Scientists / Open questions
Cédric Villani
Partial differential equations; optimal transport; kinetic theory
2010 Fields Medal
5 questions
Completing the Bolley–Guillin–Villani quantitative concentration program: dependent data, time discretization, phase-space truncation, and unbounded interaction curvature
Convexity of injectivity domains from the Ma–Trudinger–Wang condition in the focal case, and the attendant stability and second-order structure questions
Fine structure of the optimal trace-Sobolev curve : attainment, rigidity of the ball comparison, endpoint asymptotics, and the combined inequality (Maggi–Villani open problems)
Continuity of optimal transport maps on Riemannian manifolds: the (CI)+(MTW) characterization beyond dimension two, transport continuity on products of small perturbations of the 2-sphere, and the (MTW) versus hierarchy
Well-posedness, smoothing and rates of H-solutions for the spatially homogeneous non-cutoff Boltzmann and Landau–Coulomb equations, and their microscopic derivation (open problems of Villani, 1998)