Difference between revisions of "Analysis Seminar"

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'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
  
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
+
The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
 +
Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
  
If you wish to invite a speaker please contact  Brian at street(at)math
+
Zoom links will be sent to those who have signed up for the Analysis Seminar List.  If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. If you are from an institution different than UW-Madison, please send as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
  
===[[Previous Analysis seminars]]===
+
If you'd like to suggest speakers for the fall semester please contact David and Andreas.
  
 
= Analysis Seminar Schedule =
 
= Analysis Seminar Schedule =
Line 16: Line 15:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 10
+
|September 21, VV B139
| José Madrid
+
| Dóminique Kemp
| UCLA
+
| UW-Madison
|[[#José Madrid On the regularity of maximal operators on Sobolev Spaces ]]
+
|[[#Dóminique Kemp Decoupling by way of approximation ]]
| Andreas, David
+
|  
 
|-
 
|-
|Sept 13 (Friday, B139)
+
|September 28, VV B139
| Yakun Xi
+
| Jack Burkart
| University of  Rochester
+
| UW-Madison
|[[#Yakun Xi Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
+
|[[#Jack Burkart Transcendental Julia Sets with Fractional Packing Dimension ]]
| Shaoming
+
|  
 
|-
 
|-
|Sept 17
+
|October 5, Online
| Joris Roos
+
| 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
|[[#Joris Roos L^p improving estimates for maximal spherical averages ]]
+
|[[#Changkeun Oh Decoupling inequalities for quadratic forms and beyond ]]
| Brian
+
|  
 
|-
 
|-
|Sept 20 (2:25 PM Friday, Room B139 VV)
+
|October 29, TBA
| Xiaojun Huang
+
| Alexandru Ionescu (Colloquium)
| Rutgers University–New Brunswick
+
| Princeton University
|[[#linktoabstract  |  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
+
|[[#linktoabstract  |  Title ]]
| Xianghong
 
 
|-
 
|-
|Oct 1
+
|November 2, VV B139
| Xiaocheng Li
+
| Liding Yao
 
| UW Madison
 
| UW Madison
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
+
|[[#Liding Yao |   An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains ]]
| Simon
+
|  
|-
 
|Oct 8
 
| Jeff Galkowski
 
| Northeastern University
 
|[[#Jeff Galkowski  |  Concentration and Growth of Laplace Eigenfunctions ]]
 
| Betsy
 
 
|-
 
|-
|Oct 15
+
|November 9, VV B139
| David Beltran
+
| Lingxiao Zhang
 
| UW Madison
 
| UW Madison
|[[#David Beltran Regularity of the centered fractional maximal function ]]
+
|[[#linktoabstract Title ]]
| Brian
+
|  
 
|-
 
|-
|Oct 22
+
|November 12, TBA
| Laurent Stolovitch
+
| Kasso Okoudjou (Colloquium)
| University of Côte d'Azur
+
| Tufts University
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
+
|[[#linktoabstract |   Title ]]
| Xianghong
 
 
|-
 
|-
|<b>Wednesday Oct 23 in B129</b>
+
|November 16, VV B139
|Dominique Kemp
+
| Rahul Parhi
|Indiana University
+
| UW Madison (EE)
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
+
|[[#linktoabstract  |   Title ]]
|Betsy
+
|  
 
|-
 
|-
|Oct 29
+
|November 30, VV B139
| Bingyang Hu
+
| Alexei Poltoratski
 
| UW Madison
 
| UW Madison
|[[#Bingyang Hu |   Sparse bounds of singular Radon transforms]]
+
|[[#linktoabstract |   Title ]]
| Street
+
|  
 
|-
 
|-
|Nov 5
+
|December 7
| Kevin O'Neill
+
| Person
| UC Davis
+
| Institution
|[[#Kevin O'Neill A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
+
|[[#linktoabstract Title ]]
| Betsy
+
|  
 
|-
 
|-
|Nov 12
+
|December 14
| Francesco di Plinio
+
| Tao Mei
| Washington University in St. Louis
+
| Baylor University
|[[#Francesco di Plinio Maximal directional integrals along algebraic and lacunary sets]]
+
|[[#linktoabstract Title ]]
| Shaoming
+
|  
 
