Abstract: All Painlevé equations can be written as the motion of a particle under a time dependent potential, and as such they admit a natural generalisation to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). In this talk, I will show that these many-particles Hamiltonian systems admit an isomonodromic formulation, thus answering to a question raised by Takasaki. The isomonodromic formulation can be used, in combination with discrete Schlesinger transforms, to produce solutions; time permitting I will illustrate the method for the case of the second Painlevé equation. This is a joint work with Marco Bertola and Vladimir Roubtsov.
Alexi Morin-Duchesne
