2002/2003 Analysis Seminar

During the Winter term of 2003, Analysis Seminar web page was maintained by Vojkan Jaksic; the list of the talks is located here

Fall 2002

August 14, Burnside 920, 2pm

R. Brooks (Technion)

Random Constructions of Riemann Surfaces I

August 19, Burnside 920, 2pm

R. Brooks (Technion)

Random Constructions of Riemann Surfaces II.

Friday, August 30, Burnside 920, 11am

R. Brooks (Technion)

Graphs and Isospectrality.

Friday, September 6, Burnside 920, 2:30-3:30pm

J. Derezinski (Warsaw)

Simple models of the infrared problem

Abstract: I will describe a class of simple but non-trivial models of quantum systems that exhibit the so-called infrared problem. I will describe a number of open problems that are, I believe, physically interesting as well as mathematically elegant and challenging. The talk should be accessible to anybody with just a basic knowledege of quantum theory.

Thursady, September 19, Burnside 934, 2:30-3:30pm

E. Liflyand (Bar Ilan)

Asymptotic behavior of the Fourier transform.

Abstract: In certain problems of analysis asymptotic behavior of the Fourier transform is needed rather than rough estimates from above of the decay of it at infinity. Some results of this sort are known but there are open problems in analysis the solution of which strongly depends on improvement of such results, namely, either wider classes of functions in one and several dimensions to be involved, or the boundary of domains in which functions of several variables are supported to be of lower smoothness. We study both possibilities. We first consider the Fourier transform of functions as close to those of merely of bounded variation as possible. As for geometric conditions on domains we try to pose minimal assumptions in addition to convexity. Each new result of such type being obtained and applied to corresponding problems in analysis will lead to generalization of those.

Friday, September 27, Burnside 920, 1:30-2:30pm (joint seminar with Geometry)

B. Shiffmann (Johns Hopkins)

Newton polytopes and statistical patterns in polynomials

Abstract: Random polynomials of several complex variables have a rich array of statistical properties. For instance, the Newton polytope of a polynomial affects the distribution of its zeros: each polytope has an associated "classically allowed region" where zeros tend to congregate in a uniform distribution (asymptotically as the polytope is dilated). In the complementary "classically forbidden region," zeros have an exotic distribution and simultaneous zeros tend to be sparse. Our methods involve the Poincare-Lelong formula, Szego kernels, complex oscillatory integrals over polytopes, and a formula of Khovanskii-Pukhlikov for lattice sums. (This talk is based on joint work with Steve Zelditch.)

Friday, October 18, Burnside 920, 1:30-2:30pm (joint seminar with Geometry)

W. Minicozzi (Johns Hopkins)

The structure of embedded minimal disks in 3-manifolds

Abstract: I will describe ongoing joint work with Toby Colding on the structure of embedded minimal disks in a fixed Riemannian 3-manifold M. The focus will be on the case where M=R^3 and we show convergence to a foliation with minimal leaves (except in the trivial case where the curvature is bounded).

Tuesday, October 22, Burnside 1205, 5-6pm

S. Zelditch (Johns Hopkins)

Quantum ergodicity of boundary values of eigenfunctions

Abstract: The purpose of my talk is to outline a proof of a new result obtained jointly with Andrew Hassell (ANU) that L2-normalized boundary values (i.e. Cauchy data) u_j^{\flat} of eigenfunctions of the Laplacian on piecewise smooth convex domains \Omega with corners and with ergodic billiards are quantum ergodic. In other words, that (A_{h_j} u_j^{\flat}, u_j^{\flat} ) \to \int_{B^* \partial \Omega} \sigma_A d \mu_B in density one, for all semiclassical pseudodifferential operators on \partial \Omega. The relevant notion of boundary values u_j^{\flat} depends on the boundary condition B, as does the classical limit measure d\mu_B. Our methods cover Dirichlet, Neumann, Robin and more general boundary conditions. The proof is based on the analysis of boundary layer potentials and their boundary restrictions as quantizations of the billiard map.

Thursday, October 24, Burnside 934, 2:30-3:30pm

J. Stalker (Princeton)

Dispersion near spherical black holes

Abstract: In order to understand the dispersion of of radiation near a black hole one needs to devolop a theory of wave equations with slowly decaying potentials. I will give a brief description how these potentials arise and then discuss some recent work with Shadi Tahvildar-Zadeh, Fabrice Planchon, Matei Machedon, and Sergiu Klainerman on these problems.

Friday, November 1, Burnside 920, 2:30-3:30pm

A. Baranov (St.Petersburg State Univ.)

Bernstein's inequality in the de Branges spaces and shift coinvariant subspaces

Abstract:In this talk we are concerned with estimates of the differentiation operator (so-called Bernstein-type inequalities) in the de Branges spaces of entire functions and in the shift-coinvariant subspaces of the Hardy class in the upper half-plane.

Friday, November 22, Burnside 920, 2:30-3:30pm

N.Nikolski (Universite Bordeaux and Michigan State)

The Riesz turndown collar, polynomial free interpolation, and functional calculus

Abstract: Solving numerically some initial value problems, one needs efficient estimates of powers ||T^n|| for operators satisfying the Ritt and Tadmor type conditions. Using the Riesz turndown collar theorem we give the needed uniform estimates in terms of asymptotic behaviour of the resolvent. It happens that the polynomial calculus ||p(T)||< C(N)||p|| of a given degree N behaves differently. Namely, we show that the best possible constant C(N) has the order of log(N) as N->infty. The last result depends on a description of (finite) subsets of the unit disc, where H^infty free interpolation is possible by polynomials of degree N with a uniform norm control.

December 5, Burnside 920, 2:30-3:30pm

P. Deift (Courant)

Long-time behavior of solutions of the nonlinear Schroedinger equation with rough initial data

Abstract: This is joint work with Xin Zhou. The speaker will show how to use steepest descent methods for Riemann-Hilbert problems to analyze the long-time behavior of solutions of the NLS equation with initial data in a weighted Sobolev space. This information is needed, in particular, for the perturbation theory of NLS on the line.

During the Winter term of 2003, Analysis Seminar web page was maintained by Vojkan Jaksic; the list of the talks is located here