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Explore the mathematical foundations of Schramm Loewner evolution (SLE) and its applications to multiple random curves in this 50-minute lecture. Delve into Oded Schramm's groundbreaking 1999 introduction of SLE as a non-self-crossing random curve driven by Brownian motion through Loewner's transform, originally developed to mathematically describe random interfaces in 2D critical lattice models. Examine how these models satisfy both conformal invariance and the domain Markov property. Focus on the law of multiple curves that maintain conformal invariance and domain Markov property, utilizing Dubédat's commutation relation framework. Gain insights into the sophisticated mathematical structures underlying critical phenomena in statistical mechanics and their connection to complex analysis through this advanced mathematical exposition.
