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Explore advanced concepts in conformal field theory through this comprehensive lecture delivered at the Galileo Galilei Institute, covering sophisticated mathematical frameworks and theoretical principles that govern scale-invariant quantum field theories. Delve into the intricate mathematical structures underlying conformal transformations, correlation functions, and operator product expansions while examining their applications in theoretical physics. Master the computational techniques essential for analyzing conformal field theories, including the conformal bootstrap approach and its role in constraining physical theories. Investigate the deep connections between conformal field theory and critical phenomena, string theory, and condensed matter physics through rigorous mathematical derivations and physical interpretations. Learn to apply conformal mapping techniques and understand the role of central charges in characterizing different conformal field theories. Examine specific examples of conformal field theories and their realizations in various physical systems, from statistical mechanics models to quantum gravity applications.
