Université Paris 6

The aim of this talk is to explain how to axiomatize Hilbert's Theorem 90, in the setting of (the cohomology with finite coefficients of) profinite groups. I shall first explain the general framework.  It includes, in particular, the use of divided power modules over Witt vectors; a process which appears to be of independent interest in the theory of modular representations. I shall then give several applications to Galois cohomology, notably to the problem of lifting mod p Galois representations (or more accurately: torsors under these) modulo higher powers of p. I'll also explain the connection with the Bloch-Kato conjecture in Galois cohomology, proved by Rost, Suslin and Voevodsky. This is joint work in progress with Charles De Clercq.

Random Walks in Dirichlet Environment play a special role among random walks in random environments since the annealed law corresponds to the law of an edge oriented reinforced random walks. We will give few results concerning the ballistic behaviour of these walks and some properties of the asymptotic velocity. We will also compare the behaviour of these walks with general random walks in random environments in the limit of small disorder