# 2000/2001 Analysis Seminar

## Fall 2000

### Monday, September 18, 2:30-3:30, Burnside 1205:

**Andrea Fraser** (Univ. of New South Wales)
Multiplier Operators on the Heisenberg Group

### Friday, September 29, 2:30-3:30, Burnside 920:

**Nick Varopoulos** (Univ. Paris VI & I.U.F.)
Potential Theory on Lipschitz domains

### Friday, October 6, 2:30-3:30, Burnside 920:

**Iosif Polterovich** (CRM & ISM)
Geometry and combinatorics of the heat kernel
**Abstract**: Heat kernel asymptotics were subject to an
intensive study in spectral
geometry for many years. They contain a lot of geometric information such
as dimension, volume, scalar curvature etc. However, closed formulas
existed only for the first few heat kernel coefficients (called heat
invariants) and their complexity was growing very rapidly. We suggest a
new method for computation of heat invariants based on a result due to S.
Agmon and Y. Kannai. Using certain combinatorial identities we present
explicit formulas for all heat invariants in terms of powers of the
Laplacian and the distance function. As another application we obtain new
explicit expressions for the Korteweg-de Vries hierarchy.

### SPECIAL SEMINAR: Thursday, October 12, 2:30-3:30, Burnside 920

**Vojkan Jaksic** (Ottawa/Johns Hopkins)
Spectral Structure of Anderson Type Hamiltonians

### Friday, October 13, 2:30-3:30, Burnside 920:

**Ilia Binder** (Harvard)
Harmonic measure and polynomial Julia sets
**Abstract**
(ps,
pdf)

### Friday, October 20 and Saturday, October 21

Conference "Adrien Douady" at CRM

### SPECIAL SEMINAR: Thursday, October 26, 2:30-3:30, Burnside 920

**Felix Finster** (Max Planck Institute)
Curvature Estimates in Asymptotically Flat Manifolds of
Positive Scalar Curvature
**Abstract**
After a brief mathematical introduction to general relativity,
the concept of energy in curved space-time is discussed. The total energy
and momentum of an asymptotically flat manifold are introduced. The
positive energy theorem, the positive mass theorem, and the Riemannian
Penrose inequality are stated and briefly explained.
The main part of the talk is concerned with the question if and in
which sense the total mass of an asymptotically flat manifold controls the
Riemannian curvature tensor. For the proof of the resulting curvature
estimates, we work with Dirac spinors and use a Weitzenboeck formula as
well as an integration-by-parts argument for the second derivatives of the
spinors.

### A series of Four one-hour lectures

**Paul Koosis** (McGill)
The two boundary Harnack principles for Lipschitz domains
(I) Monday, October 30, 2:30-3:30, Burnside 920
(II) Wednesday, November 1, 2:30-3:30, Burnside 920
(III) Monday, November 6, 2:30-3:30, Burnside 920
(IV) **Extra lecture:** Wednesday, 8 November, 2:30pm - 3:30pm,
Burnside 920

### SPECIAL SEMINAR: Friday, November 3, 2:30-3:30, Burnside 920

**Duong Phong** (Columbia)
Degeneracies and Stability

### Friday, November 10, 2:30-3:30, Burnside 920

**Der-Chen Chang** (Georgetown)
On the boundary of Fourier and complex analysis: the
Pompeiu problem

### Saturday, November 18

54^{th} Quebec Mathematics Colloquium at Concordia

### Friday, November 24, 2:30-3:30, Burnside 920

**Jie Xiao** (Concordia)
Representation Theorems for Q Spaces
**Abstract** (ps,
pdf)
### Friday, December 1, 2:30-3:30, Burnside 920

**Ivo Klemes** (McGill)
Finite Toeplitz matrices and sharp Littlewood conjectures
**Abstract:**
I will describe some very simple questions
about strings of zeros and ones and the matrix
of their forward shifts. The answers would have a
bearing on certain p-norm inequalities for exponential
sums proposed by Hardy and Littlewood in the 1920s.
(This will be an "analysis" version of a talk given
in the Matrix Seminar last year, but with a couple
of new counterexamples).

