The Heisenberg Algebra in Symplectic Algebraic Geometry

Part of geometric representation theory involves constructing representations of algebras on the cohomology of algebraic varieties. A great example of such a construction is the work of Nakajima and Grojnowski, who independently constructed an action of a Heisenberg algebra on the singular cohomology of the Hilbert Scheme of points on a complex surface. Of course, there is no reason to stop at cohomology, as it is also natural to consider other vector spaces (such equivariant K-theory) or even categories (such as the derived category of coherent sheaves, or the Fukaya category). In this talk we will attempt to describe a small part of the tremendous structure (braid groups, vertex operators, affine Lie algebras,...) that emerges when one considers the symmetries of categories associated to the Hilbert scheme.

Date

Speakers

Anthony Licata

Affiliation

Institute for Advanced Study; Member, School of Mathematics