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.