Irregular isomonodromic deformations: Hamiltonian aspects and algebraic description of the phase space

This thesis is dedicated to the isomonodromic deformation equations on the Riemann sphere with the punctures of an arbitrary Poincaré rank (regular and irregular isomonodromic problems). Such deformations are closely related to the Painlevé equations and Garnier systems, as well as to the moduli space of flat connections over the Riemann sphere with boundaries. In this thesis we investigate the symplectic and Poisson aspects of such systems. For an arbitrary isomonodromic problem we introduce the universal un-reduced phase space, which gives a unique approach for the reduction and can be used to obtain the Hamiltonians for the reduced deformation equations in a systematic way (we provide an example of the Okamoto Hamiltonians for the classical Painlevé equations). Moreover we give an algebraic description of the phase space for the cases of the classical Painlevé equations. We also investigate the quantization of the universal un-reduced phase space and it's relation to the quasi-classical solutions of the \(KZ\) equations written in the terms of the classical isomonodromic \(\tau\)-function.