|-
 
|-
|Nov 13 (Wednesday)
+
|February 1
| Xiaochun Li
+
| Person
| UIUC
+
| Institution
|[[#Xiaochun Li Roth's type theorems on progressions]]
+
|[[#linktoabstract Title ]]
| Brian, Shaoming
+
|  
 
|-
 
|-
|Nov 19
+
|February 8
| Joao Ramos
+
| Person
| University of Bonn
+
| Institution
|[[#Joao Ramos Fourier uncertainty principles, interpolation and uniqueness sets ]]
+
|[[#linktoabstract Title ]]
| Joris, Shaoming
+
|  
 
|-
 
|-
|Jan 21
+
|February 15
| No Seminar
+
| Person
 +
| Institution
 +
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Friday, Jan 31, 4 pm, B239, Colloquium
+
|February 22
| Lillian Pierce
+
| Person
| Duke University
+
| Institution
|[[#Lillian Pierce On Bourgain’s counterexample for the Schrödinger maximal function ]]
+
|[[#linktoabstract Title ]]
| Andreas, Simon
+
|  
 
|-
 
|-
|Feb 4
+
|March 1
| Ruixiang Zhang
+
| Person
| UW Madison
+
| Institution
|[[#Ruixiang Zhang  |  Local smoothing for the wave equation in 2+1 dimensions ]]
 
| Andreas
 
|-
 
|Feb 11
 
| Zane Li
 
| Indiana University
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
|  
 
|-
 
|-
|Feb 18
+
|March 8
| Sergey Denisov
+
| Brian Street
 
| UW Madison
 
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|Feb 25
+
|March 15: No Seminar
| Michel Alexis
+
| Person
| Local
+
| Institution
|[[#Michel Alexis The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
+
|[[#linktoabstract Title ]]
| Denisov
+
|  
 
|-
 
|-
|Mar 3
+
|March 23
| William Green
+
| Person
| Rose-Hulman Institute of Technology
+
| Institution
|[[#William Green Dispersive estimates for the Dirac equation ]]
+
|[[#linktoabstract Title ]]
| Betsy
+
|  
 
|-
 
|-
|Mar 10
+
|March 30
| Yifei Pan
+
| Person
| Indiana University-Purdue University Fort Wayne
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
 
|-
 
|Mar 17
 
| Spring Break!
 
|
 
|
 
 
|  
 
|  
 
|-
 
|-
|Mar 24
+
|April 5
| Oscar Dominguez
+
| Person
| Universidad Complutense de Madrid
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Andreas
+
|  
 
|-
 
|-
|Mar 31
+
|April 12
| Brian Street
+
| Person
| University of Wisconsin-Madison
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Local
+
|  
 
|-
 
|-
|Apr 7
+
|April 19
| Hong Wang
+
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|<b>Monday, Apr 13</b>
+
|April 22, Colloquium
|Yumeng Ou
+
|Detlef Müller
|CUNY, Baruch College
+
|University of Kiel
|[[#linktoabstract  |  TBA ]]
+
|[[#linktoabstract  |  Title ]]
|Zhang
+
|  
 
|-
 
|-
|Apr 14
+
|April 29
| Tamás Titkos
+
| Person
| BBS University of Applied Sciences & Rényi Institute
+
| Institution
|[[#linktoabstract  |  Distance preserving maps on spaces of probability measures ]]
+
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|Apr 21
+
|May 3
| Diogo Oliveira e Silva
+
| Person
| University of Birmingham
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
|  
 
|-
 
|-
|Apr 28
+
|Date
| No Seminar
+
| Person
|-
+
| Institution
|May 5
 
|Jonathan Hickman
 
|University of Edinburgh
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Andreas
+
|  
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===José Madrid===
+
===Dóminique Kemp===
  
Title: On the regularity of maximal operators on Sobolev Spaces
+
Decoupling by way of approximation
  
Abstract:  In this talk, we will discuss the regularity properties (boundedness and
+
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 $M$.
continuity) of the classical and fractional maximal
 
operators when these act on the Sobolev space W^{1,p}(\R^n). We will
 
focus on the endpoint case p=1. We will talk about
 
some recent results and current open problems.
 
