## 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

1999/2000 Seminars

2000/2001 Seminars

2001/2002 Seminars

2002/2003 Seminars