Discrete Integrable Systems and Factorization of Matrices
Anton Dzhamay There are many ways in which one can define "integrability" of a discrete dynamical system, e.g., consistency conditions, algebraic complexity, singularity confinement, Painlevé property. In this talk we explain the Lagrangian approach to integrability, due to J. Moser and A. Veselov, which relates integrable dynamics to the refactorization transformations of matrix functions. We use this approach to obtain the Lagrangian description of discrete Painlevé equations (specifically, dPV). Along the way we explain what discrete Painlevé equations are, why they are interesting, how can they be obtained from the isomonodromy transformations of discrete linear systems (and what is meant by a monodromy in the discrete case), and also the beautiful classification of Painleve equation, due to H. Sakai, through the Cremona transformations of rational surfaces.