Oscar Randal-Williams: Tautological rings for high-dimensional manifolds.

The cohomology of the classifying space BDiff(M) of the group of diffeomorphisms of a manifold M may be considered as the ring of characteristic classes of smooth fibre bundles with fibre M. This ring is difficult to understand, but when M is an orientable surface the close connection between BDiff(M) and the moduli space of Riemann surfaces means that a lot is known. In this case, algebraic geometers have found it productive to focus not on all the cohomology but a certain subring, the "tautological ring", containing the geometrically interesting classes. One can make a similar definition for manifolds of higher dimension. I will explain all these terms, and discuss some recent results on the large scale structure of these tautological rings. This is joint work with Ilya Grigoriev and Soren Galatius.