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)
+
|September 28, VV B139
| Yakun Xi
+
| Jack Burkart
| University of  Rochester
+
| UW-Madison
|[[#linktoabstract Title ]]
+
|[[#Jack Burkart Transcendental Julia Sets with Fractional Packing Dimension ]]
| Shaoming
+
|  
 
|-
 
|-
|Sept 17
+
|October 5, Online
| Joris Roos
+
| Giuseppe Negro
| UW Madison
+
| University of Birmingham
|[[#linktoabstract Title ]]
+
|[[#Giuseppe Negro Stability of sharp Fourier restriction to spheres ]]
| Brian
+
|  
 
|-
 
|-
|Sept 20 (2:25 PM Friday)
+
|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
+
|  
 
|-
 
|-
|Sept 24
+
|October 19, Online
| Person
+
|Itamar Oliveira
| Institution
+
|Cornell University
|[[#linktoabstract Title ]]
+
|[[#Itamar Oliveira A new approach to the Fourier extension problem for the paraboloid ]]
| Sponsor
+
|  
 
|-
 
|-
|Oct 1
+
|October 26, VV B139
| Xiaocheng Li
+
| Changkeun Oh
 
| UW Madison
 
| UW Madison
 +
|[[#Changkeun Oh  |  Decoupling inequalities for quadratic forms and beyond ]]
 +
|
 +
|-
 +
|October 29, TBA
 +
| Alexandru Ionescu (Colloquium)
 +
| Princeton University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Simon
 
 
|-
 
|-
|Oct 8
+
|November 2, VV B139
| Jeff Galkowski
+
| Liding Yao
| Northeastern University
+
| UW Madison
|[[#linktoabstract Title ]]
+
|[[#Liding Yao An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains ]]
| Betsy
+
|  
 
|-
 
|-
|Oct 15
+
|November 9, VV B139
| David Beltran
+
| Lingxiao Zhang
 
| UW Madison
 
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Brian
+
|  
 
|-
 
|-
|Oct 22
+
|November 12, TBA
| Laurent Stolovitch
+
| Kasso Okoudjou (Colloquium)
| University of Nice Sophia-Antipolis
+
| Tufts University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
 
 
|-
 
|-
|<b>Wednesday Oct 23 in B129</b>
+
|November 16, VV B139
|Dominique Kemp
+
| Rahul Parhi
|Indiana University
+
| UW Madison (EE)
|tbd | tbd
+
|[[#linktoabstract  |   Title ]]
|Betsy
+
|  
 
|-
 
|-
|Oct 29
+
|November 30, VV B139
| Bingyang Hu
+
| Alexei Poltoratski
 
| UW Madison
 
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|Nov 5
+
|December 7
| Kevin O'Neill
+
| Person
| UC Davis
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
+
|  
 
|-
 
|-
|Nov 12
+
|December 14
| Francesco di Plinio
+
| Tao Mei
| Washington University in St. Louis
+
| Baylor University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Shaoming
+
|  
 
|-
 
|-
|Nov 19
+
|February 1
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Nov 26
+
|February 8
| No Seminar
+
| Person
|  
+
| Institution
|
+
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
 
|-
 
|-
|Dec 3
+
|February 15
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
|Dec 10
 
| No Seminar
 
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Jan 21
+
|February 22
| No Seminar
+
| Person
 +
| Institution
 +
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Jan 28
+
|March 1
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 +
|-
 +
|March 8
 +
| Brian Street
 +
| UW Madison
 +
|[[#linktoabstract  |  Title ]]
 +
|
 
|-
 
|-
|Feb 4
+
|March 15: No Seminar
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Feb 11
+
|March 23
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Feb 18
+
|March 30
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Feb 25
+
|April 5
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Mar 3
+
|April 12
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Mar 10
+
|April 19
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
|Mar 17
 
| Spring Break!
 
|
 
|
 
 
|  
 
|  
 
|-
 
|-
|Mar 24
+
|April 22, Colloquium
| Oscar Dominguez
+
|Detlef Müller
| Universidad Complutense de Madrid
+
|University of Kiel
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Andreas
+
|  
 
|-
 
|-
|Mar 31
+
|April 29
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
|-
 
|Apr 7
 
| Reserved
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Street
 
 
|-
 
|-
|Apr 14
+
|May 3
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
 
|-
 
|-
|Apr 21
+
|Date
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
|-
 
|Apr 28
 
| No Seminar
 
|
 
|
 
|
 
 
|-
 
|-
 
|}
 
|}
  
 
=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.
 
  
===Xiaojun Huang===
+
===Jack Burkart===
  
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
+
Transcendental Julia Sets with Fractional Packing Dimension
  
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.
+
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.
  
===Name===
+
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
+
===Giuseppe Negro===
  
Abstract
+
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.
  
===Name===
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
Title
+
===Rajula Srivastava===
  
Abstract
+
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.
  
===Name===
+
===Itamar Oliveira===
  
Title
+
A new approach to the Fourier extension problem for the paraboloid
  
Abstract
+
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===
  
===Name===
+
Decoupling inequalities for quadratic forms and beyond
  
Title
+
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
+
===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=
 
=Extras=
 +
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
 +
 +
 +
Graduate Student Seminar:
 +
 +
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html

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