2 p.m. - 3 p.m. Location: JO 4.614
University of Northern Colorado
The Beautiful Geometry of Discrete Painlevé Equations
Discrete Painlevé equations (dPe) is a special class of non-autonomous discrete integrable systems. They originally appeared (in the theory of orthogonal polynomials and in theory of quantum gravity) as nonlinear second-order recursions that converge to classical differential Painlevé equations in some continuous limits. Further, dPe describe discrete symmetries of differential Painlevé equations. Many examples of dPe appear as deautonomizations of autonomous discrete integrable mappings, such as the QRT mappings. Similar to classical Painlevé equations, discrete Painlevé equations describe properties of discrete analogues of integrable models of random matrix type and so they play an important role in the emerging field of integrable probability. Geometric theory of differential Painlevé equations has been initiated by K. Okamoto in 1980s and it had been extended to the discrete case by H. Sakai in 2000. Sakai proposed a classification scheme for discrete Painlevé equations that is based on the birational geometry of rational algebraic surfaces equipped with actions of affine Weyl groups (a generalization of the classical theory of del Pezzo surfaces). These geometric description turned out to be the key to understanding properties of discrete Painlevé equations and for construction of various families of special solutions. The goal of my talk would be to give an accessible introduction into this beautiful class of ideas.
Sponsored by the Department of Mathematical Sciences