As is classical, the heat equation is a parabolic partial differential equation that dscribes the distribbution of heat (or variation in temperature) in a given region over time. As the prototypical parabolic partial differential equation and via its connection with Brownian motion in probability theory, the heat equation is of fundamental importance in diverse scientific fields, including (Riemannian) geometry, Lie groups, mathematical physics, graph theory, up to mechanics or biology.

During the last decades, the gift of ubiquity of the heat equation has developed to approach various geometric and functional inequalities. Starting from the elementary example of Hölder’s inequality, the talk will feature illustrations of the heat flow method to families of integral inequalities, Sobolev inequalities and more refined isoperimetric theorems in Euclidean and Riemannian spaces. The method developed recently in discrete Boolean analysis motivated by questions in theoretical computer science.