Cohomology of some diffeomorphism groups Seminar

I will explain the state of the art concerning the cohomology of the classifying spaces of the diffeomorphism groups of the manifolds #^g S^n x S^n. For n at least 3 the stable (as g tends to infinity) cohomology of these classifying spaces has been computed by Galatius and myself, so I will focus on the unstable cohomology instead. When n=1 the unstable cohomology has been studied in detail by algebraic geometers, as in this case these classifying spaces model Riemann's moduli space. They have found it productive to focus not on all the cohomology but a certain subring, the "tautological ring", containing the geometrically interesting classes. I will recall this background, then explain ongoing work with Galatius and Grigoriev which explains some aspects of the large scale structure of these tautological rings for n > 1.