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.

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.

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**

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.

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.

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.

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.

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.

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

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.

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.

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.