Algebraic analysis of integrable hierarchies II
This is a continuation of "Algebraic analysis of integrable hierarchies I" in the last semester, September - December 2025.
Among integrable systems the KP hierarchy and its cousins are quite important. In the previous course, we defined the KP hierarchy and showed that it can be regarded as a dynamical system on an infinite dimensional Grassmann manifold, following Mikio Sato's idea.
In the present course, after reviewing the previous course, we shall continue the study of the KP hierarchy and discuss the bilinear identities of the tau function, symmetries of the solution space. Then we will study other hierarchies (the mKP hierarchy, the Toda lattice hierarchy) as well.
Among integrable systems the KP hierarchy and its cousins are quite important. In the previous course, we defined the KP hierarchy and showed that it can be regarded as a dynamical system on an infinite dimensional Grassmann manifold, following Mikio Sato's idea.
In the present course, after reviewing the previous course, we shall continue the study of the KP hierarchy and discuss the bilinear identities of the tau function, symmetries of the solution space. Then we will study other hierarchies (the mKP hierarchy, the Toda lattice hierarchy) as well.
讲师
日期
2026年03月09日 至 06月08日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周五 | 09:50 - 11:25 | A3-2-201 | ZOOM 01 | 928 682 9093 | BIMSA |
修课要求
Undergraduate analysis and algebra. The contents of the previous course "Algebraic analysis of integrable hierarchies I" are used, but we will briefly review them at the beginning of this course.
课程大纲
1. Review of the previous semester (definition of the KP hierarchy, the Sato Grassmann manifold, interpretation of the KP hierarchy as a dynamical system on the Sato Grassmann manifold)
2. Bilinear identities of the KP hierarchy (review of Pluecker coordinates and Pluecker embeddings, dual Sato Grassmann manifold, the tau function, various bilinear identities)
3. Symmetry of the KP hierarchy (symmetry on the Sato Grassmann manifold, the Fock space, vertex operators)
4. Infinite dimensional flag manifolds and the mKP hierarchy
5. Toda lattice hierarchy
2. Bilinear identities of the KP hierarchy (review of Pluecker coordinates and Pluecker embeddings, dual Sato Grassmann manifold, the tau function, various bilinear identities)
3. Symmetry of the KP hierarchy (symmetry on the Sato Grassmann manifold, the Fock space, vertex operators)
4. Infinite dimensional flag manifolds and the mKP hierarchy
5. Toda lattice hierarchy
参考资料
Miwa, T., Jimbo, M. and Date, E., Solitons —Differential equations, symmetries and infinite-dimensional
algebras—. Iwanami, Tokyo, (1993), in Japanese; Engl. translation. Cambridge Tracts in Mathematics, 135. Cambridge University Press, Cambridge, (2000).
Sato, M and Noumi, M, Soliton Equations and Universal Grassmann Manifold, (in Japanese), Sophia University, Tokyo, Mathematical Lecture Note 18, (1984)
https://digital-archives.sophia.ac.jp/repository/view/repository/20200107004?lang=en
Sato Mikio Lecture Notes (note taken by Umeda), RIMS Lecture Notes 5, Kyoto University,
https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/215756?locale=en
Takebe, T., A note on modified KP hierarchy and its (yet another) dispersionless limit. Lett. Math. Phys. 59 (2002), 157–172.
Ueno, K. and Takasaki, K., Toda lattice hierarchy, , Adv. Studies in Pure Math. 4 (1984) 1–95.
algebras—. Iwanami, Tokyo, (1993), in Japanese; Engl. translation. Cambridge Tracts in Mathematics, 135. Cambridge University Press, Cambridge, (2000).
Sato, M and Noumi, M, Soliton Equations and Universal Grassmann Manifold, (in Japanese), Sophia University, Tokyo, Mathematical Lecture Note 18, (1984)
https://digital-archives.sophia.ac.jp/repository/view/repository/20200107004?lang=en
Sato Mikio Lecture Notes (note taken by Umeda), RIMS Lecture Notes 5, Kyoto University,
https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/215756?locale=en
Takebe, T., A note on modified KP hierarchy and its (yet another) dispersionless limit. Lett. Math. Phys. 59 (2002), 157–172.
Ueno, K. and Takasaki, K., Toda lattice hierarchy, , Adv. Studies in Pure Math. 4 (1984) 1–95.
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
Takashi Takebe 从事数学物理可积系统方向的研究。 2023年8月前，他在俄罗斯莫斯科国立研究大学高等经济学院数学系担任教授，并于2023年9月加入北京雁栖湖应用数学研究院任研究员一职。