Introduction to Equivariant Cohomology - William Graham

Special Year Seminar I
2:00pm|Simonyi 101
Topic: Introduction to Equivariant Cohomology
Speaker: William Graham
Affiliation: Institute for Advanced Study
Date: January 22, 2025 

Equivariant cohomology was introduced in the 1960s by Borel, and has been studied by many mathematicians since that time. The talks will be an introduction to some of this work. We will focus on torus-equivariant cohomology (as well as Borel-Moore homology and Chow groups), and examples related to flag varieties and Schubert varieties. After describing basic definitions and properties of these theories, we will consider localization theorems for torus applications, which relate the equivariant theories for a space to those of the fixed point locus.

We will define the equivariant multiplicity of a variety at a torus-fixed point. As an application of the localization theorems we will give a formulation of the integration formula (related to the Bott residue formula) which works for singular varieties. The equivariant cohomology of the flag variety is of interest both in geometry and combinatorics.

We will describe the equivariant cohomology of the flag variety from the point of view of convolution, as well as the use of divided difference operators to obtain representatives of Schubert classes in cohomology, and Bott-Samelson resolutions to obtain formulas for equivariant multiplicities.

Equivariant cohomology can be used to obtain information about singularities at torus-fixed points, and we will discuss the relation of equivariant cohomology to multiplicities, smoothness and rational smoothness. If time permits we may discuss joint work with Scott Larson on weighted flag varieties.