Geometric Aspects of Integrability
Kindly note that from May 13 to June 10, every Tuesday, Prof. Koroteev will add one more lecture on 9:50-12:15.
The ZOOM ID on every Tuesday will be different, please refer to the following information:
2025-05-13, 09:50-12:15, A7-307 (Zoom 17: 442 374 5045)
2025-05-15, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-05-20, 09:50-12:15, A7-307 (Zoom 17: 442 374 5045)
2025-05-22, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-05-27, 09:50-12:15, A7-307 (Zoom 17: 442 374 5045)
2025-05-29, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-06-03, 09:50-12:15, A7-307 (Zoom 16: 468 248 1222)
2025-06-05, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-06-10, 09:50-11:25, A7-307 (Zoom 16: 468 248 1222)
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
The ZOOM ID on every Tuesday will be different, please refer to the following information:
2025-05-13, 09:50-12:15, A7-307 (Zoom 17: 442 374 5045)
2025-05-15, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-05-20, 09:50-12:15, A7-307 (Zoom 17: 442 374 5045)
2025-05-22, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-05-27, 09:50-12:15, A7-307 (Zoom 17: 442 374 5045)
2025-05-29, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-06-03, 09:50-12:15, A7-307 (Zoom 16: 468 248 1222)
2025-06-05, 13:30-15:05, A14-203 (Zoom 16: 468 248 1222)
2025-06-10, 09:50-11:25, A7-307 (Zoom 16: 468 248 1222)
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
讲师
日期
2025年02月20日 至 06月10日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 13:30 - 15:05 | A14-203 | Zoom 16 | 468 248 1222 | BIMSA |
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。