EVENT DETAILS AND ABSTRACT

Title: Gauge theory and homotopical representation theory
Speaker: Dan Ramras
Speaker Info: Vanderbilt
Brief Description:
Special Note:
Abstract:

The deformation K-theory spectrum of a discrete group G, introduced by Gunnar Carlsson, can be thought of as a homotopy theoretical analogue of the representation ring. Computations suggest that for many infinite discrete groups G with compact classifying spaces, deformation K-theory will agree with topological K-theory of BG (but only above the rational cohomological dimension of G minus 2). The relationship with K-theory is reminiscent of the Atiyah-Segal theorem, while the failure in low dimensions is precisely analogous to the low dimensional failure in the Quillen-Lichtenbaum conjecture (which relates algebraic K-theory to etale K-theory). When BG = M is a manifold, the relationship between deformation K-theory and topological K-theory can be interpreted in terms of gauge theory on principal bundles over M. This perspective, together with work of Tyler Lawson, has been used to calculate deformation K-theory for products of (possibly non-orientable) surfaces. I'll explain these results and, time permitting, ongoing joint work with Tom Baird which explicitly produces the low-dimensional failure in many cases.