# 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)

### 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

54th 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 Cn
Abstract: Stein manifolds are maybe the main object in the field of several complex variables. For example, an open set W in Cn 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 Cn 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

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 Rn 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

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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