Abstract: I will start with defining the notion of equivariant cohomology for a group action on a topological space. It is a ring that encodes information both about the topology of the space as well as the action of the group. Often equivariant cohomology is easier to compute and one can recover the usual cohomology of a space from its equivariant cohomology. I will discuss an important class of examples of smooth varieties/manifolds with torus action (called GKM manifolds after Goresky-Kottwitz-MacPherson) such that there is an elegant purely combinatorial description of the equivariant cohomology (in terms of the so-called momentum graph of the action). Examples include flag varieties and toric varieties. In the next week's talk, I will make a connection between the Chiniese remainder theorem (for polynomial ring) and the cohomology surjectivity problem, that is, when the restriction map from the cohomology of a space to a subspace is surjective (recent joint work with J. B. Carrell). The talk will be accessible to graduate students and is aimed to be an introduction to equivariant cohomology. Some basic background from topology is needed.

Thackeray 427