BIMSA

Kindly note that from April 8 to May 20, every Tuesday, Prof. Koroteev will add one more lecture on 9:50-11:25.
The ZOOM ID on every Tuesday will change to 442 374 5045 (PW: BIMSA) and the classroom is on A7-307.

In this course, I will explain how quantum and classical integrable systems arise from algebrogeometric constructions. In particular, I will discuss space of opers and their deformations on the projective line and how this space leads to both quantum spin chains (XXX, XXZ, XYZ) and classical many-body systems (Calogero, Ruijsenaars, etc). The two types of systems are related to each other via so-called quantum/classical duality which is an integrable systems avatar of the Geometric Langlands correspondence. The topics will include
1. (q-)Opers on the projective line
2. QQ-systems, Bethe Ansatz
3. The ODE/IM Correspondence
4. Quantum/Classical duality
5. Elliptic integrable systems
6. Enumerative algebraic geometry with connections to integrability

2025年02月20日至 06月26日

Weekday	Time	Venue	Online	ID	Password
周四	13:30 - 15:05	A14-203	Zoom 16	468 248 1222	BIMSA

In this course, I will explain how quantum and classical integrable systems arise from algebro-geometric constructions. In particular, I will discuss space of opers and their deformations on the projective line and how this space leads to both quantum spin chains (XXX, XXZ, XYZ) and classical many-body systems (Calogero, Ruijsenaars, etc). The two types of systems are related to each other via so-called quantum/classical duality which is an integrable systems avatar of the Geometric Langlands correspondence.

The topics will include

(q-)Opers on the projective line.
a. Definitions, connections, q-connections, differential/difference equations
b. (q-) Oper conditions. Miura opers, Z-twisted Miura opers
QQ-systems, Bethe Ansatz
a. QQ-systems from q-oper conditions.
b. Bethe Ansatz equations from QQ-equations.
c. Extended QQ-systems. Toroidal q-opers.
The ODE/IM Correspondence
a. Review of Bazhanov-Lukyanov-Zamolodchikov work
b. Reformulation in terms of affine/q-opers
Quantum/Classical duality
a. Opers of type A. qWronskians from QQ-relations
b. Spectral curves of many-body systems from q-Wronskian relation
Elliptic integrable systems
a. XYZ chain, Elliptic Calogero/RS models.
b. The diamond of integrability
Enumerative algebraic geometry with connections to integrability
a. Quantum equivariant K-theory of Nakajima quiver varieties. qKZ/dynamical equations
b. Bethe equations from enumerative counts
c. Many-body systems from enumerative counts.

Advanced Undergraduate , Graduate , 博士后 , Researcher

公开

英文

My education begain in Russia where I learned math and physics at Moscow Insitute of Physics and Technology. I started my research career as a theoretical physicist after obtaining my PhD from University of Minnesota in 2012. At first, my research focus was drawn to various aspects of supersymmetric gauge theories and string theory. However, I have always been drawn to pure abstract mathematics since my student days. Since around 2017 I have been a full time mathematician.

My current research is focused on the interaction between enumerative algebraic geometry, geometric representation theory and integrable systems. In general I work on physical mathematics which nowadays represents a large part of modern math. A significant amount of problems that are studied by mathematicians comes from string/gauge theory. More recently I began to study number theory and how it is connected to other branches of mathematics.

If you are postdoc or a graduate student in Beijing area and you are interested in working with me contact me via email.