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
+
|  
 
|-
 
|-
|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
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
+
|[[#linktoabstract |   Title ]]
| Simon
+
|  
 
|-
 
|-
|Oct 8
+
|October 29, TBA
| Jeff Galkowski
+
| Alexandru Ionescu (Colloquium)
| Northeastern University
+
| Princeton University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
 
 
|-
 
|-
|Oct 15
+
|November 2, VV B139
| David Beltran
+
| Liding Yao
 
| UW Madison
 
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Brian
+
|  
 
|-
 
|-
|Oct 22
+
|November 9, VV B139
| Laurent Stolovitch
+
| Lingxiao Zhang
| University of Nice Sophia-Antipolis
 
|[[#linktoabstract  |  Title ]]
 
| Xianghong
 
|-
 
|<b>Wednesday Oct 23 in B129</b>
 
|Dominique Kemp
 
|Indiana University
 
|tbd | tbd
 
|Betsy
 
|-
 
|Oct 29
 
| Bingyang Hu
 
 
| UW Madison
 
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
|  
 
|-
 
|-
|Nov 5
+
|November 12, TBA
| Kevin O'Neill
+
| Kasso Okoudjou (Colloquium)
| UC Davis
+
| Tufts University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Betsy
 
 
|-
 
|-
|Nov 12
+
|November 16, VV B139
| Francesco di Plinio
+
| Rahul Parhi
| Washington University in St. Louis
+
| UW Madison (EE)
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Shaoming
+
|  
 
|-
 
|-
|Nov 19
+
|November 30, VV B139
| Person
+
| Alexei Poltoratski
| Institution
+
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
|Nov 26
 
| No Seminar
 
|
 
|
 
 
|  
 
|  
 
|-
 
|-
|Dec 3
+
|December 7
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
|Dec 10
 
| No Seminar
 
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Jan 21
+
|December 14
| No Seminar
+
| reserved
|
 
|
 
|
 
|-
 
|Jan 28
 
| Person
 
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|  
|-
 
|Feb 4
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Feb 11
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Feb 18
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Feb 25
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
 
|-
 
|-
|Mar 3
+
|Date
 
| Person
 
| Person
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
 
|-
 
|Mar 10
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Mar 17
 
| Spring Break!
 
|
 
|
 
 
|  
 
|  
 
|-
 
|-
|Mar 24
+
 
| Oscar Dominguez
 
| Universidad Complutense de Madrid
 
|[[#linktoabstract  |  Title ]]
 
| Andreas
 
|-
 
|Mar 31
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Apr 7
 
| Reserved
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Street
 
|-
 
|Apr 14
 
| Person
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Sponsor
 
|-
 
|Apr 21
 
| Person
 
| Institution
 
|[[#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.
 
  
===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.
 
  
===Xiaojun Huang===
+
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.
  
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
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.
+
===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===
  
===Xiaocheng Li===
+
A new approach to the Fourier extension problem for the paraboloid
  
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
+
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.
  
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
  
===Name===
+
=Extras=
  
Title
+
[[Blank Analysis Seminar Template]]
  
Abstract
 
  
 +
Graduate Student Seminar:
  
===Name===
+
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html
 
 
Title
 
 
 
Abstract
 
 
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 

Latest revision as of 11:15, 17 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 Title
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 reserved 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.

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