Let V be an infinite vector space over a finite field k and let be an approximate homomorphism from V to End(V), i.e. the rank of f(x+y)-f(x)-f(y) is uniformly bounded by some constant r. We show that there is a homomorphism g such that f-g is of rank uniformly bounded by R(r).

We introduce a notion of a approximate cohomology groups and interpret this result as a computation of H^1 in a special case. Our proof uses the inverse theorem for the Gowers norms over finite fields; and an independent proof of this result (and some natural generalizations of it) would lead to a new proof of the inverse theorem. Joint work with D. Kazhdan

I will outline some of the ideas behind the developments in ergodic theory and combinatorial number theory starting with Szemeredi's theorem on arithmetic progressions in dense subsets of the integers and its ergodic theoretic proof by Furstenberg, and culminating with the recent proof of the Hardy-Littlewood estimates for the number of solutions to systems of linear equations of finite complexity.