Global Lie Theory: Wild Riemann Surfaces and Their Character Varieties

Explore the evolution of topological symplectic structures and their applications in global Lie theory through this comprehensive lecture. Delve into the construction and development of these structures, from their origins in the late 1990s to the general, purely algebraic approach of the 2010s. Examine the generalization of Riemann surfaces to wild Riemann surfaces and their character varieties. Investigate the connections between these symplectic varieties and quantum groups, including the Drinfeld-Jimbo quantum group and G-braid group actions. Learn about the classification of these varieties as global analogues of Lie groups and the emerging theory of Dynkin diagrams. Discover how these two-forms integrate with the Bottacin-Markman Poisson structure on meromorphic Higgs bundle moduli spaces to create wild nonabelian Hodge hyperkähler manifolds. Explore the relationship between these hyperkähler metrics and special Lagrangian fibrations, as well as their connection to Painlevé equations and classical integrable systems. Gain insights into the applications of these concepts in theoretical physics, including Seiberg-Witten theory and the Gaudin model. Uncover the historical context and mathematical developments that link these ideas to abelian functions, spectral curves, and harmonic theory.