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Explore advanced mathematical concepts in this comprehensive workshop covering nonlocal partial differential equations and their applications across geometry, physics, and probability theory. Delve into specialized topics including plasma collisions through the Landau equation, second-order conformally invariant elliptic equations, and quantization of measures using gradient flow approaches. Study entropy methods for hypocoercive BGK and Fokker-Planck equations, regularity theory for the Boltzmann equation in bounded domains, and information theoretic inequalities for stable densities. Examine propagation of chaos for homogeneous Landau equations, construction of weak solutions through approximation methods, and gradient flow structures for quantum evolution equations. Investigate global regularity of quadratic diffusion-reaction systems, nonlocal equations from multiple perspectives, and the Kac master equation with entropy considerations. Learn about nonlinear fractional parabolic equations in bounded domains, curves and surfaces with constant nonlocal mean curvature, and nonlinear aggregation-diffusion processes in various regimes. Master fractional Poincaré inequalities on manifolds, nonlinear tools in fractional settings, and Villani's constructive approach to equilibrium convergence rates. Conclude with an in-depth analysis of the De Giorgi conjecture for the half-Laplacian in dimension 4, providing a thorough foundation in cutting-edge mathematical research at the intersection of analysis, geometry, and mathematical physics.
