Heat equations: Geometric methods and applications

This proposal aims to continue a successful direction developed over the last two years to understand the behaviour of heat-type partial differential equations. Our new methods involve control of moduli of continuity, isoperimetric profiles, and related objects using maximum principle arguments, and have already been applied to prove important long-standing conjectures in spectral theory and geometric analysis. We will continue to develop these methods to produce outcomes in understanding the regularity theory of partial differential equations, the behaviour of important geometric evolution equations, and the analysis of eigenvalues.