Algebraic analysis of integrable hierarchies II
Hi all, the lecture on May 22 will be postponed.
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.
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.
Lecturer
Date
9th March ~ 12th 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 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.