Seminario "Hamiltonian structure of rational isomonodromic deformation systems"

Isomonodromic deformations are the linearized description of several interesting nonlinear ODEs and PDEs, chief amongst which the six Painlevé equations that found application in the last decades to a  wide range of problems, from condensed matter to number theory.

Their Hamiltonian structure has been investigated thoroughly, and yet the literature (and even the experts in the field) is often less than clear on certain structural issues, for example on the question of how, exactly, the Painlevé equations (and more generally the isomonodromic systems) are “de-autonomizations” of corresponding autonomous classical integrable systems. 

In the talk I will clarify the nature of the isomonodromic deformations as the sum of a Hamiltonian vector field and an additional term which determines a Poisson flat connection that can be assimilated to, or thought of as, the "explicit derivative'' with respect to the Birkhoff invariants. This implies the existence of a quotient manifold of the space of rational matrices by the action of this flat connection.