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

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The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
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The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
+
Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
  
Zoom links will be sent to those who have signed up for the Analysis Seminar List.  If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
+
Zoom links will be sent to those who have signed up for the Analysis Seminar List.  If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. If you are from an institution different than UW-Madison, please send as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
  
If you'd like to suggest speakers for the spring semester please contact David and Andreas.
+
If you'd like to suggest speakers for the spring semester please contact David and Andreas.
  
 
= Analysis Seminar Schedule =
 
= Analysis Seminar Schedule =
Line 15: Line 15:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Date
+
|September 21, VV B139
| Person
+
| Dóminique Kemp
| Institution
+
| UW-Madison
|[[#linktoabstract  |  Title ]]
+
|[[#Dóminique Kemp  |  Decoupling by way of approximation ]]
| Sponsor
+
|
 +
|-
 +
|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
 +
|[[#Changkeun Oh  |  Decoupling inequalities for quadratic forms and beyond ]]
 +
|
 +
|-
 +
|October 29, Colloquium, Online
 +
| 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
 +
|[[#Lingxiao Zhang  |  Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition ]]
 +
|
 +
|-
 +
|November 12, Colloquium, Online
 +
| Kasso Okoudjou
 +
| Tufts University
 +
|[[#Kasso Okoudjou  |  An exploration in analysis on fractals ]]
 +
|-
 +
|November 16, VV B139
 +
| Rahul Parhi
 +
| UW Madison (EE)
 +
|[[#Rahul Parhi  |    On BV Spaces, Splines, and Neural Networks ]]
 +
| Betsy
 +
|-
 +
|November 30, VV B139
 +
| Alexei Poltoratski
 +
| UW Madison
 +
|[[#Alexei Poltoratski  |  Pointwise convergence for the scattering data and non-linear Fourier transform. ]]
 +
|
 +
|-
 +
|December 7, Online
 +
| John Green
 +
| The University of Edinburgh
 +
|[[#John Green  |  Estimates for oscillatory integrals via sublevel set estimates ]]
 +
|
 +
|-
 +
|December 14, VV B139
 +
| Tao Mei
 +
| Baylor University
 +
|[[#Tao Mei  |  Fourier Multipliers on free groups ]]
 +
| Shaoming
 +
|-
 +
|Winter break
 +
|
 +
|
 +
|-
 +
|February 8, VV B139
 +
|Alexander  Nagel
 +
| UW Madison
 +
|[[#Alex Nagel  |  Global estimates for a class of kernels and multipliers with multiple homogeneities]]
 +
|
 +
|-
 +
|February 15, Online
 +
| Sebastian Bechtel
 +
| Institut de Mathématiques de Bordeaux
 +
|[[#Sebastian Bechtel  | Square roots of elliptic systems on open sets]]
 +
|
 +
|-
 +
|Friday,  February 18,  Colloquium, VVB239
 +
| Andreas Seeger
 +
| UW Madison
 +
|[[#Andreas Seeger | Spherical maximal functions and fractal dimensions of dilation sets]]
 +
|
 +
|-
 +
|February 22, VV B139
 +
|Tongou Yang
 +
|University of British Comlumbia
 +
|[[#linktoabstract  |  Restricted projections along $C^2$ curves on the sphere ]]
 +
| 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 ]]
 +
|
 +
|-
 +
|March 8, VV B139
 +
| Brian Street
 +
| UW Madison
 +
|[[#Brian Street  |  Maximal Subellipticity ]]
 +
|
 +
|-
 +
|March 15: No Seminar
 +
|
 +
|
 +
|
 +
|
 +
|-
 +
|March 22
 +
| Laurent Stolovitch
 +
| University of Cote d'Azur
 +
|[[#linktoabstract  |  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
 +
|[[#Betsy Stovall  |  On extremizing sequences for adjoint Fourier restriction to the sphere ]]
 +
|
 +
|-
 +
|April 5, Online
 +
|Malabika Pramanik
 +
|University of British Columbia
 +
|[[#Malabika Pramanik |  Dimensionality and Patterns with Curvature]]
 +
|
 +
|-
 +
|April 12, VV B139
 +
| Hongki Jung
 +
| IU Bloomington
 +
|[[#Hongki Jung  |  A small cap decoupling for the twisted cubic ]]
 +
| Shaoming
 
