Parabolic equations applied to deform geometric objects such as metrics, surfaces, curves, or maps. Such equations arise naturally in many contexts:  Well-known elliptic problems in geometry (such as the minimal surface equation and the harmonic map equation) have natural parabolic analogues, and frequently the parabolic approach is a useful way of exploring these elliptic problems. The parabolic flows also arise, for example, in models of phase boundaries. Finally, these equations have useful properties which make them highly applicable both in practical problems (image processing, for example) and in proving results in other areas of mathematics. They have been used to prove new results in global differential geometry, as well as isoperimetric inequalities of various kinds.  Recently they were used by Perelman in his proof of the Poincaré conjecture.

We will begin by introducing some differential geometry and summarising some of the important results in global differential geometry. Then we will introduce some of the important examples of geometric evolution equations, such as the motion of hypersurfaces by their mean curvature and the deformation of Riemannian metrics by their Ricci curvature.  We will develop techniques to understand the behaviour of these equations, and treat numerous examples of the application of flows to global differential geometry. We will aim to finish with a sketch of Perelman's proof of the Poincaré conjecture.