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Title:Noncommutative Painlev茅 equations and systems of Calogero type.

础产蝉迟谤补肠迟:听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 Painlev茅 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 Painlev茅 equations. This solves a conjecture posed by T. Takasaki in 2010.