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Explore the fascinating connections between probability theory and differential geometry in this Bourgain Lecture that examines how random curves bridge universality phenomena and Kähler geometry. Discover how universality in probability theory leads to common limits for random processes despite microscopic differences, exemplified by random walks converging to Brownian motion regardless of individual step laws. Learn about the Schramm-Loewner Evolution (SLE) as the universal scaling limit of interfaces in conformally invariant 2D systems and its central role in 2D random conformal geometry, probabilistic approaches to 2D quantum gravity, and conformal field theory. Investigate the Weil-Petersson Teichmüller space formed by relatively regular simple curves and its unique Kähler geometry described through group structures and infinitesimal curve variations. Understand how these seemingly disparate mathematical worlds connect through the Loewner energy, which serves as a bridge between probabilistic and geometric perspectives. Examine recent developments linking Loewner energy to renormalized volume in hyperbolic 3-space, motivated by the AdS/CFT holographic principle, and explore applications and future directions in this emerging interdisciplinary field that combines probability theory, complex analysis, and differential geometry.
