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Explore the mathematical properties of diffusion transport maps through the lens of log-semiconcavity creation in this 30-minute conference lecture. Delve into the fundamental challenge of finding regular transport maps between measures, a crucial task in generative modeling and functional inequality transfer. Learn about Caffarelli's contraction theorem, which demonstrates that optimal transport maps from Gaussian to uniformly log-concave measures are globally Lipschitz, and understand why optimality is not essential for this analysis. Examine alternative transport maps derived from diffusion processes, as pioneered by Kim and Milman, and discover how these approaches expand beyond traditional optimal transport frameworks. Focus on the establishment of lower bounds for log-semiconcavity along heat flow for asymptotically log-concave measures, and understand how these bounds lead to Lipschitz constraints for heat flow maps. Gain insights into the stability implications of these mathematical constructions and their broader applications in probability theory and analysis. The presentation draws from collaborative research with Louis-Pierre Chaintron and Giovanni Conforti, offering cutting-edge perspectives on probabilistic mass transport theory.
