A major development in twentieth-century analysis was the discovery that uniformly elliptic and parabolic equations enforce Hölder continuity even when their coefficients are merely measurable. Starting from Hilbert’s nineteenth problem, I will trace the development of two major traditions in regularity theory: the energy methods of De Giorgi, Nash and Moser for equations in divergence form, and the geometric and probabilistic methods of Landis, Aleksandrov, Krylov and Safonov for equations in non-divergence form. I will then describe a new chapter in this story: the extension to kinetic equations combining diffusion in velocity with transport in position. Here, regularity in position must emerge through the interaction of transport and diffusion. I will discuss joint work on kinetic De Giorgi methods and conclude with AI-assisted progress on kinetic Aleksandrov estimates and kinetic Krylov–Safonov theory.

A wine reception in the Central Core will follow this talk.