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This page was the home of the **X**traordinary cohomology theory & **K**-theory **C**ollective **D**iscussion group^{*} for the fall 2012 semester.

All none of the talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).

__Thursday, May 2 -- 200-230 Lane History Corner__ (Stanford)
** 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 *E*_{n}-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 *Sym*^{k}(M). One should think of this as a local-to-global principle for homological stability.

*Sander Kupers*

** 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.

*Sebastian Thyssen*

^{*} the xkcd group -- making xkcd not stand for nothing since 2010!