The correspondence between integrable systems and enumerative geometry

started roughly 25 years ago in the works of Givental and his collaborators,

studying quantum cohomology and quantum K-theory. Around 10 years ago,

physicists Nekrasov and Shatashvili proposed an unexpected relation between

quantum K-theory and quantum integrable systems based on quantum groups

within their studies of 3-dimensional gauge theories. Their bold proposal

led to a lot of interesting developments in mathematics, bringing a new life

to older ideas of Givental, and enriching it with flavors of geometric

representation theory via the results of Braverman, Maulik, Nakajima,

Okounkov and many others. In this talk I will focus on recent breakthroughs

in the subject, leading to the proper mathematical understanding of

Nekrasov-Shatashvili original papers as well as some other subsequent

conjectures made by physicists. Our main illustration of such a relation is

an interplay between equivariant quantum K-theory of the cotangent bundles

to Grassmanians and the Heisenberg XXZ spin chain. We will also discuss

relation of equivariant quantum K-theory of cotangent bundles of flag

varieties and many-body integrable systems of Ruijsenaars-Schneider and

Toda.

### Math-Physics Joint Seminar

Thursday, March 14, 2019 - 4:30pm

#### Anton Zeitlin

LSU