# David Drasin

Title: "Positive harmonic functions in T-automorphic domains" (joint with V Azarin and P Poggi-Corradini)

When one leave nice domains in the plane (Lipschitz, for example), the number

of Martin boundary  points can be very large at a given boundary point.

We show that if the domain $D$ is homogemeous with respect to the group action

$z\to Tz$ for some fixed $T>1$, then $D$

always has one distinguished boundary point at $\infty$, corresponding to

a Martin function of ${\sl finite order\,}$ at $\infty$; there may be

others as well.

The order, $\pho(D)$ of this Martin function has geometric significance

for $D$, and plays a role in the spectral analysis of certain non-selfadjont

operators on the torus.

# John Garnett

Title: "Analytic Capacity, Cauchy Integrals, Bilipschitz Maps and Cantor Sets"

I will give a survey of the theory of analytic capacity

and its connections to the Cauchy integral and Menger

curvature, including recent work of Tolsa, Volberg and

others.

# Victor Havin

Title: "Variations on a theme by Beurling&Malliavin."

The theme is the B&M Multiplier Theorem which may be interpreted as a fact

of Fourier Analysis (existence of a non-zero L^2 function on the line with

bounded spectrum and prescribed majorant of the modulus). This theorem

will be discussed in connection with similar problems for shift

coinvariant ("model") subspaces of the Hardy H^2 space on the line. The

talk is based on a joint work with Javad Mashreghi.

Jean-Pierre Kahane

Title: "Old and new results on partial sums of Fourier series"

The purpose of the talk is to review a few known facts on partial sums of

ordinary Fourier series of integrable functions, to mention a few

questions, and to see what applies to Fourier-Walsh series.

## Ivo Klemes

Title: "Extension of a lemma of Gohberg and Krein"

We study sufficient conditions for obtaining p-norm inequalities for vectors in n dimensions.

Specifically, certain conditions involving the elementary symmetric polynomials of n variables are generalized.

## Robert Milson

Title: "Spectral deformations and zeroes of orthogonal polynomials"

In my talk I will recall some results on the factorization

of second order self-adjoint operators on the line.  Such

factorizations allow for the deformation of the operator's discrete

spectrum.  By considering particular operators, it is possible to

derive results about the zeroes of orthogonal polynomials.

# Louis Nirenberg

Title: "Distance function to the boundary and Hamilton-Jacobi equations"

We study the singular set of the distance function to the boundary in a

smooth domain in n dimensions.Then we investigate the singular set of

solutions of Hamilton -Jacobi equations.This involves the distance function

relative to a Finsler metric.The talk will be expository.

## Henrik Pedersen

Title: "Entire functions and logarithmic sums"

Abstract: The set of polynomials with sufficiently small logarithmic

sums is a normal family in the complex plane. This result was obtained

by Koosis (published in 1966) and applied to weighted approximation on

the set of integers. During my Ph.D. work with Koosis another proof of

the result making systematic use of least superharmonic majorants was

found. I shall give an idea of the proof and mention some extensions.

## Thomas Ransford

Title: "Cyclic vectors for the Dirichlet space"

In 1949, Beurling showed that a function $f$ in the Hardy space $H^2$ is

cyclic if and only if it is an outer function. (By the term cyclic, we mean

that the closed $z$-invariant subspace generated by $f$ is the whole space.)

\par

The corresponding problem for the Dirichlet space $\cal D$  still lacks

such a complete solution. In 1984, Brown and Shields proved that, if $f$ is

cyclic for $\cal D$, then (i) it is an outer function, and (ii) the zero

set of $f^*$ (on the circle) has capacity zero. They further conjectured

that (i) and (ii) together imply that $f$ is cyclic. I shall discuss some

of the progress made towards proving their conjecture. Part of this is

joint work with Omar El-Fallah and Karim Kellay

## Misha Sodin

Title: "Growth, zeroes, and area estimates. Variations on the theme"

We'll discuss recent results pertaining to the following topics:

1. Topological control of harmonic functions and area of the positivity

set.

2. High-energy Laplace-Beltrami eigenfunctions on smooth compact

surfaces.

3. Dimension-free estimates for volumes of sublevel sets of polynomials

and analytic functions of many variables.

4. Zeroes and lower bounds for quasianalytically smooth functions.

Bang's degree.

The talk is based on joint works with F. Nazarov, L. Polterovich and A.

Volberg.

## Sergei Treil

Title: "The Operator Corona Theorem and Geometry of Holomorphic Vector Bundles"

In this talk I will discuss the connection between the

operator corona problem and geometry of holomorphic vector bundles. This

will lead to some new results in the operator corona, as well as to new

open problems related to the corona problem in planar domains or in

several complex variables.

Michael Wilson

Title: "Functions and Square Functions"

We will try to give a friendly overview of some problems in one-parameter

Littlewood-Paley theory (usually involving weights) which have been opened

up (or, sometimes, closed down) in the last twenty years or so.

# Reem Yassawi

Title: "Asymptotic randomisation of measures by Cellular Automata on Sofic Shifts"

Let A be a finite set, and L be an infinite lattice (e.g. $L = Z^D$). A cellular automaton (CA) is a continuous transformation $T : A^L\to A^L$ which commutes with all shifts. Many cellular automata `asymptotically randomize’ $A^L$, in the sense that many initial probability measures µ on $A^L$ converge weak* to the uniformly distributed measure η in the forward time averages, i.e.
$$\lim_{N\to\infty}(1/N)\sum_{i=0}^{N-1}\mu\circ T^{-i}=\eta$$
Results so far exist for Markov measures with full support. We discuss recent results on asymptotic randomisation for measures supported on Sofic Subshifts and Markov Subgroups. This is joint work

with Alejandro Maass, Servet Martinez, and Marcus Pivato.