Non-linear heat equations have played an important role in differential geometry and topology over the last decades. Broadly speaking, a geometric quantity or structure on a manifold is evolved in a canonical way towards an optimal one, that is, we deform a manifold into another one with nicer properties.

 During the lectures we will focus on the curve-shortening flow and its higher dimensional analogue, the mean curvature flow, which deform a curve

(hypersurface) in its normal direction with speed equal to the curvature (mean curvature) at each point. Analitically, this process is described by a weakly parabolic system of partial differential equations for the local embedding map of the evolving hypersurfaces. At the curvature level it looks like a reaction-diffusion system. The reaction part, which is cubic in the curvatures, generally forces the formation of singularities (points near which the curvature blows up) in finite time. The diffusion part, involving the Laplace-Beltrami operator of the moving hypersurface, shares many properties with the heat equation; in particular, it tends to balance differences e.g. of the curvature on the manifold (so only with the diffusion effect the curvature will eventually tend to a constant). As these two effects are competing, we need a combination of techniques of analysis and geometry to control the behaviour of the flow.

 The lectures will start with the easiest case of evolution of simple closed curves in the plane.