### Friday, December 8, 2:30-3:30, Burnside 920

**Dmitry Jakobson** (McGill)
Limits of eigenfunctions on flat tori
**Abstract:** We study limits of eigenfunctions of the Laplacian
on arithmetic flat tori. We classify such limits in dimension 2, and
provide some L^p bounds in dimensions 3, 4, 5, 6. We also relate certain
questions about limits in dimension n to analogous questions about
eigenfunctions in dimensions (n-2) and (n-3). A crucial ingredient
of the proofs is a geometric lemma which describes a property of simplices
of codimension one in R^n whose vertices are lattice points on spheres.
We also discuss a generalization of a
two-dimensional result due to Cooke and Zygmund to higher dimensions,
as well as related questions concerning L^p/L^2 bounds for eigenfunctions.

### Wednesday, December 13, 2:30-3:30, Burnside 920

**Eduardo S. Zeron** (Inst. Politecnico Nacional Mexico)
Complement of Stein sets in C^{n}
**Abstract:**
Stein manifolds are maybe the main object in the field of
several complex variables. For example, an open set W in C^{n}
is Stein if
there is a function analytic on W which cannot be extended outside W.
There are several characterizations of Stein manifolds (and in particular
of Stein open sets), some of them use Dolbeaut cohomology or
plurisubharmonic functions. In this talk, we want to show that an open set
W in C^{n} is Stein if and only if polynomials have the local maximum
modulus principle on the complement of W.

## Winter 2001

### Friday, January 19, 1:00-2:00, UdeM, Pav. Andre-Aisenstadt, salle 4336

**Richard Fournier** (CRM)
Sur un problème de Bohr

### Friday, January 19, 3:00-4:00, Burnside 920

**D. Korotkin** (Concordia)
Isomonodromic deformations, theta-functions and self-dual Einstein
equations
**Abstract:**
We solve a class of matrix Riemann-Hilbert problems explicitly
in terms of theta-functions and Szego kernel on Riemann surfaces.
The results give effectivization of Hitchin's description of
self-dual SU(2)-invariant Einstein metrics.

### Friday, January 26, 2:30-3:30, Burnside 920

**Pengfei Guan** (McMaster)
On Christoffel-Minkowski Problems
**Abstract:**
we consider a problem of prescribing
surface area functions for convex bodies. In the
extremal cases, they correspond to classical Christoffel
and Minkowski problems respectively. The problem
for the imtermediate cases has been open for serval
decades. This problem in differential geometry will
be treated through Hessain equations on its support
functions. Here we employ recently developed PDE techniques
to obtain a general sufficient condition for the solution
to the problem.

### Friday, February 2, 2:30-3:30, Burnside 920

**Yuri Khidirov** (Concordia)
Degree theory for variational inequalities and index of
K-critical point
**Abstract:**
We refer to the classical finite-dimensional topological degree theory.
Then we construct a variant of degree theory, so-called K-degree theory,
that satisfies all basic properties and is applicable to investigation of
variational inequalities. We demonstrate a simple formula for index of
nondegenerate
K-critical point for a polyhedral cone. Finally, we formulate some results
concern number of solutions of variational inequalities.

### CRM-ISM Colloquium: Friday, February 2, 4:00-5:00, UdeM,
Pav. Andre-Aisenstadt, salle 6214

**Damien Roy** (Université d'Ottawa)
Interpolation en plusieurs variables

### Wednesday, February 7, 2:30-3:30, Burnside 920

**Javad Mashreghi** (McGill)
On application of entire functions of exponential type

### Friday, February 16, 13h00
U de M, DMS, Pav. André-Aisenstadt, salle 5183

**Paul Gauthier** (Université de Montréal)
La réciproque du théorème sur la correspondance
à la frontière
**Résumé:** Le théorème de Osgood-Carathéodory
affirme que toute
transformation conforme entre deux domaines de Jordan
se prolonge à
un homéomorphisme des fermetures. Nous montrons une espéce
de réciproque.

### Wednesday, February 28, 3:30-4:30, Burnside 1214 (Note room change!)

**Roland Speicher** (Queens)
Free probability theory and random matrices
**Abstract:**
Free probability was introduced by Dan Voiculescu
about 15 years ago in order to get some insight into the
structure of special operator algebras. Since then it has
turned out that this theory has an interesting structure of
its own and possesses a lot of links with quite different
fields, like combinatorics, random matrices or statistical
physics. I will give an introduction into free probability
theory and show some typical results by looking more closely
onto free Brownian motion and free diffusion. In particular,
I will stress the relation of these objects with random
matrices.
The talk focuses on the analytic and probabilistic properties
of the theory - no knowledge about operator algebras is needed.

