Dan Ramras from Vanderbilt University speaking about "Gauge theory and homotopical representation theory"

Abstract: Deformation K-theory serves as a homotopy theoretical analogue of the representation ring of a group. Computations suggest that for many infinite discrete groups G with compact classifying spaces, deformation K-theory agrees 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. When BG = M is a manifold, this relationship 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 surfaces. I'll explain these results as well as joint work with Tom Baird which (conjecturally) explains the low-dimensional discrepancy between representation theory and K-theory.