|-
 
|-
|Date
+
|Friday, April 15, Colloquium, VV B239
| Person
+
| Bernhard Lamel
| Institution
+
| Texas A&M University at Qatar
|[[#linktoabstract Title ]]
+
|[[#Bernhard Lamel Convergence and Divergence of Formal Power Series Maps ]]
| Sponsor
+
| Xianghong
 
|-
 
|-
|Date
+
|April 19, Online
| Person
+
| Carmelo Puliatti
| Institution
+
| Euskal Herriko Unibertsitatea
|[[#linktoabstract Title ]]
+
|[[#Carmelo Puliatti Gradients of single layer potentials for elliptic operators 
| Sponsor
+
with coefficients of Dini mean oscillation-type ]]
 +
| David
 +
|
 +
|
 
|-
 
|-
|Date
+
|April 25-26-27, Distinguished Lecture Series
| Person
+
|Larry Guth
| Institution
+
|MIT
|[[#linktoabstract |   Title ]]
+
|[[#Larry Guth | Reflections on decoupling and Vinogradov's mean value problem. ]]
| Sponsor
 
 
|-
 
|-
|Date
+
|April 25, 4:00 p.m., Lecture I, VV B239
| Person
+
|
| Institution
+
|
|[[#linktoabstract  |   Title ]]
+
|[[#linktoabstract  | Introduction to decoupling and Vinogradov's mean value problem ]]
| Sponsor
 
 
|-
 
|-
|Date
+
|April 26, 4:00 p.m., Lecture II, Chamberlin 2241
| Person
+
|
| Institution
+
|
|[[#linktoabstract  |   Title ]]
+
|[[#linktoabstract  | Features of the proof of decoupling  ]]
| Sponsor
 
 
|-
 
|-
|Date
+
|April 27, 4:00 p.m., Lecture III, VV B239
| Person
+
|
| Institution
+
|
|[[#linktoabstract  |   Title ]]
+
|[[#linktoabstract  |   Open problems ]]
| Sponsor
+
|  
 +
|
 
|-
 
|-
|Date
+
|
| Person
+
|
| Institution
+
|
|[[#linktoabstract  |  Title ]]
+
|
| Sponsor
 
 
|-
 
|-
|Date
+
|Talks in the Fall semester 2022:
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
 
|-
 
|-
|Date
+
|September 20,  PDE and Analysis Seminar
| Person
+
|Andrej Zlatoš
| Institution
+
|UCSD
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
| Hung Tran
 
|-
 
|-
|Date
+
|Friday, September 23, 4:00 p.m., Colloquium
| Person
+
|Pablo Shmerkin
| Institution
+
|University of British Columbia
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|Shaoming and Andreas
 
|-
 
|-
|Date
+
|September 24-25, RTG workshop in Harmonic Analysis
| Person
+
|
| Institution
+
|
|[[#linktoabstract  |  Title ]]
+
|
| Sponsor
+
|Shaoming and Andreas
 
|-
 
|-
|Date
+
|Tuesday, November 8,
| Person
+
|Robert Fraser
| Institution
+
|Wichita State University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
| Shaoming and Andreas
|-
 
 
 
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Name===
+
===Dóminique Kemp===
  
Title
+
Decoupling by way of approximation
  
Abstract
+
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===
  
===Name===
+
Transcendental Julia Sets with Fractional Packing Dimension
  
Title
+
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.
  
Abstract
+
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===
  
===Name===
+
Stability of sharp Fourier restriction to spheres
  
Title
+
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.
  
Abstract
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
 +
===Rajula Srivastava===
  
===Name===
+
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
Title
+
We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.
  
Abstract
+
===Itamar Oliveira===
  
 +
A new approach to the Fourier extension problem for the paraboloid
  
===Name===
+
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.
  
Title
+
===Changkeun Oh===
  
Abstract
+
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
  
=[[Previous_Analysis_seminars]]=
+
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 $K$.
  
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
+
===Kasso Okoudjou===
  
=Abstracts=
+
An exploration in analysis on fractals
===Alexei Poltoratski===
 
  
Title: Dirac inner functions
+
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.
  
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.
+
===Rahul Parhi===
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential
 
operators and the non-linear Fourier transform.
 
