Geometric Aspects of Integrability
Integrable systems represent a ubiquitous part of modern mathematical physics. Recently, new algebraic and geometric ideas have emerged in mathematical literature which shed light on the origin of integrability. The goal of the course is to familiarize students with basic concepts of integrable systems as well as introduce new geometric ideas.
Lecturer
Date
20th February ~ 26th June, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 00:00 - 00:00 | - | - | - |
Syllabus
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.
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.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
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.
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.