n the talk I will study the spectral properties of a class of
SturmLiouville-type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, one can construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. I will show that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively. A key ingredient in the proof is the so- called Wonderland theorem.
The talk is based on joint work with Malcolm Brown and Karl
Michael Schmidt.

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.