Fall 2021 and Spring 2022 Analysis Seminars

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

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	Person	Institution	Title	Sponsor
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

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