I'll describe the notion of topological conformal field theory, which is like a 2-dimensional TQFT except we put in higher homotopical information to take account of the topology of moduli spaces of Riemann surfaces. Open TCFTs are basically the same as Calabi-Yau (or cyclic) A-infinty categories; this is the categorical version of the ribbon graph decomposition of moduli space. I'll also discuss how to construct a closed TCFT from an open. This gives operations on the Hochschild homology of a Calabi-Yau category coming from the homology of moduli space.
For X a smooth separated scheme of finite type over a field k, and for G a linear algebraic group over k, we construct homology-type functors H_i(X,G) from cycles in the simplicial scheme BG x X. When X=Spec(k) and k is algebraically closed, these groups are the ordinary homology groups of the discrete group of k-rational points of G. We construct a spectral sequence beginning with our groups and converging to the etale cohomology of the simplicial scheme BG, thus relating this theory to the study of Friedlander's Generalized Isomorphism Conjecture. We conclude with some calculations in the case where X is the spectrum of the real numbers.
Motivated by a construction of Benson, Krause and Schwede for differential graded algebras I will define the universal Toda bracket of a ring spectrum R. This is a certain cohomology class in a graded version of the topological Hochschild cohomology of the ring of homotopy groups of R. The class contains information about the question whether a module over the homotopy groups of R arises as the homotopy groups of an R-module spectrum. As an example, I will consider the real K-theory spectrum. I will also indicate how to handle higher universal Toda brackets for periodic ring spectra.
We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the number of copies of RP^2 in the decomposition of the surface. Using the Madsen-Weiss machinery, we compute the stable rational cohomology of these groups.
(Joint work with Kasper Andersen.) A p-compact groups is a homotopical version of a Lie group, but with all the structure concentrated at a single prime p. I'll state and outline a proof of the classification of 2-compact groups, hence completing the classification at all primes (the case of odd primes is proven in earlier work by Andersen-Grodal-Moller-Viruel). The theorem says that there is a 1-1-correspondence between connected 2-compact groups and root data over the 2-adic integers.
A Hopf algebra developed by Andre Henriques and the speaker and used for computing the $tmf$ homology of a space or spectrum is presented. This is done using a variant of the Adams spectral sequence in the world of $tmf$-module spectra. Following a presentation of this, it will be used to show that the $tmf$ homology of the cofiber of the transfer map $B\Sigma_3->S^0$ is a torsion free indecomposable module over $tmf_*$.
Recent calculations of surgery obstruction groups makes it possible to classify completely all topological manifolds doubly covered by by S^1xS^n, n>2. I will begin by describing this classification and then go on to study closer an infinite family of such manifolds in dimension 4, all homotopy equivalent to RP^4#RP^4.