Hamiltonian Structure of Rational Isomonodromic Deformation Systems

Explore the Hamiltonian structure of rational isomonodromic deformation systems in this mathematical physics seminar. Delve into the world of nonlinear differential equations generating isomonodromic deformations of linear systems with isolated singular points, including the famous Painlevé transcendents. Discover their applications in various physics domains, such as partition function calculations, dimensionally reduced quantum gravity, random matrix spectral distributions, and field theories. Examine the intricate relationship between Hamiltonian structure and integrability properties of these systems. Gain insights into the isomonodromic deformation dynamics for generic rational covariant derivative operators on the Riemann sphere with irregular singularities. Understand how these dynamics relate to isospectral Hamiltonian systems and explore the role of deformation parameters as Casimir elements. Investigate the connection between Birkhoff invariants, spectral invariant Hamiltonians, and the fundamental meromorphic differential on the associated spectral curve.