The famous Oseledets theorem states that if gn is a station-
ary sequence of m × m matrices, then with probability 1 there is a (random) basis in R m such that for any vector x the asymptotic behaviour of ||gn . . . g1 x|| is the same as that for one of the vectors from this basis. The fact that the sequence is stationary is crucial for the existence of such a basis. I shall consider the product of non-identically distributed independent matrices and will explain under what conditions one can prove the existence of distinct Lyapunov exponents as well as the Oseledetss dichotomy (or rather multihotomy) of the space.

Predicted by Einstein in 1924, the Bose-Einstein condensation is a
striking phase transition that takes place in certain systems of
quantum bosonic particles. The dependence of the critical
temperature on the interparticle interactions has been a controversial
issue in the physics community. I will review the mathematical setting
and the literature, and I will describe rigorous upper bounds for the
critical temperature of dilute systems. This upper bound is expected
to be sharp in 2D but not in 3D. (This is joint work with R. Seiringer.)

Normally hyperbolic smooth trapped sets are structurally
stable and occur in many interesting situations: for instance
for Kerr black hole metrics. We show that the corresponding
semiclassical resonances (e.g. quasinormal modes for Kerr
black holes) are separated from the real axis which has
consequences for decay of waves and other phenomena.

The proof is a simple example of techniques
used in the semiclassical study of quantum resonances
and I hope to present it in a self-contained way.

The talk will discuss Jacobi matrices with periodic right
limits and corresponding square-summable variations. In particular, it will illuminate the optimality of a recent theorem of Kaluzhny-Shamis by showing that a recent conjecture of BreuerLastSimon is wrong.

We study the recurrence and ergodicity for the billiard in infinite polygons, either $Z$-periodic or $Z^2$-periodic. In the $Z$-periodic case the results are quite complete. In the more difficult $Z^2$-periodic case we obtain partial results and discuss suggestive examples. This is joint work with J.P. Conze.