University of Chicago
Department of Mathematics

Third Namboodiri Lecture. For information, see http://www.math.uchicago.edu/lectures.html#Namboodiri

An E1 ring spectrum is what is usually called an A-infinity ring spectrum or just "S-algebra". An E1 ring spectrum has a nice stable category of modules. An E2 structure gives this category a monoidal product. An E3 structure gives this monoidal category a braided monoidal structure. If the E3 structure extends to an E4 structure, the braided monoidal structure is a symmetric monoidal structure. If time and interest permits, I will explain how this is related to Deligne's Hochschild Cohomology conjecture (proved) and the new generalization it suggests (unproved).

We prove that the map from the stable mapping class group of surfaces to the stable automorphism group of free groups induces an infinite loop map on the classifying spaces after plus construction. To show this, we introduce automorphisms of free groups with boundaries and use certain 3-dimensional manifolds to prove an appropriate homological stability theorem for these groups.

The purpose of this talk is to describe some methods to compute the cohomology of functors on categories in which every endomorphism of an object is an isomorphism. This is motivated by open problems in the modular representation theory of finite groups, such as Alperin's weight conjecture which is shown to admit a formulation in terms of equivariant Bredon cohomology.

I will describe a spectrum Q(N) which is built out of several copies of TMF, using isogenies of elliptic curves. I will then explain how the K(2)-local sphere is built out of Q(N) and its Gross-Hopkins dual, at least when p=3 and N=2. This recovers a resolution of the K(2)-local sphere recently produced by Goerss, Henn, Mahowald, and Rezk, and explains some phenomena that occurs in their work.

Thesis defense

No seminar (Adrian Albert Lecture) http://www.math.uchicago.edu/lectures.html#Albert

Cancelled