Difference between revisions of "Analysis Seminar"
(→Analysis Seminar Schedule) |
(→Abstracts) |
||
Line 249: | Line 249: | ||
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. | 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. | ||
+ | |||
===Dominique Kemp=== | ===Dominique Kemp=== | ||
− | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature | + | ==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. | 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. |
Revision as of 08:59, 26 September 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 | Title | Brian |
Oct 22 | Laurent Stolovitch | University of Nice Sophia-Antipolis | Title | 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 | Title | Betsy |
Nov 12 | Francesco di Plinio | Washington University in St. Louis | Title | Shaoming |
Nov 19 | Person | Institution | Title | Sponsor |
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 |
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.
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.
Name
Title
Abstract