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Explore the remarkable properties and applications of holomorphic motions in this 42-minute conference talk from IPAM's New Interactions Between Probability and Geometry Workshop. Discover how these mathematical objects, originally introduced by Mane, Sad and Sullivan in the 1980s for holomorphic dynamics, have found applications across diverse areas including Kleinian groups, Teichmüller theory, and quasiconformal mappings. Learn about recent breakthrough results on how various mathematical quantities vary under holomorphic motions, including logarithmic capacity, Minkowski/Packing/Hausdorff dimension, and area measure. Understand how a new unified methodological approach provides fresh insights into celebrated theorems about quasiconformal mappings, particularly Astala's work on area and dimension distortion and Smirnov's research on quasicircle dimensions. Gain insight into cutting-edge research that bridges probability theory and geometric analysis through the lens of complex analysis and dynamical systems.
