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This lecture explores the probabilistic approach to Conformal Field Theories (CFT), which are believed to describe universal behavior of physical systems at second order phase transition points. Delve into the mathematical foundations of CFTs, which play a crucial role in Quantum Field Theories by describing properties at small and large length scales. Learn about the rich mathematical structure of two-dimensional CFTs first uncovered by physicists Belavin, Polyakov, and Zamolodchicov in 1983, and understand how their path integral formulation connects to Graeme Segal's geometric approach to conformal bootstrap. The lecture particularly focuses on two prominent CFTs: Liouville CFT, central to Liouville Quantum Gravity and random surfaces theory, and Wess-Zumino-Witten models with their rich representation theoretical content. This is the first part of a series presented by Antti Kupiainen at the Hausdorff Center for Mathematics.
