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.
讲师
日期
2025年02月20日 至 06月26日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 00:00 - 00:00 | - | - | - |
课程大纲
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.
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
我的教育始于俄罗斯，在莫斯科物理技术学院学习数学和物理。移居美国后，我于2012年在明尼苏达大学获得博士学位，开始了理论物理学家的研究生涯。起初，我的研究聚焦于超对称规范理论和弦理论的多个方面。然而，自学生时代起，我一直对纯粹抽象数学充满兴趣。约从2017年起，我成为全职数学家。我当前的研究侧重于枚举代数几何、几何表示论与可积系统之间的互动。总的来说，我致力于物理数学的研究，这在当今代表了现代数学的重要组成部分。许多数学家研究的问题来源于弦理论/规范理论。最近，我开始研究数论及其与数学其他分支的联系。如果你是北京地区的博士后或研究生，并有意与我合作，请通过电子邮件联系我。