Lagrangian correspondences in Schubert calculus

The collection of Schubert varieties provides a particularly nice basis for the equivariant cohomology of a partial flag variety G/P. Some classical questions in Schubert calculus involve understanding the product of such basis elements, as well as their restriction (in cohomology) to other partial flag varieties. Recent work considers this in the context of upgrading G/P to its cotangent bundle, and the Schubert classes to the collection of Segre-Schwartz-MacPherson classes. I will discuss a puzzle approach to the structure constants for these classes when restricting in cohomology from type A to type C Grassmannians. We compute this expansion using Lagrangian correspondences and the machinery of quantum integrable systems. This is joint work with Allen Knutson and Paul Zinn-Justin.