GAAG seminar - Equivariant cohomology and rings of functions

In topology of manifolds, there is a long history of theorems which relate global topological invariants to local behavior of vector fields, functions or group actions. This dates back to Poincare-Hopf, but many important results have been proved since, with Morse theory, Bott residue formula for complex manifolds and Białynicki-Birula decomposition for algebraic varieties being notable examples.

The classical work of Carrell–Liebermann and Akyildiz goes further and sees the whole cohomology ring as a ring of functions on a thick point, zero of a vector field. In a joint work with Tamas Hausel we generalize the work of Brion and Carrell (for Borel of SL_2) and show how to see the equivariant cohomology as a ring of function for quite general setups.

I will start the talk by recalling the classical results, and present the newest developments in the field. This allows us to see geometrically the spectra of equivariant cohomology of classical spaces like flag or Schubert varieties. I will also show how we might be able to tackle the cohomology rings of spherical varieties, extract some information for singular varieties, and how much can be said when we replace cohomology with K-theory.