Motivated by the infinite horizon discounted problem, we study the convergence of solutions of the Hamilton Jacobi equation when the discount vanishes. If the Aubry set consists in a
finite number of hyperbolic critical points, we give an explicit expression for the limit.
A lattice of rank N is called well-rounded (abbreviated WR) if its minimal vectors span R^N. WR lattices are extremely important for discrete optimization problems. In this
talk, I will discuss the distribution of WR lattices in R^2, specifically concentrating
on WR sublattices of Z^2. Studying the structure of the set C of similarity classes of
these lattices, I will show that elements of C are in bijective correspondence with
certain ideals in Gaussian integers, and will construct an explicit parametrization of
lattices in each such similarity class by elements in the corresponding ideal. I will
then use this parameterization to investigate some basic analytic properties of zeta
function of WR sublattices of Z^2.
We will informally discuss a seemingly paradoxical consequence of the Lwenheim-Skolem theorem: if ZFC is consistent, there is a countable model of set theory. We will reexamine our intuitive notion of uncountablity and reach a mathematically satisfying resolution.
