsev
+
|Pablo Shmerkin
| Cornell
+
|University of British Columbia
| [[#linkofabstract | TBA]]
+
|[[#linktoabstract  |   Title ]]
| Andreas, Betsy
+
|Shaoming and Andreas
 
|-
 
|-
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
+
|September 24-25, RTG workshop in Harmonic Analysis
 
|
 
|
 
|
 
|
 
|
 
|
|Betsy, Andreas
+
|Shaoming and Andreas
 
|-
 
|-
 +
|Tuesday, November 8,
 +
|Robert Fraser
 +
|Wichita State University
 +
|[[#linktoabstract  |  Title ]]
 +
| Shaoming and Andreas
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Brian Street===
+
===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===
  
Title:  Convenient Coordinates
+
Decoupling inequalities for quadratic forms and beyond
  
Abstract:  We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields.  We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic).  By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps.  When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger.  When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".
+
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.
  
===Hiroyoshi Mitake===
+
===Alexandru Ionescu===
  
Title:  Derivation of multi-layered interface system and its application
+
Polynomial averages and pointwise ergodic theorems on nilpotent groups
  
Abstract:  In this talk, I will propose a multi-layered interface system which can
+
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.
be formally derived by the singular limit of the weakly coupled system of
 
the Allen-Cahn equation. By using the level set approach, this system can be
 
written as a quasi-monotone degenerate parabolic system.
 
We give results of the well-posedness of viscosity solutions, and study the  
 
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
 
  
===Joris Roos===
+
===Liding Yao===
  
Title: A polynomial Roth theorem on the real line
+
An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
  
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.
+
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.
  
===Michael Greenblatt===
+
===Lingxiao Zhang===
  
Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces
+
Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
  
Abstract:  A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.
+
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 $K$.
  
===David Beltran===
+
===Kasso Okoudjou===
  
Title:  Fefferman Stein Inequalities
+
An exploration in analysis on fractals
  
Abstract:  Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.
+
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.
  
===Jonathan Hickman===
+
===Rahul Parhi===
  
Title: Factorising X^n.
+
On BV Spaces, Splines, and Neural Networks
  
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.
+
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.
  
===Xiaochun Li ===
+
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.
  
Title:  Recent progress on the pointwise convergence problems of Schrodinger equations
+
This is joint work with Robert Nowak.
  
Abstract:  Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3.  This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and  the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.
+
===Alexei Poltoratski===
  
===Fedor Nazarov=== 
+
Title: Pointwise convergence for the scattering data and non-linear Fourier transform.
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
+
Abstract: This talk is about applications of complex and harmonic analysis in spectral and scattering theory
conjecture is sharp.
+
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: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
+
===John Green===
the norm of the Hilbert transform on the line as an operator from $L^1(w)$
 
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work
 
with Andrei Lerner and Sheldy Ombrosi.
 
  
===Stefanie Petermichl===
+
Estimates for oscillatory integrals via sublevel set estimates.
Title: Higher order Journé commutators
 
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
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.
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
 
more basic terms, let us consider a function on the unit circle in its
 
Fourier representation. Let P_+ denote the projection onto
 
non-negative and P_- onto negative frequencies. Let b denote
 
multiplication by the symbol function b. It is a classical theorem by
 
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
 
only if b is in an appropriate space of functions of bounded mean
 
oscillation. The necessity makes use of a classical factorisation
 
theorem of complex function theory on the disk. This type of question
 
can be reformulated in terms of commutators [b,H]=bH-Hb with the
 
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
 
as in the real variable setting, in the multi-parameter setting or
 
other, these classifications can be very difficult.
 
  
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
+
===Tao Mei===
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
 
spaces of bounded mean oscillation via L^p boundedness of commutators.
 
We present here an endpoint to this theory, bringing all such
 
characterisation results under one roof.
 
  
The tools used go deep into modern advances in dyadic harmonic
+
Fourier Multipliers on free groups.
analysis, while preserving the Ansatz from classical operator theory.
 
  
===Shaoming Guo ===
+
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: Parsell-Vinogradov systems in higher dimensions
 
  
Abstract:
+
===Alex Nagel===
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
 
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
 
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
 
  
===Naser Talebizadeh Sardari===
+
Global estimates for a class of kernels and multipliers with multiple homogeneities
  
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
+
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.
  
Abstract:  Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>,  and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math>  in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove  a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given  eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.
+
===Sebastian Bechtel===
  
===Xianghong Chen===
+
Square roots of elliptic systems on open sets
  
Title:  Some transfer operators on the circle with trigonometric weights
+
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.
  
Abstract:  A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.
+
===Tongou Yang===
  
===Bobby Wilson===
+
Restricted projections along $C^2$ curves on the sphere
  
Title: Projections in Banach Spaces and Harmonic Analysis
+
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.
  
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
+
===Po Lam Yung===
  
===Andreas Seeger===
+
Revisiting an old argument for Vinogradov's Mean Value Theorem
  
Title: Singular integrals and a problem on mixing flows
+
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.
  
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
+
===Brian Street===
  
===Dong Dong===
+
Maximal Subellipticity
  
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
+
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.
  
Abstract:  This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.
+
===Laurent Stolovitch===
  
===Sergey Denisov===
+
Classification of reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional hyperbolic type
  
Title: Spectral Szegő  theorem on the real line
+
The aim of this joint work with Martin Klimes is twofold:
  
Abstract:  For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.
+
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.
  
===Ruixiang Zhang===
+
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 $\mathbb{C}^2$.
  
Title:  The (Euclidean) Fractal Uncertainty Principle
+
===Betsy Stovall===
  
Abstract:  On the real line, a  version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work  the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).
+
On extremizing sequences for adjoint Fourier restriction to the sphere
  
===Detlef Müller===
+
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.
  
Title: On Fourier restriction for a non-quadratic hyperbolic surface
+
===Malabika Pramanik===
  
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about  hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically  in the presence of a perturbation term,  and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.
+
https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf
  
===Winfried Sickel===
+
===Hongki Jung===
  
Title: On the regularity of compositions of functions
+
A small cap decoupling for the twisted cubic
  
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math>
+
===Bernhard Lamel===
we associate the composition operator
 
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>.
 
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.
 
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and  Slobodeckij spaces <math>W^s_p</math>.
 
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math>
 
such that the composition operator maps such a space <math>E</math> into itself.
 
  
===Martina Neuman===
+
Convergence and Divergence of Formal Power Series Maps
  
Title:  Gowers-Host-Kra norms and Gowers structure on Euclidean spaces
+
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.
  
Abstract:  The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.
+
===Carmelo Puliatti===
  
===Jill Pipher===
+
Gradients of single layer potentials for elliptic operators 
 +
with coefficients of Dini mean oscillation-type
  
TitleMathematical ideas in cryptography
+
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 $A$.
  
Abstract:  This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,
+
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and
including homomorphic encryption.
+
Xavier Tolsa.
  
===Lenka Slavíková===
+
===Larry Guth===
  
Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria
+
Series title: Reflections on decoupling and Vinogradov's mean value problem.
  
AbstractIt is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.
+
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.
  
===Xianghong Gong===
 
  
Title: Smooth equivalence of deformations of domains in complex euclidean spaces
+
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.
  
Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.
+
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.
  
===Ebru Toprak===
+
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.
  
Title:  Dispersive estimates for massive Dirac equations
+
=[[Previous_Analysis_seminars]]=
  
Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
 
=Extras=
 
=Extras=
 +
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
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Graduate Student Seminar:
 +
 +
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html

Latest revision as of 04:00, 5 July 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
Tuesday, November 8, Robert Fraser Wichita State University Title 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