"Distinguished varieties" are a special class of algebraic
curves in C^2 that exit the bidisk through the distinguished boundary
(aka the torus). We shall discuss connections with the polynomials
that define these curves and polynomials with no zeros on the bidisk,
and use a powerful "sums of squares" formula (actually a two variable
version of the Christoffel-Darboux formula for orthogonal polynomials)
to a prove a determinantal representation of distinguished varieties.
As an application of our approach, we will prove a certain bounded
analytic "extension" theorem.

I will first give a brief introduction of the connection between
a Hamilton-Jacobi equation and the Aubrey-Mather theory. This is the so
called weak KAM theory. An extremely interesting project in weak KAM
theory is to understand what kind of dynamical information is encoded in
the
effective Hamiltonian. I will present a result about the connection
between linear pieces on level curves of the effective Hamiltonian and the
structure of correspondent Aubry sets.

The Schwarzchild, respectively the Kerr space-times are solutions for the vacuum Einstein equation which model a spherically symmetric, respectively a rotating black hole. In this talk I will discuss the decay properties of solutions to the linear wave equation on
such backgrounds.

Suppose g is a square integrable function on the real line. The principal shift invariant space, , generated by g is the closure of the span of the system
B ={g(.-k): k an integer}. These spaces are most important in many areas of Analysis. This is particulrly true in the theory of Wavelets. We begin by describing a very simple method for obtaining the basic properties of and the systems B.
The systems obtained by applying, in addition to the integral translations, also the integral modulations (these are the multiplication of a function by exp(-2pinx)) are known as the Gabor systems. By using the Zak transform we show how the same methods can be used to study the basic properties of the Gabor systems and their span.
We will define the Zak transform and explain all this
in a very simple way that will be easily understood by all who know only a "smidgeon" of mathematics. A bit more challenging will be the explanation how all this can be extended to general locally compact abelian groups and their duals.
This is joint work with E. Hernandez, H. Sikic and E. N.
Wilson.