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This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the fall 2012 semester.
All none of the talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).
1:00-2:00 -- Topological chiral homology and homological stability
I will give a short introduction to topological chiral homology and then sketch the proof of an application that is the subject of recent joint work with Jeremy Miller: we prove that if an En-algebra A satisfies homological stability then for an open framed n-dimensional manifold M topological chiral homology of M with coefficients in A also satisfies homological stability. For example, if A is the natural numbers, then we obtain homological stability for the symmetric powers Symk(M). One should think of this as a local-to-global principle for homological stability.
2:30-3:30 -- Higher q-expansions and TAF-character maps
In number theory, the q-expansion map takes modular forms to their Fourier expansions. This has a topological realization, as a map between cohomology theories from topological modular forms to Tate K-theory. This is an example of a "transchromatic" character map, a higher height generalization of the Chern character from complex K-theory to rational cohomology.
In this talk, I will give some background and then discuss work in progress to obtain a generalization of the q-expansion map for automorphic forms on complex hyperbolic spaces. The main conjecture is that there is an associated topological realization as a map between different versions of the cohomology theory of topological automorphic forms.
* the xkcd group -- making xkcd not stand for nothing since 2010!