Geometry of Discrete Integrable Systems: QRT Maps and Discrete Painlevé Equations

Abstract

Many interesting examples of discrete integrable systems can be studied from the geometric point of view. In this talk we will consider two classes of examples of such system: autonomousand non-autonomousé. We introduce some geometric tools to study these systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and construction of a birational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic. This talk is based on joint work with Stefan Carsteaand Tomoyuki Takenawa.

Speaker Intro

Anton Dzhamay got his undergraduate eduction in Moscow where he graduated from the Moscow Institute of Electronics and Mathematicsin 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.