Difference between revisions of "Analysis Seminar"

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'''Analysis Seminar
+
'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
'''
  
Line 16: Line 16:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 11
+
|Sept 10
| Simon Marshall
+
| José Madrid
| UW Madison
+
| UCLA
|[[#Simon Marshall | Integrals of eigenfunctions on hyperbolic manifolds ]]
+
|[[#José Madrid |   On the regularity of maximal operators on Sobolev Spaces ]]
|  
+
| Andreas, David
 
|-
 
|-
|'''Wednesday, Sept 12'''
+
|Sept 13 (Friday, B139)
| Gunther Uhlmann 
+
| Yakun Xi
| University of Washington
+
| University of Rochester
| Distinguished Lecture Series
+
|[[#Yakun Xi  |  Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
| See colloquium website for location
+
| Shaoming
 
|-
 
|-
|'''Friday, Sept 14'''
+
|Sept 17
| Gunther Uhlmann 
+
| Joris Roos
| University of Washington
+
| UW Madison
| Distinguished Lecture Series
+
|[[#Joris Roos  |  L^p improving estimates for maximal spherical averages ]]
| See colloquium website for location
+
| Brian
 
|-
 
|-
|Sept 18
+
|Sept 20 (2:25 PM Friday, Room B139 VV)
| Grad Student Seminar
+
| Xiaojun Huang
|  
+
| Rutgers University–New Brunswick
|
+
|[[#linktoabstract  |  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
|
+
| Xianghong
 
|-
 
|-
|Sept 25
+
|Sept 24
| Grad Student Seminar
+
| Person
|
+
| Institution
|
+
|[[#linktoabstract  |  Title ]]
|
+
| Sponsor
 
|-
 
|-
|Oct 9
+
|Oct 1
| Hong Wang
+
| Xiaocheng Li
| MIT
+
| UW Madison
|[[#Hong Wang |   About Falconer distance problem in the plane ]]
+
|[[#Xiaocheng Li An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
| Ruixiang
+
| Simon
|-
 
|Oct 16
 
| Polona Durcik
 
| Caltech
 
|[[#Polona Durcik |  Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
 
| Joris
 
 
|-
 
|-
|Oct 23
+
|Oct 8
| Song-Ying Li
+
| Jeff Galkowski
| UC Irvine
+
| Northeastern University
|[[#Song-Ying Li Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
+
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
| Xianghong
+
| Betsy
|-
 
|Oct 30
 
|Grad student seminar
 
|
 
|
 
|
 
 
|-
 
|-
|Nov 6
+
|Oct 15
| Hanlong Fang
+
| David Beltran
 
| UW Madison
 
| UW Madison
|[[#HanlongFang A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
+
|[[#David Beltran Regularity of the centered fractional maximal function ]]
 
| Brian
 
| Brian
 
|-
 
|-
||'''Monday, Nov. 12, B139'''
+
|Oct 22
| Kyle Hambrook
 
| San Jose State University
 
|[[#Kyle Hambrook  |  Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
 
| Andreas
 
|-
 
|Nov 13
 
 
| Laurent Stolovitch
 
| Laurent Stolovitch
| Université de Nice - Sophia Antipolis
+
| University of Côte d'Azur
|[[#Laurent Stolovitch  |   Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
+
|[[#Laurent Stolovitch  | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
|Xianghong
+
| Xianghong
 
|-
 
|-
|Nov 20
+
|<b>Wednesday Oct 23 in B129</b>
| Grad Student Seminar
+
|Dominique Kemp
|  
+
|Indiana University
|[[#linktoabstract  |   ]]
+
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
|  
+
|Betsy
 
|-
 
|-
|Nov 27
+
|Oct 29
| No Seminar
+
| Bingyang Hu
|  
+
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|  
+
| Street
 
|-
 
|-
|Dec 4
+
|Nov 5
| No Seminar
+
| Kevin O'Neill
|  
+
| UC Davis
|[[#linktoabstract Title ]]
+
|[[#Kevin O'Neill A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
|  
+
| Betsy
 
|-
 
|-
|Jan 22
+
|Nov 12
| Brian Cook
+
| Francesco di Plinio
| Kent
+
| Washington University in St. Louis
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Street
+
| Shaoming
 
