Math 7310: Topics in Algebra -- Quiver varieties

Prof. Allen Knutson
MWF 12:20 in Malott 205

The course description:
"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.
The principal topics in this course, in order, will be
  • Topology of Grassmannians and flag manifolds, Schubert classes
  • First approach to puzzle rules for multiplication of Grassmannian Schubert classes -- associativity proof
  • Equivariant cohomology
  • Equivariant puzzle rules -- scattering diagram proof
  • Nakajima quiver varieties, e.g. cotangent bundles of flag manifolds
  • Maulik-Okounkov's "stable envelope" construction of relations between quiver varieties
  • The action of the Yangian on cohomology of type A quiver schemes
  • Stable classes in equivariant cohomology of quiver varieties
  • Puzzle rules for multiplying stable classes -- geometric proof (at long last)
  • Open problems
  • I am writing a book on the subject, the first 42 pp here (updated 10/29): Course notes