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.
Lecturer
Date
9th March ~ 8th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Friday | 09:50 - 11:25 | A3-2-201 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
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.
Syllabus
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
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
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.
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.