Lecture 1: Geometric heat equations

Starting from the linear heat equation the lecture introduces geometric heat equations such as the curve shortening flow, the mean curvature flow of hypersurfaces and the Ricci flow of Riemannian metrics. It is shown how these quasi-linear parabolic systems can be used to deform geometric objects into more uniform, recognizable shapes, leading to classification results such as the proof of the Poincaré conjecture. Particular emphasis will be on the necessary interplay between geometric concepts and analytical estimates.