Curve Counting, Geometric Representation Theory, and Quantum Integrable Systems
Definition and basic properties of quantum cohomology. Quantum cohomology of projective spaces and of the Grassmannians. Andrei OKOUNKOV
Course Description
My goal in these lectures will be to explain, focusing on the simplest example of cotangent bundles of Grassmannian, how counting rational curves is certain algebraic varieties is related to several branches of mathematics pioneered and developed here in Kyoto, especially to the quantum group analysis of integrable spin chains and to the geometric realization of quantum groups provided by the Nakajima varieties.
This connection was discovered by Nekrasov and Shatashvili and the example of the Grassmannians is really the most basic example in which the theory can be fully explained. If time permits, I will try to describe the general contours of the theory, as we see them today.

 Definition and basic properties of quantum cohomology. Quantum cohomology of projective spaces and of the Grassmannians.

Andrei OKOUNKOV
Nov. 21, 2017 1:59:36 English

 Review of equivariant cohomology and equivariant Ktheory in general, and in the example of the Grassmann varieties. Equivariant quantum cohomology of the cotangent bundles T^*Gr.

Andrei OKOUNKOV
Nov. 21, 2017 2:03:50 English

 The action of sl(2) on the cohomology of T^*Gr and the representationtheoretic description of its equivariant quantum cohomology. The Yangian of gl(2) and the equivalence between quantum cohomology and a spin chain.

Andrei OKOUNKOV
Nov. 21, 2017 2:03:13 English

 Bethe Ansatz: the eigenvalues and eigenvectors in the spin chain, and their geometric meaning

Andrei OKOUNKOV
Nov. 21, 2021 2:05:24 English

 Beyond sl(2) and Grassmannians: an introduction to the enumerative geometry and representationtheoretic meaning of general Nakajima varieties.

Andrei OKOUNKOV
Nov. 21, 2017 2:01:09 English
Details
 Year/Term
 2017 / Intensive, Second semester
 Date
 November 13th to November 21st, 2017
 Faculty/
Graduate School  Graduate School of Science
 Language
 English
 Instructor name
 Andrei OKOUNKOV（Distinguished Visiting Professor, Kyoto University）
Related Courses
Stephen Theriault
Graduate School of Science
2021Boris FEIGIN
Graduate School of Science
2019Gennadi Kasparov
Graduate School of Science
2019Nigel Higson
Graduate School of Science
2019