I will present a criterion for the validity of the central limit theorem
for a class of dependent random variables and then I will discuss some applications of
it on random, boundary homogenization problems of nonlinear PDEs such nonlinear
parabolic ones and Navier walls.

In 1969, Hirsch posed the following problem: given a diffeomorphism of a manifold and a hyperbolic set for the diffeomorphism, describe the topology of the hyperbolic set and the dynamics of the diffeomorphism for this set. We solve the problem when the hyperbolic set is a closed 3-manifold.

One possible approach to the study of the geometry of the Gibbs measure in the Sherrington-Kirkpatrick
type models (for example, the chaos and ultrametricity problems) is based on the analysis of the free energy
on several replicas of the system under some constraints on the distances between replicas. In general, this
approach runs into serious technical difficulties, but we were able to make some progress in the setting of the
spherical p-spin SK models where many computations become more explicit.