A basic way to probe a cohomology theory is through its complexification and the resulting Chern character map. Features of the cohomology theory can then be transferred to this complexification, e.g., pushforwards determine Riemann-Roch factors. Field theories also have a good notion of character theory via their partition functions, which are functions on certain moduli spaces. Constructions in field theories similarly endow partition functions with additional structure. I'll apply these ideas to elliptic cohomology and 2-dimensional supersymmetric field theories, showing how the characters of these objects are essentially the same. Moreover, a closer look at the additional structures sets up several suggestive connections between physical and topological constructions.

Complex cobordism and its relatives, the Johnson-Wilson theories, E(n), carry an action of C_2 by complex conjugation. Taking fixed points of the latter yields Real Johnson-Wilson theories, ER(n). They are generalizations of real K-theory and are similarly amenable to computations. We will describe their properties, survey recent work on the ER(n)-cohomology of some well-known spaces, and describe how this brings new information to bear on the immersion problem for real projective spaces. This is joint work with Nitu Kitchloo and W. Stephen Wilson.

In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle, showing in particular that the non-multiplicativity of the signature modulo 8 is detected by an Arf invariant. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles.

Due to plane delays the following scheduled talk was cancelled.

Let M be a manifold, and let Mod(M) be its mapping class group. The Nielsen realization problem for diffeomorphisms asks, "Can a given subgroup G < Mod(M) be lifted to the diffeomorphism group Diff(M)?" This question about group actions is related to a question about flat connections on fiber bundles with fiber M. In the case M is a closed surface, the answer is "yes" for finite G (by work of Kerckhoff) and "no" for G=Mod(M) (by work of Morita). For most infinite G < Mod(M), we have no idea. I will discuss some obstructions that can be used to show that certain groups don't lift. Some of this work is joint with Nick Salter.