Painlevé Equations and Symmetries – II
This course is a second semester of the introductory course in the modern theory of Painlevé equations with the emphasis on the structure of symmetries (Bäcklund transformations) of these equations and construction of special solutions. Among the topics that we consider are the notion of the Okamoto Space of Initial Conditions, Hamiltonian structure of Painlevé equations, the Isomonodromic Deformations approach, and other selected topics.
Lecturer
Date
9th September ~ 4th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 13:30 - 15:05 | A3-1-103 | ZOOM 04 | 482 240 1589 | BIMSA |
Prerequisite
Undergraduate-level classes in Differential Equations, Complex Analysis, and Abstract Algebra are necessary, some basic knowledge of Algebraic Geometry and Elliptic Functions would be helpful. I will try to make the course as self-contained as possible. Since this course is a continuation (still moderately self-contained) on the Spring semester, I will assume the knowledge of: solving differential equations in the complex domain using the method of Frobenius (including the resonant cases), properties of the Gauss Hypergeometric Equation (including the connection formulae and the computation of the monodromy)
Reference
1. Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida; From Gauss to Painlevé: A Modern Theory of Special Functions; Aspects of Mathematics (ASMA) vol 16, Springer 1991
2. Robert Conte and Micheline Musette; The Painlevé Handbook (Mathematical Physics Studies); 2nd Edition, Springer 2020
3. Valerii I. Gromak, Ilpo Laine, and Shun Shimomura; Painlevé Differential Equations in the Complex Plane (De Gruyter Studies in Mathematics, 28); De Gruyter 2002
4. Masatoshi Noumi; Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004.
5. Kazuo Okamoto; Studies on the Painlevé Equations I-V
6. Kazuo Okamoto; Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé; Japan. J. Math. (N.S.) 5(1979).
7. Kenji Kajiwara, Masatoshi Noumi, and Yasuhiko Yamada; Geometric aspects of Painlevé equations, J. Phys. A 50 (2017), no. 7, 073001, 164.
8. Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
2. Robert Conte and Micheline Musette; The Painlevé Handbook (Mathematical Physics Studies); 2nd Edition, Springer 2020
3. Valerii I. Gromak, Ilpo Laine, and Shun Shimomura; Painlevé Differential Equations in the Complex Plane (De Gruyter Studies in Mathematics, 28); De Gruyter 2002
4. Masatoshi Noumi; Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004.
5. Kazuo Okamoto; Studies on the Painlevé Equations I-V
6. Kazuo Okamoto; Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé; Japan. J. Math. (N.S.) 5(1979).
7. Kenji Kajiwara, Masatoshi Noumi, and Yasuhiko Yamada; Geometric aspects of Painlevé equations, J. Phys. A 50 (2017), no. 7, 073001, 164.
8. Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Anton Dzhamay received his undergraduate education in Moscow where he graduated from the Moscow Institute of Electronics and Mathematics (MIEM) in 1993. He got his PhD from Columbia University under the direction of Professor Igor Krichever in 2000. After having postdoc and visiting positions at the University of Michigan and Columbia University, Anton moved to the University of Northern Colorado, getting tenure in 2011, becoming a Full Professor in 2016, and now transitioning to the Emeritus status in 2025. In 2023–2024 Anton was also a Visiting Professor at BIMSA, he became a permanent BIMSA faculty in Summer 2024 . His research interests are focused on the application of algebro-geometric techniques to integrable systems. Most recently he has been working on discrete integrable systems, Painlevé equations, and applications.