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Explore the mathematical foundations and recent advances in Toda Conformal Field Theories through this 56-minute lecture that examines their solvability and connection to broader mathematical frameworks. Delve into how these two-dimensional CFTs extend the well-known Liouville theory while maintaining conformal invariance and enhanced symmetry properties encoded by W-algebras. Discover the rigorous probabilistic framework underlying these models, which incorporates vectorial Gaussian Free Fields and Gaussian Multiplicative Chaos. Learn about cutting-edge research progress in solving these complex mathematical structures, with particular emphasis on translating representation theory results from W-algebras into concrete statements about Toda correlation functions. Understand how this translation provides a systematic approach to addressing the Mukhin-Varchenko conjecture for Selberg-type integrals, drawing from collaborative research findings that bridge abstract algebraic concepts with practical computational methods in conformal field theory.
