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This lecture is the third part of a series exploring the extension of De Giorgi-Nash-Moser theory to hypoelliptic PDEs in kinetic theory. Dive into the regularity theory of kinetic equations with rough coefficients, building on the groundbreaking work of De Giorgi (1958) and Nash (1959) that solved Hilbert's 19th problem. Explore how this major contribution to 20th century PDE analysis, which established Hölder regularity for elliptic and parabolic PDEs with merely measurable coefficients, is being extended to hypoelliptic PDEs in kinetic theory. Follow Professor Clément Mouhot from the University of Cambridge as he discusses recent advances by researchers including Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-Mouhot-Vasseur, and others. The lecture particularly emphasizes quantitative robust methods based on trajectory construction and their connections to control theory and hypocoercivity, featuring work with collaborators Dieter, Hérau, Hutridurga, Niebel, and Zacher. The prototypical case examined is the Kolmogorov equation (kinetic Fokker-Planck equation) with rough coefficient matrices in kinetic diffusion.
