Geometry of Discrete Integrable Systems and Painlevé Equations
This course is an introduction to some modern ideas of the geometric approach to two-dimensional discrete integral systems. We focus on two main classes of such systems — the QRT maps that are autonomous discrete systems, and discrete Painlevé equations, that are non-autonomous. We see how resolving indeterminacies of the dynamics naturally leads to appearance of rational algebraic surfaces as configuration spaces. The dynamics then defines a linear map of the Picard lattices of the surfaces, which contains a lot of information about the system. In particular, in the discrete Painlevé case this linear map completely characterizes the equation via binational representations of certain affine Weyl groups. We explain the relationship between differential and discrete Painlevé equations and the geometric classification scheme of Painlevé equations due to H. Sakai. Finally, we show how the geometric approach can be used in some applications in the theory of orthogonal polynomials and in integrable probability.
Lecturer
Date
27th February ~ 24th May, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 13:30 - 15:05 | A3-1-103 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
The course is designed to be essentially 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.
Syllabus
Topic I: The QRT mapping as an example of a discrete integrable system
1. Involutions on a biquadratic curve on P1 x P1.
2. Pencil of biquadratic curves and the definition of a QRT dynamics.
3. Basepoints and their resolution using the blowup procedure.
4. An example of a QRT map.
5. Rational elliptic surfaces. Regular and singular fibers.
6. Divisors and divisor classes. The Picard lattice of a QRT surface.
7. Linearization of the dynamic on the Picard lattice.
Topic II: Deautonomization of a QRT mapping and examples of discrete Painlevé equations
1. Deautonomization of a QRT mapping along a chosen fiber. Generalized Halphen surfaces.
2. The anti-canonical divisor class. The surface and the symmetry sub-lattices of the Picard lattice.
3. Affine Cartan matrices and Dynkin diagrams.
4. The Period Mapping and the root variables.
5. Discrete Painlevé dynamic on the root variables and symmetry root lattices.
Topic III: From birational representations of affine Weyl groups to discrete Painlevé equations
1. Reflection action of symmetry roots on the Picard lattice and induced birational maps.
2. Affine Weyl groups and extended affine Weyl groups.
3. Translation elements in affine Weyl groups and their decompositions.
4. Constructing discrete Painlevé equations from the algebraic data.
Topic IV: Differential Painlevé equations and their symmetries.
1. Hamiltonian form of Painlevé equations. The Okamoto space of Initial Conditions.
2. Bäcklund transformations and symmetry groups of Painlevé equations.
3. Translation elements and discrete Painlevé equations.
Topic V: Point configurations and Sakai's classification scheme
1. Eight-point configurations on P1 x P1 and their degenerations.
2. The Sakai geometric classification scheme. Elliptic, multiplicative, and additive discrete Painlevé equations.
3. Continuous limits.
Topic VI: Selected extra topics
1. Involutions on a biquadratic curve on P1 x P1.
2. Pencil of biquadratic curves and the definition of a QRT dynamics.
3. Basepoints and their resolution using the blowup procedure.
4. An example of a QRT map.
5. Rational elliptic surfaces. Regular and singular fibers.
6. Divisors and divisor classes. The Picard lattice of a QRT surface.
7. Linearization of the dynamic on the Picard lattice.
Topic II: Deautonomization of a QRT mapping and examples of discrete Painlevé equations
1. Deautonomization of a QRT mapping along a chosen fiber. Generalized Halphen surfaces.
2. The anti-canonical divisor class. The surface and the symmetry sub-lattices of the Picard lattice.
3. Affine Cartan matrices and Dynkin diagrams.
4. The Period Mapping and the root variables.
5. Discrete Painlevé dynamic on the root variables and symmetry root lattices.
Topic III: From birational representations of affine Weyl groups to discrete Painlevé equations
1. Reflection action of symmetry roots on the Picard lattice and induced birational maps.
2. Affine Weyl groups and extended affine Weyl groups.
3. Translation elements in affine Weyl groups and their decompositions.
4. Constructing discrete Painlevé equations from the algebraic data.
Topic IV: Differential Painlevé equations and their symmetries.
1. Hamiltonian form of Painlevé equations. The Okamoto space of Initial Conditions.
2. Bäcklund transformations and symmetry groups of Painlevé equations.
3. Translation elements and discrete Painlevé equations.
Topic V: Point configurations and Sakai's classification scheme
1. Eight-point configurations on P1 x P1 and their degenerations.
2. The Sakai geometric classification scheme. Elliptic, multiplicative, and additive discrete Painlevé equations.
3. Continuous limits.
Topic VI: Selected extra topics
Reference
[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] 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
[4] Masatoshi Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004.
[5] Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
[6] Igor R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, 3rd edn (2013)(Heidelberg: Springer)
[2] Johannes J. Duistermaat, Discrete Integrable Systems: QRT Maps and Elliptic Surfaces (Springer Monographs in Mathematics) 2010 (NewYork: Springer).
[3] 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
[4] Masatoshi Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004.
[5] Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), no. 1, 165–229.
[6] Igor R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, 3rd edn (2013)(Heidelberg: Springer)
Audience
Graduate
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.