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
| UW Madison
+
| University of Birmingham
|[[#Joris Roos L^p improving estimates for maximal spherical averages ]]
+
|[[#Giuseppe Negro Stability of sharp Fourier restriction to spheres ]]
| Brian
+
|  
 
|-
 
|-
|Sept 20 (2:25 PM Friday, Room B139 VV)
+
|October 12, VV B139
| Xiaojun Huang
+
|Rajula Srivastava
| Rutgers University–New Brunswick
+
|UW Madison
|[[#linktoabstract A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
+
|[[#Rajula Srivastava Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]]
| Xianghong
+
|  
 
|-
 
|-
|Oct 1
+
|October 19, Online
| Xiaocheng Li
+
|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
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
+
|[[#Changkeun Oh |   Decoupling inequalities for quadratic forms and beyond ]]
| Simon
+
|  
 
|-
 
|-
|Oct 8
+
|October 29, TBA
| Jeff Galkowski
+
| Alexandru Ionescu (Colloquium)
| Northeastern University
+
| Princeton University
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
+
|[[#linktoabstract Title ]]
| Betsy
 
 
|-
 
|-
|Oct 15
+
|November 2, VV B139
| David Beltran
+
| Liding Yao
 
| UW Madison
 
| UW Madison
|[[#David Beltran Regularity of the centered fractional maximal function ]]
+
|[[#linktoabstract Title ]]
| Brian
+
|  
|-
 
|Oct 22
 
| Laurent Stolovitch
 
| University of Côte d'Azur
 
|[[#Laurent Stolovitch  | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
 
| Xianghong
 
|-
 
|<b>Wednesday Oct 23 in B129</b>
 
|Dominique Kemp
 
|Indiana University
 
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
 
|Betsy
 
 
|-
 
|-
|Oct 29
+
|November 9, VV B139
| Bingyang Hu
+
| Lingxiao Zhang
 
| UW Madison
 
| UW Madison
|[[#Bingyang Hu |   Sparse bounds of singular Radon transforms]]
+
|[[#linktoabstract |   Title ]]
| Street
+
|  
 
|-
 
|-
|Nov 5
+
|November 12, TBA
| Kevin O'Neill
+
| Kasso Okoudjou (Colloquium)
| UC Davis
+
| Tufts University
|[[#Kevin O'Neill A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
+
|[[#linktoabstract Title ]]
| Betsy
 
 
|-
 
|-
|Nov 12
+
|November 16, VV B139
| Francesco di Plinio
+
| Rahul Parhi
| Washington University in St. Louis
+
| UW Madison (EE)
|[[#Francesco di Plinio Maximal directional integrals along algebraic and lacunary sets]]
+
|[[#linktoabstract Title ]]
| Shaoming
+
|  
 
|-
 
|-
|Nov 13 (Wednesday)
+
|November 30, VV B139
| Xiaochun Li
+
| Alexei Poltoratski
| UIUC
+
| UW Madison
|[[#Xiaochun Li Roth's type theorems on progressions]]
+
|[[#linktoabstract Title ]]
| Brian, Shaoming
+
|  
 
|-
 
|-
|Nov 19
+
|December 7
| Joao Ramos
+
| Person
| University of Bonn
+
| Institution
|[[#Joao Ramos Fourier uncertainty principles, interpolation and uniqueness sets ]]
+
|[[#linktoabstract Title ]]
| Joris, Shaoming
+
|  
 
|-
 
|-
|Nov 26
+
|December 14
| No Seminar
+
| Tao Mei
|  
+
| Baylor University
|
+
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
 
|-
 
|-
|Dec 3
+
|February 1
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
|Dec 10
 
| No Seminar
 
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Jan 21
+
|February 8
| No Seminar
+
| Person
 +
| Institution
 +
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Jan 28
+
|February 15
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Feb 4
+
|February 22
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Feb 11
+
|March 1
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Feb 18
+
|March 8
| Sergey Denisov
+
| Brian Street
 
| UW Madison
 
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|Feb 25
+
|March 15: No Seminar
| Dmitry Chelkak
+
| Person
| Ecole Normale, Paris
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Denisov
+
|  
 
|-
 
|-
|Mar 3
+
|March 23
| William Green
+
| Person
| Rose-Hulman Institute of Technology
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#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
| Reserved
+
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|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.
+
=[[Previous_Analysis_seminars]]=
  
===Xiaocheng Li===
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
+
=Extras=
  
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.
+
[[Blank Analysis Seminar Template]]
  
  
===Xiaochun Li===
+
Graduate Student Seminar:
  
Title: Roth’s type theorems on progressions
+
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html
 
 
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===
 
 
 
<b>Concentration and Growth of Laplace Eigenfunctions</b>
 
 
 
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.
 
 
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 

Latest revision as of 07:59, 20 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 Title
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.

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Graduate Student Seminar:

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