Analysis and Probability seminar

Abstract: The heat equation stands as a cornerstone in mathematics, physics, and various applied fields, describing the distribution of heat (or diffusion of particles) over time. A fundamental question associated with this equation is the Dirichlet problem, which seeks solutions given boundary conditions—a challenge that links the behavior of the heat equation to the geometry of underlying spaces. In recent collaborative work with S. Bortz, S. Hofmann, and J-M. Martell, we address two longstanding conjectures on the L^p Dirichlet problem for the heat equation, caloric measure, and parabolic uniform rectifiability. First, we establish that, for parabolic Lipschitz graphs, the solvability of the L^p Dirichlet problem is equivalent to parabolic uniform rectifiability. Second, we show that, in the broader setting of parabolic Ahlfors-David regular boundaries, the solvability of the L^p Dirichlet problem necessitates parabolic uniform rectifiability. This talk will outline these results, highlighting the interplay between the heat equation and geometric regularity.