### Friday, March 2, 2:30-4:00, Burnside 920

**F. Nazarov** (Michigan State and Minnesota, St. Paul)
The geometric KLS lemma, dimension-free estimates for the
distribution of values of polynomials, and distribution of
zeroes of random analytic functions.
**Abstract:**
The goal of the talk is to attract the attention of the
reader to one simple dimension-free geometric inequality that can be
proved using the classical needle decomposition technique. This inequality
allows to derive quite sharp dimension-free estimates for the distribution
of values of polynomials in convex subsets in R^{n} in a simple and
elegant way. Such estimates, in their turn, lead to a surprising result
about the distribution of zeroes of random analytic functions; informally
speaking, we show that for simple families of analytic functions, there
exists a ``typical'' distribution of zeroes such that the portion of the
family occupied by the functions whose distribution of zeroes deviates
from that typical one by some fixed amount is about
Const*e^{-(size of the deviation)}.

### Monday, March 12, 2:30-3:30, Burnside 920

**S.V. Khrushchev** (St. Petersburg and Purdue)
Continued Fractions and Schur's Algorithm
in Orthogonal Polynomials on the Unit Circle

### Friday, March 16, 13h00, U de M, DMS, Pav. André-Aisenstadt,
salle 5183

**Richard Fournier** (CRM et Dawson)
Les zéros à la frontière des solutions d'équations
fonctionnelles II

### CRM-ISM Colloquium: Friday, March 16, 4:00-5:00, UQAM,
Pavillon Sherbrooke, Salle SH-3420

**Serge Lang** (Yale)
Heat kernels, theta functions and zeta functions

### Friday, March 23, 13h00, U de M, DMS, Pav. André-Aisenstadt,
salle 5183

**Javad Mashregi** (McGill)
On Marshall's Theorem
**Résumé:** Let H^{\infty} denote the Banach space
of bounded holomorphic functions in the unit disc. Marshall's Theorem
asserts that the unit sphere of H^{\infty} is the closed convex hull
of Blaschke products.

### Friday, March 30, 13:30, U de M, DMS, Pav. André-Aisenstadt,
salle 5340

**Dominic Rochon** (U de M)
Thesis Defence: Dynamique bicomplexe et theoreme de Bloch pour
fonctions hyperholomorphes

### Friday, April 6, 2:30-3:45, Burnside 920

**A. Eremenko** (Purdue)
Rational functions with real critical points,
with applications to real enumerative geometry

### Friday, April 20, 2:30-3:30, Burnside 920

**A. Bourget** (McGill)
Asymptotic statistics for the zeros of Lamé ensemble

### Friday, April 27, 13:00, U de M, DMS, Pav. André-Aisenstadt,
salle 5340

**Kihel Karim** (CRM)
Plongements projectifs de surfaces de Riemann compactes
**Résumé:** Il est impossible de représenter une
surface de Riemann compacte comme surface dans l'espace euclidien
complexe C^n. Il est donc naturel d'essayer de la représenter
comme surface dans l'espace projectif P^n.
### Friday, May 11, 13:00, U de M, DMS, Pav. André-Aisenstadt,
salle 5340

**Roth, Oliver** (U. Michigan)
Pontryagin's maximum principle in geometric function theory

### Friday, June 1, 13:00, U de M, DMS, Pav. André-Aisenstadt,
salle 5183

**Paul Gauthier** (UdeM)
La fonction zeta de Riemann et les cercles de remplissage
**Résumé:** Le th\'eor\`eme de Picard affirme que toute fonction
enti\`ere
non-constante prend toute valeur, avec au plus une valeur exceptionnellle. On
montre
que la fonction
zeta de Riemann a la m\^eme propri\'et\'e dans la bande critique. Selon
l'hypoth\`ese
de Riemann, la valeur z\'ero serait cette (unique) valeur exceptionnelle.

### Thursday, August 2, 13:30, U de M, DMS, Pav. André-Aisenstadt,
salle 5183

**Victor Havin** (St. Petersburg)
Sur la s\'eparation des singularit\'es de fonctions analytiques
born\'ees

### Monday, August 6, 13:30, U de M, DMS, Pav. André-Aisenstadt,
salle 5183

**Victor Khatskevich** (Ort Braude College, Israel)
Abel-Schroeder equations for linear fractional maps of operator

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