Let T be a compact torus. Goresky, Kottwitz and Macpherson showed that for a large and interesting class of T-equivariant projective varieties X, the equivariant cohomology ring H_T^*(X) is may be encoded in a graph, now called the GKM-graph, with vertices corresponding to the fixed points of X and edges labeled by the weights, Hom(T, U(1)). In this lecture, we explain how the GKM construction can be generalized to any finite T-CW complex. This generalization gives rise to new mathematical objects: GKM-hypergraphs and GKM-sheaves. If time permits, we will show how these methods were used to resolve a conjecture concerning the moduli space of flat connections over a non-orientable surface.

For more information, contact Khalid Bou-Rabee khalidb[at]umich(dot)edu