Geometry of Discrete Integrable Systems and Painlevé Equations

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.

Abstract

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 Intro.

Anton Dzhamay got his undergraduate eduction 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 where he is now a Full Professor. His research interests are focused on the application of algebro-geometric techniques to integrable systems. Most recently he's been working on discrete integrable systems, Painlevé equations, and applications.