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 joined BIMSA as a professor in September 2023. Before coming to BIMSA he worked as a professor at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow, Russia, from April 2009 till August 2023, as an associate professor at the Department of Mathematics, the Faculty of Science, of Ochanomizu University in Tokyo, Japan, from April 1999 till March 2009, and as an assistant professor at the Graduate School of Mathematical Sciences of the University of Tokyo, Japan, from October 1991 till March 1999.
He also performed his research at the University of California, Berkeley, USA, from September 1995 till August 1997, supported by the Overseas Research Fellowships of the Japan Society for the Promotion of Science (JSPS).
He graduated the Department of Mathematics, the Faculty of Science, of the University of Tokyo, Japan in March 1987 and was a graduate student of the Graduate School of Sciences from April 1987 till September 1991. He got the degree of Ph.D. (Mathematical Sciences) in 1995.
He also studied in Leningrad, the USSR, from October 1990 till September 1991 as an exchange student between the USSR and Japan.
He also performed his research at the University of California, Berkeley, USA, from September 1995 till August 1997, supported by the Overseas Research Fellowships of the Japan Society for the Promotion of Science (JSPS).
He graduated the Department of Mathematics, the Faculty of Science, of the University of Tokyo, Japan in March 1987 and was a graduate student of the Graduate School of Sciences from April 1987 till September 1991. He got the degree of Ph.D. (Mathematical Sciences) in 1995.
He also studied in Leningrad, the USSR, from October 1990 till September 1991 as an exchange student between the USSR and Japan.