Moduli spaces of representations

Daniel Ramras (New Mexico State University)

Abstract: Given a discrete group G, one may form the representation varieties Hom(G, U(n)) and the (coarse) moduli spaces Hom(G, U(n))/U(n), which are semi-algebraic sets. The Atiyah-Segal construction relates the representation varieties to the K-theory of the classifying space BG, and there are more mysterious relationships between the moduli spaces and the integral cohomology of G. For instance, work of Tyler Lawson provides a spectral sequence in representation theory analogous to the classical Atiyah-Hirzebruch spectral sequence. I'll discuss examples including surface groups, where Yang-Mills theory plays a key role, and crystallographic groups. In the case of surface groups, I'll explain some progress (joint work with Sean Lawton) on a conjecture relating these ideas to Goldman's symplectic form.