Tutte seminar - Anna Bertiger

The Rim Hook Rule: Relating Quantum Cohomology of the Grassmannian to Ordinary Cohomology of the Grassmannian

Abstract:

I will present the cohomology ring of the Grassmannian of k planes in $\mathbb{C}^n$, roughly a ring with generators related to subspaces of the Grassmannian and product related to overlaps up to translation. It turns out this is a nice polynomial ring with many combinatorial properties. The same is true of the quantum cohomology ring of the Grassmannian, which has the same generators with a product given by curves joining the subspaces, rather than overlap. These two geometrically motivated rings, which seem very different are related by a result of Bertram, Ciocan-Fontanine and Fulton. I will explain this result, and also a new equivariant version of this result. I intend for the talk to be friendly, with all of the geometric notions defined.
 
This is joint work with Liz Beazley and Kaisa Taipale.