  
===Yakun Xi===
+
===Jack Burkart===
  
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
+
Transcendental Julia Sets with Fractional Packing Dimension
  
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.
+
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.
  
===Joris Roos===
+
In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.
  
Title: L^p improving estimates for maximal spherical averages
+
===Giuseppe Negro===
  
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
+
Stability of sharp Fourier restriction to spheres
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.
 
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.
 
  
 +
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.
  
===Joao Ramos===
+
===Rajula Srivastava===
  
Title: Fourier uncertainty principles, interpolation and uniqueness sets
+
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that
+
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.
  
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
+
===Itamar Oliveira===
  
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.
+
A new approach to the Fourier extension problem for the paraboloid
  
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$
+
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.
  
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.
+
===Changkeun Oh===
  
===Xiaojun Huang===
+
Decoupling inequalities for quadratic forms and beyond
  
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
+
In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.
  
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.
+
===Liding Yao===
  
===Xiaocheng Li===
+
An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
  
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
+
Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.
  
Abstract:  We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.
+
=[[Previous_Analysis_seminars]]=
  
 +
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
===Xiaochun Li===
+
=Extras=
  
Title:  Roth’s type theorems on progressions
+
[[Blank Analysis Seminar Template]]
  
Abstract:  The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.
 
  
===Jeff Galkowski===
+
Graduate Student Seminar:
  
<b>Concentration and Growth of Laplace Eigenfunctions</b>
+
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html
 
 
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.
 
 
 
===David Beltran===
 
 
 
Title: Regularity of the centered fractional maximal function
 
 
 
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.
 
 
 
This is joint work with José Madrid.
 
 
 
===Dominique Kemp===
 
 
 
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
 
 
 
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone.  Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.
 
 
 
 
 
===Kevin O'Neill===
 
 
 
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
 
 
 
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.
 
 
 
===Francesco di Plinio===
 
 
 
<b>Maximal directional integrals along algebraic and lacunary sets </b>
 
 
 
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas  of Parcet-Rogers and of  Nagel-Stein-Wainger.
 
 
 
===Laurent Stolovitch===
 
 
 
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
 
 
 
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.
 
 
 
===Bingyang Hu===
 
 
 
<b>Sparse bounds of singular Radon transforms</b>
 
 
 
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.
 
 
 
===Lillian Pierce===
 
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
 
 
 
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices.
 
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”
 
 
 
===Ruixiang Zhang===
 
 
 
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
 
 
 
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate  gains a fractional  derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.
 
 
 
 
 
===William Green===
 
 
 
<b> Dispersive estimates for the Dirac equation </b>
 
 
 
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds.  Dirac formulated a hyberbolic system of partial differential equations
 
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.
 
 
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations.  We will survey recent work on time-decay estimates for the solution operator.  Specifically the mapping properties of the solution operator between L^p spaces.  As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution.  We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay.  The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).
 
 
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 

Latest revision as of 15:56, 24 October 2021

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 fall 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, TBA Alexandru Ionescu (Colloquium) Princeton University Title
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 Title
November 12, TBA Kasso Okoudjou (Colloquium) Tufts University Title
November 16, VV B139 Rahul Parhi UW Madison (EE) Title
November 30, VV B139 Alexei Poltoratski UW Madison Title
December 7 Person Institution Title
December 14 Tao Mei Baylor University Title
February 1 Person Institution Title
February 8 Person Institution Title
February 15 Person Institution Title
February 22 Person Institution Title
March 1 Person Institution Title
March 8 Brian Street UW Madison Title
March 15: No Seminar Person Institution Title
March 23 Person Institution Title
March 30 Person Institution Title
April 5 Person Institution Title
April 12 Person Institution Title
April 19 Person Institution Title
April 22, Colloquium Detlef Müller University of Kiel Title
April 29 Person Institution Title
May 3 Person Institution Title
Date Person Institution Title

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.

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.

Previous_Analysis_seminars

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

Extras

Blank Analysis Seminar Template


Graduate Student Seminar:

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