  
===Polona Durcik and Joris Roos===
+
On BV Spaces, Splines, and Neural Networks
  
Title: A triangular Hilbert transform with curvature, I & II.
+
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.
  
Abstract: The triangular Hilbert is a two-dimensional bilinear singular
+
In this talk, I will discuss my research which aims to rectify this
originating in time-frequency analysis. No Lp bounds are currently
+
issue by understanding the regularity properties of neural networks in a
known for this operator.
+
similar vein to classical methods based on splines and wavelets. In
In these two talks we discuss a recent joint work with Michael Christ
+
particular, we will show that neural networks are optimal solutions to
on a variant of the triangular Hilbert transform involving curvature.
+
variational problems over BV-type function spaces defined via the Radon
This object is closely related to the bilinear Hilbert transform with
+
transform. These spaces are non-reflexive Banach spaces, generally
curvature and a maximally modulated singular integral of Stein-Wainger
+
distinct from classical spaces studied in analysis. However, in the
type. As an application we also discuss a quantitative nonlinear Roth
+
univariate setting, neural networks reduce to splines and these function
type theorem on patterns in the Euclidean plane.
+
spaces reduce to classical univariate BV spaces. If time permits, I will
The second talk will focus on the proof of a key ingredient, a certain
+
also discuss approximation properties of these spaces, showing that they
regularity estimate for a local operator.
+
are, in some sense, "small" compared to classical multivariate spaces
 +
such as Sobolev or Besov spaces.
  
===Andrew Zimmer===
+
This is joint work with Robert Nowak.
  
Title:  Complex analytic problems on domains with good intrinsic geometry
+
===Alexei Poltoratski===
  
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).
+
Title: Pointwise convergence for the scattering data and non-linear Fourier transform.
  
===Hong Wang===
+
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.
  
Title: Improved decoupling for the parabola
+
===John Green===
  
Abstract: In 2014, Bourgain and Demeter proved the  $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. 
+
Estimates for oscillatory integrals via sublevel set estimates.
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$.  This is joint work with Larry Guth and Dominique Maldague.
 
  
===Kevin Luli===
+
In many situations, oscillatory integral estimates are known to imply sublevel set estimates in a stable manner. Reversing this implication is much more difficult, but understanding when this is true is helpful for understanding scalar oscillatory integral estimates. We shall motivate a line of investigation in which we seek to reverse the implication in the presence of a qualitative structural assumption. After considering some one-dimensional results, we turn to the setting of convex functions in higher dimensions.
  
Title: Smooth Nonnegative Interpolation
+
===Tao Mei===
  
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E  \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets.  
+
Fourier Multipliers on free groups.
  
===Niclas Technau===
+
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: Number theoretic applications of oscillatory integrals
+
===Alex Nagel===
  
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.
+
Global estimates for a class of kernels and multipliers with multiple homogeneities
  
===Terence Harris===
+
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.
  
Title: Low dimensional pinned distance sets via spherical averages
+
===Sebastian Bechtel===
  
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.
+
Square roots of elliptic systems on open sets
  
===Yuval Wigderson===
+
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.
  
Title: New perspectives on the uncertainty principle
+
===Tongou Yang===
  
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.
+
Restricted projections along $C^2$ curves on the sphere
  
===Oscar Dominguez===
+
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.
  
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
+
===Po Lam Yung===
  
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.
+
Revisiting an old argument for Vinogradov's Mean Value Theorem
  
===Tamas Titkos===
+
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.
  
Title: Isometries of Wasserstein spaces
+
===Brian Street===
  
Abstract: Due to its nice theoretical properties and an astonishing number of
+
Maximal Subellipticity
applications via optimal transport problems, probably the most
 
intensively studied metric nowadays is the p-Wasserstein metric. Given
 
a complete and separable metric space $X$ and a real number $p\geq1$,
 
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection
 
of Borel probability measures with finite $p$-th moment, endowed with a
 
distance which is calculated by means of transport plans \cite{5}.
 
