Its, AlexanderProkhorov, AndreiBleher, PavelEremenko, AlexandreTarasov, Vitaly2019-07-182019-07-182019-08https://hdl.handle.net/1805/19905http://dx.doi.org/10.7912/rygf-2h27http://dx.doi.org/10.7912/C2/2405Indiana University-Purdue University Indianapolis (IUPUI)We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.en-USAttribution 3.0 United Statesisomonodromic tau functionconnection problemHamiltonian systemsclassical actionRiemann-Hilbert correspondenceisomonodromic deformationsquasihomogeneous functionsPainlevé equationsThesis10.7912/rygf-2h27