2011 Chern Lectures

Andrei OkounkovThe 2011 Chern Lectures will be given by Andrei Okounkov, Princeton University, on April 5, 7, 12, and 14 (4 pm).

Department of Mathematics, University of California, Berkeley, presents

The 2011 Chern Lectures
Andrei Okounkov
Princeton University

2 Leconte Hall on April 5 and April 12
60 Evans Hall on April 7 and April 14

Time: 4 pm on all days

Quantum groups and quantum cohomology

Abstract: Quantum cohomology is a deformation of the classical cohomology algebra of an algebraic variety X that takes into account enumerative geometry of rational curves in X. A great deal is known about its structure for special X. For example, Givental and Kim described the quantum cohomology of flag manifolds in terms of certain quantum integrable systems, namely Toda lattices. A general vision for a connection between quantum cohomology and quantum integrable systems recently emerged in supersymmetric gauge theories, in particular in the work of Nekrasov and Shatashvili. Mathematically, the relevant class of varieties X to consider appears to be the so-called equivariant symplectic resolutions. These include, for example, cotangent bundles to compact homogeneous varieties, as well as Hilbert schemes of points and more general instanton moduli spaces. In my lectures, which will be based on joint work with Davesh Maulik, I will construct certain solutions of the Yang-Baxter equation associated to symplectic resolutions as above. The associated quantum integrable system will be identified with the quantum cohomology of X.