  
The main aim of our research project is to reveal the structure of the
+
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.
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although
 
$\mathrm{Isom}(X)$ embeds naturally into
 
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding
 
turned out to be surjective in many cases (see e.g. [1]), these two
 
groups are not isomorphic in general. Kloeckner in [2] described
 
the isometry group of the quadratic Wasserstein space
 
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$
 
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$
 
is extremely rich. Namely, it contains a large subgroup of wild behaving
 
isometries that distort the shape of measures. Following this line of
 
investigation, in \cite{3} we described
 
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and
 
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.
 
  
In this talk I will survey first some of the earlier results in the
+
===Laurent Stolovitch===
subject, and then I will present the key results of [3]. If time
 
permits, I will also report on our most recent manuscript [4] in
 
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)
 
and D\'aniel Virosztek (IST Austria).
 
  
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein
+
Classification of reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional hyperbolic type
spaces: isometric rigidity in negative curvature}, International
 
Mathematics Research Notices, 2016 (5), 1368--1386.
 
  
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean
+
The aim of this joint work with Martin Klimes is twofold:
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di
 
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.
 
  
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of
+
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.
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373
 
(2020), 5855--5883.
 
  
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
+
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 $\mathbb{C}^2$.
Wasserstein spaces: The Hilbertian case}, submitted manuscript.
 
  
[5] C. Villani, \emph{Optimal Transport: Old and New,}
+
===Betsy Stovall===
(Grundlehren der mathematischen Wissenschaften)
 
Springer, 2009.
 
  
===Shukun Wu===
+
On extremizing sequences for adjoint Fourier restriction to the sphere
  
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
+
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.
  
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem.
+
===Malabika Pramanik===
  
 +
https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf
  
===Jonathan Hickman===
+
===Hongki Jung===
  
Title: Sobolev improving for averages over space curves
+
A small cap decoupling for the twisted cubic
  
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity.  Joint with D. Beltran, S. Guo and A. Seeger.
+
===Bernhard Lamel===
  
===Hanlong Fang===
+
Convergence and Divergence of Formal Power Series Maps
  
Title: Canonical blow-ups of Grassmann manifolds
+
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:  We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification.
+
===Carmelo Puliatti===
  
===Bingyang Hu===
+
Gradients of single layer potentials for elliptic operators 
 +
with coefficients of Dini mean oscillation-type
  
Title: Some structure theorems on general doubling measures.
+
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: In this talk, we will first  several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely,  we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.
+
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and 
 +
Xavier Tolsa.
  
===Krystal Taylor===
+
===Larry Guth===
  
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting
+
Series title: Reflections on decoupling and Vinogradov's mean value problem.
  
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections.  
+
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 proofThe 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.
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage pointsIn particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.
 
  
===Dominique Maldague===
 
  
Title: A new proof of decoupling for the parabola
+
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: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.
+
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.
  
===Diogo Oliveira e Silva===
+
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: Global maximizers for spherical restriction
+
=[[Previous_Analysis_seminars]]=
  
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
===Oleg Safronov===
+
=Extras=
 
 
Title: Relations between discrete and continuous spectra of differential operators
 
 
 
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation:  one needs to consider two operators one of which is obtained  from the other
 
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.
 
 
 
===Ziming Shi===
 
 
 
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary
 
 
 
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$  and $s>1/p$.
 
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work  of X. Gong.
 
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate.
 
Joint work with Liding Yao.
 
 
 
===Xiumin Du===
 
 
 
Title: Falconer's distance set problem
 
 
 
Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.
 
  
===Etienne Le Masson===
 
 
Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces
 
 
Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods.
 
Joint work with Tuomas Sahlsten.
 
 
===Theresa Anderson===
 
 
Title: Dyadic analysis (virtually) meets number theory
 
 
Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.
 
 
===Nathan Wagner===
 
 
Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness
 
 
Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.
 
 
===David Beltran===
 
 
Title: Sobolev improving for averages over curves in $\mathbb{R}^4$
 
 
Abstract: Given a smooth non-degenerate space curve (that is, a smooth curve whose n-1 curvature functions are non-vanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$-Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the non-trivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.
 
 
===Yumeng Ou===
 
 
Title: On the multiparameter distance problem
 
 
Abstract: In this talk, we will describe some recent progress on the Falconer distance problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{|x-y|: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent work on this problem, which also includes some new results on the multiparameter radial projection theory of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.
 
 
=Extras=
 
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
  

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