Ruijsenaars duality in many-particle analogs of the Painlevé equation

The talk will discuss duality in many-particle analogs of the Painlevé equation introduced by K.Takasaki. Recently, M.Bertola, M.Cafasso, and V.Rubtsov provided an isomonodromic description for given system using the procedure of Hamiltonian reduction. In the talk I will tell about this reduction, with simplest example being self-dual rational Calogero-Moser model. Then basic notions of the theory of the Painlevé equations and monodromic deformations, as well as classical Calogero-Painlevé correspondence for 2-particle systems will be introduced. Further, we will show how to extend this correspondence on the case of many-particle systems using the procedure of Hamiltonian reduction for matrix analogs of the Painlevé equations and which dual systems appear from applying this procedure to marix Painlevé equations. At the end I will discuss relation of many-particle Calogero-Painlevé correspondence to reductions matrix integrable equations in partial derivatives by means of an example of matrix mKdV. Joint work with V.Rubtsov.