# 2011 Chern Lectures

*The 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

**Venue:**

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 know 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.