Math 7310: Topics in Algebra -- Quiver varieties

"Geometric representation theory" is misnamed; it is a collection of examples of groups acting on the homology of various varieties. One of the two best examples of such actions is Nakajima's 1994 action of a Lie group on the top homology groups of "quiver varieties", where the Lie group and the varieties are both defined from a (rather generalized) Dynkin diagram. This prompts the question of what should act on the total homology or K-theory of quiver varieties, and it turns out to be the corresponding "quantized loop algebra".

Such representations had already appeared in statistical mechanics, where they provide tools for "completely integrating" some interesting stat mech problems. The connection of these "R-matrices" to the quiver variety geometry was spelled out in 2012 by Maulik and Okounkov. In the last couple of years Zinn-Justin and I have been using these tools to solve old problems in the cohomology rings of Grassmannians and other flag manifolds, a circle of problems called "Schubert calculus".

All of this will be built from the ground up. I will assume basic knowledge of homology and cohomology (though equivariant cohomology and K-theory would be nice too), and some exposure to Lie groups and their representations. Some results in algebraic geometry will be taken on faith.