Discrete Painlevé Equations
This course is an introduction to the geometric theory of discrete Painlevé equations and, more generally, discrete integrable systems. Discrete Painlevé equations originally appeared as two-dimensional non-linear recurrence relations that have a continuous limit to one of classical differential Painlevé equations. Gradually it became clear that the theory of discrete Painlevé equations is more general and very algebro-geometric -- they are nothing but translations in some birational representations of certain affine Weyl groups. Discrete Painlevé equations and their autonomous limits, the so-called QRT maps, are in fact prototypical examples of two-dimensional discrete integrability. They also plan important roles in various applications, such as the theory of orthogonal polynomials and computation of gap probabilities in certain determinants probabilistic models.
The plan of the course is to start with QRT maps and use them to introduce the main algebro-geometric tools, such as resolutions of singularities of mappings using blowups and linearizing the mapping via its action on the Picard lattice, and then gradually shift to the affine Weyl symmetry groups of certain families of rational algebraic surfaces and their birational representations as a way to obtain discrete Painlevé equations. We special feature of this course is the emphasis on the the geometric aspects of the theory.
The plan of the course is to start with QRT maps and use them to introduce the main algebro-geometric tools, such as resolutions of singularities of mappings using blowups and linearizing the mapping via its action on the Picard lattice, and then gradually shift to the affine Weyl symmetry groups of certain families of rational algebraic surfaces and their birational representations as a way to obtain discrete Painlevé equations. We special feature of this course is the emphasis on the the geometric aspects of the theory.
讲师
日期
2025年02月17日 至 05月28日
位置
Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|
周一,周三 | 13:30 - 15:05 | A3-3-301 | ZOOM 04 | 482 240 1589 | BIMSA |
修课要求
The course is self-contained, but some background in higher-level mathematics is expected. Specifically, we need some basic results from Linear Algebra, Differential Equations, Complex Analysis, Group Theory, and elementary Algebraic Geometry
课程大纲
The QRT mapping as an example of a discrete integrable system
Blowing up as a way to resolve singularities
Algebraic surfaces and the Picard group
Elliptic surfaces and singular fibers
Deautonomization and examples of discrete Painlevé equations
Affine Weyl groups and their birational representations
The period map and the root variables
Differential Painlevé equations and their symmetries
Selected extra topics and applications
Blowing up as a way to resolve singularities
Algebraic surfaces and the Picard group
Elliptic surfaces and singular fibers
Deautonomization and examples of discrete Painlevé equations
Affine Weyl groups and their birational representations
The period map and the root variables
Differential Painlevé equations and their symmetries
Selected extra topics and applications
参考资料
[1] Kenji Kajiwara, Masatoshi Noumi, and Yasuhiko Yamada, Geometric aspects of Painlevé equations, J. Phys. A 50 (2017), no. 7, 073001, 164.
[2] Johannes J. Duistermaat, Discrete Integrable Systems: QRT Maps and Elliptic Surfaces (Springer Monographs in Mathematics) 2010 (NewYork: Springer).
[3] Masatoshi Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004.
[4] Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
[5] Adrian Stefan Carstea, Anton Dzhamay, and Tomoyuki Takenawa, Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations J. Phys. A: Math. Theor. 50 (2017) 405202
[2] Johannes J. Duistermaat, Discrete Integrable Systems: QRT Maps and Elliptic Surfaces (Springer Monographs in Mathematics) 2010 (NewYork: Springer).
[3] Masatoshi Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004.
[4] Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
[5] Adrian Stefan Carstea, Anton Dzhamay, and Tomoyuki Takenawa, Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations J. Phys. A: Math. Theor. 50 (2017) 405202
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
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.