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.
Lecturer
Date
9th September ~ 9th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Friday | 09:50 - 11:25 | A3-3-201 | ZOOM 02 | 518 868 7656 | 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
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.