Algebraic analysis of integrable hierarchies I
Among integrable systems the KP hierarchy and its cousins are quite important. The solution spaces of them are
(infinite dimensional) algebraic varieties and have infinite dimensional symmetries. In this lecture, following
Sato's original idea, we show how the solution space of the KP hierarchy is related to an infinite dimensional
Grassmann manifold. If the time permits, we will study other hierarchies (the mKP hierarchy, the Toda lattice
hierarchy) as well.
(infinite dimensional) algebraic varieties and have infinite dimensional symmetries. In this lecture, following
Sato's original idea, we show how the solution space of the KP hierarchy is related to an infinite dimensional
Grassmann manifold. If the time permits, we will study other hierarchies (the mKP hierarchy, the Toda lattice
hierarchy) as well.
Lecturer
Date
15th September ~ 12th December, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Friday | 09:50 - 11:25 | A3-2-201 | ZOOM 07 | 559 700 6085 | BIMSA |
Prerequisite
Undergraduate analysis and algebra
Syllabus
1. Introduction (incomplete history of soliton theory)
2. Microdifferential operators
3. KP hierarchy
4. Finite dimensional Grassmann manifolds and differential equations
5. Sato Grassmannian
6. Sato Grassmannian and microdifferential operators
7. KP hierarchy as a dynamical system on the Sato Grassmannian
2. Microdifferential operators
3. KP hierarchy
4. Finite dimensional Grassmann manifolds and differential equations
5. Sato Grassmannian
6. Sato Grassmannian and microdifferential operators
7. KP hierarchy as a dynamical system on the Sato Grassmannian
Reference
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
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
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Takashi Takebe is a researcher of mathematical physics, in particular integrable systems. He worked as a professor at the faculty of mathematics of National Research University Higher School of Economics in Moscow, Russia, till August 2023 and joined BIMSA as a professor in September 2023.