|-
 
|-
|Jan 29
+
|Nov 19
| Trevor Leslie
+
| Joao Ramos
| UW Madison
+
| University of Bonn
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|  
+
| Joris, Shaoming
 
|-
 
|-
|Feb 5
+
|Nov 26
| No seminar
+
| No Seminar
 
|  
 
|  
 
|
 
|
|
+
|  
 
|-
 
|-
|'''Friday, Feb 8'''
+
|Dec 3
| Aaron Naber
+
| Person
| Northwestern University
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| See colloquium website for location
+
| Sponsor
 
|-
 
|-
|Feb 12
+
|Dec 10
| Shaoming Guo
+
| No Seminar
| UW Madison
 
|[[#linktoabstract  |  Title ]]
 
 
|  
 
|  
 +
|
 +
|
 
|-
 
|-
|'''Friday, Feb 15'''
+
|Jan 21
| Charles Smart
+
| No Seminar
| University of Chicago
+
|  
|[[#linktoabstract  |  Title ]]
 
| See colloquium website for information
 
|-
 
|Feb 19
 
| No seminar
 
|
 
 
|
 
|
 
|
 
|
 
|-
 
|-
|Feb 26
+
|Jan 28
 
| Person
 
| Person
 
| Institution
 
| Institution
Line 154: Line 130:
 
| Sponsor
 
| Sponsor
 
|-
 
|-
|Mar 5
+
|Feb 4
 
| Person
 
| Person
 
| Institution
 
| Institution
Line 160: Line 136:
 
| Sponsor
 
| Sponsor
 
|-
 
|-
|Mar 12
+
|Feb 11
| No Seminar
+
| Person
|
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
| Sponsor
|-
 
|Mar 19
 
|Spring Break!!!
 
|
 
|
 
|
 
 
|-
 
|-
|Mar 26
+
|Feb 18
 
| Person
 
| Person
 
| Institution
 
| Institution
Line 178: Line 148:
 
| Sponsor
 
| Sponsor
 
|-
 
|-
|Apr 2
+
|Feb 25
| Stefan Steinerberger
+
| Person
| Yale
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Shaoming, Andreas
+
| Sponsor
 
|-
 
|-
 
+
|Mar 3
|Apr 9
+
| Person
| Franc Forstnerič
+
| Institution
| Unversity of Ljubljana
 
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong, Andreas
+
| Sponsor
 
|-
 
|-
|Apr 16
+
|Mar 10
 
| Person
 
| Person
 
| Institution
 
| Institution
Line 197: Line 166:
 
| Sponsor
 
| Sponsor
 
|-
 
|-
|Apr 23
+
|Mar 17
| Person
+
| Spring Break!
 +
|
 +
|
 +
|
 +
|-
 +
|Mar 24
 +
| Oscar Dominguez
 +
| Universidad Complutense de Madrid
 +
|[[#linktoabstract  |  Title ]]
 +
| Andreas
 +
|-
 +
|Mar 31
 +
| Reserved
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
| Street
 
|-
 
|-
|Apr 30
+
|Apr 7
| Person
+
| Hong Wang
 
| Institution
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
| Street
 +
|-
 +
|Apr 14
 +
| Tamás Titkos
 +
| BBS University of Applied Sciences & Rényi Institute
 +
|[[#linktoabstract  |  Distance preserving maps on spaces of probability measures ]]
 +
| Street
 +
|-
 +
|Apr 21
 +
| Diogo Oliveira e Silva
 +
| University of Birmingham
 +
|[[#linktoabstract  |  Title ]]
 +
| Betsy
 +
|-
 +
|Apr 28
 +
| No Seminar
 +
|-
 +
|May 5
 +
|Jonathan Hickman
 +
|University of Edinburgh
 +
|[[#linktoabstract  |  Title ]]
 +
| Andreas
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Simon Marshall===
+
===José Madrid===
 +
 
 +
Title: On the regularity of maximal operators on Sobolev Spaces
 +
 
 +
Abstract:  In this talk, we will discuss the regularity properties (boundedness and
 +
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===
 +
 
 +
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
 +
 
 +
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.
 +
 
 +
===Joris Roos===
  
''Integrals of eigenfunctions on hyperbolic manifolds''
+
Title: L^p improving estimates for maximal spherical averages
  
Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity. I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
+
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
 +
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===
  
===Hong Wang===
+
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
  
''About Falconer distance problem in the plane''
+
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 E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.
+
===Xiaocheng Li===
  
===Polona Durcik===
+
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
  
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
+
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.
  
Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
+
===Jeff Galkowski===
  
 +
<b>Concentration and Growth of Laplace Eigenfunctions</b>
  
===Song-Ying Li===
+
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.
  
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
+
===David Beltran===
  
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
+
Title: Regularity of the centered fractional maximal function
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
 
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
 
Kohn Laplacian on strictly pseudoconvex hypersurfaces.
 
  
 +
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===
  
===Hanlong Fan===
+
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
  
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
+
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.
  
An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.
 
  
===Kyle Hambrook===
+
===Kevin O'Neill===
  
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
+
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
  
I will discuss my recent work on some problems concerning
+
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.
Fourier decay and Fourier restriction for fractal measures on curves.
 
  
 
===Laurent Stolovitch===
 
===Laurent Stolovitch===
  
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
+
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
  
We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$  are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.
+
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.
  
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]

Latest revision as of 08:10, 18 October 2019

Fall 2019 and Spring 2020 Analysis Seminar Series

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

If you wish to invite a speaker please contact Brian at street(at)math

Previous Analysis seminars

Analysis Seminar Schedule

date speaker institution title host(s)
Sept 10 José Madrid UCLA On the regularity of maximal operators on Sobolev Spaces Andreas, David
Sept 13 (Friday, B139) Yakun Xi University of Rochester Distance sets on Riemannian surfaces and microlocal decoupling inequalities Shaoming
Sept 17 Joris Roos UW Madison L^p improving estimates for maximal spherical averages Brian
Sept 20 (2:25 PM Friday, Room B139 VV) Xiaojun Huang Rutgers University–New Brunswick A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries Xianghong
Sept 24 Person Institution Title Sponsor
Oct 1 Xiaocheng Li UW Madison An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ Simon
Oct 8 Jeff Galkowski Northeastern University Concentration and Growth of Laplace Eigenfunctions Betsy
Oct 15 David Beltran UW Madison Regularity of the centered fractional maximal function Brian
Oct 22 Laurent Stolovitch University of Côte d'Azur Linearization of neighborhoods of embeddings of complex compact manifolds Xianghong
Wednesday Oct 23 in B129 Dominique Kemp Indiana University Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature Betsy
Oct 29 Bingyang Hu UW Madison Title Street
Nov 5 Kevin O'Neill UC Davis A Quantitative Stability Theorem for Convolution on the Heisenberg Group Betsy
Nov 12 Francesco di Plinio Washington University in St. Louis Title Shaoming
Nov 19 Joao Ramos University of Bonn Title Joris, Shaoming
Nov 26 No Seminar
Dec 3 Person Institution Title Sponsor
Dec 10 No Seminar
Jan 21 No Seminar
Jan 28 Person Institution Title Sponsor
Feb 4 Person Institution Title Sponsor
Feb 11 Person Institution Title Sponsor
Feb 18 Person Institution Title Sponsor
Feb 25 Person Institution Title Sponsor
Mar 3 Person Institution Title Sponsor
Mar 10 Person Institution Title Sponsor
Mar 17 Spring Break!
Mar 24 Oscar Dominguez Universidad Complutense de Madrid Title Andreas
Mar 31 Reserved Institution Title Street
Apr 7 Hong Wang Institution Title Street
Apr 14 Tamás Titkos BBS University of Applied Sciences & Rényi Institute Distance preserving maps on spaces of probability measures Street
Apr 21 Diogo Oliveira e Silva University of Birmingham Title Betsy
Apr 28 No Seminar
May 5 Jonathan Hickman University of Edinburgh Title Andreas

Abstracts

José Madrid

Title: On the regularity of maximal operators on Sobolev Spaces

Abstract: In this talk, we will discuss the regularity properties (boundedness and 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

Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities

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.

Joris Roos

Title: L^p improving estimates for maximal spherical averages

Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$. 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

Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries

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.

Xiaocheng Li

Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$

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.

Jeff Galkowski

Concentration and Growth of Laplace Eigenfunctions

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

Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature

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

A Quantitative Stability Theorem for Convolution on the Heisenberg Group

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.

Laurent Stolovitch

Linearization of neighborhoods of embeddings of complex compact manifolds

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.

Extras

Blank Analysis Seminar Template