Room 4336, Pav. André-Aisenstadt, 2920, ch. de la Tour, CA
Title:Noncommutative Painlevé equations and systems of Calogero type.
Abstract: The Calogero (Moser(Sutherland)) system is an autonomous integrable Hamiltonian system of n particles on the line interacting with inverse square potential (or Weierstrass-P function). The non-interacting part is a classically integrable Hamiltonian (e.g. the harmonic oscillator). The principal goal of the talk is to explain how the integrability survives if we replace the single-particle Hamiltonian by any of the Hamiltonian for the six Painlevé equations, hence turning the system into a time-dependent, interacting dynamics. In this case, as we would expect, the isospectral character of the associated Lax system, is replaced by an isomonodromic evolution that linearizes non-commutative versions (i.e. matrix-valued) of the six Painlevé equations. This solves a conjecture posed by T. Takasaki in 2010.