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Explore the mathematical foundations of conformal bootstrap theory and its groundbreaking applications in number theory and spectral geometry through this comprehensive lecture. Delve into the algebraic formulation of conformal field theory from a rigorous mathematical perspective, examining how its axioms define precise mathematical structures despite the historical challenges in realizing corresponding objects. Discover the remarkable connection between conformal bootstrap structures and spectral theory of automorphic forms, where the Hilbert space of conformal field theory transforms into a direct sum of automorphic representations and structure constants relate to L-functions. Learn how importing bootstrap methodologies into pure mathematics has generated new progress on significant open problems, including establishing bounds on spectral gaps of hyperbolic manifolds and achieving subconvex bounds on L-functions. Gain insights into cutting-edge research that bridges theoretical physics and advanced mathematics, supported by recent publications and delivered by an expert from IPhT Saclay and IHES.
