"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.