Beijing Institute of Mathematical Sciences and Applications

This course is an introduction to the geometric theory of discrete Painlevé equations and, more generally, discrete integrable systems. Discrete Painlevé equations originally appeared as two-dimensional non-linear recurrence relations that have a continuous limit to one of classical differential Painlevé equations. Gradually it became clear that the theory of discrete Painlevé equations is more general and very algebro-geometric -- they are nothing but translations in some birational representations of certain affine Weyl groups. Discrete Painlevé equations and their autonomous limits, the so-called QRT maps, are in fact prototypical examples of two-dimensional discrete integrability. They also plan important roles in various applications, such as the theory of orthogonal polynomials and computation of gap probabilities in certain determinants probabilistic models.

The plan of the course is to start with QRT maps and use them to introduce the main algebro-geometric tools, such as resolutions of singularities of mappings using blowups and linearizing the mapping via its action on the Picard lattice, and then gradually shift to the affine Weyl symmetry groups of certain families of rational algebraic surfaces and their birational representations as a way to obtain discrete Painlevé equations. We special feature of this course is the emphasis on the the geometric aspects of the theory.