The classical heat equation describes how a temperature distribution changes in time. Over time, the temperature spreads itself out more and more evenly and the temperature goes to a steady-state equilibrium. There are a number of geometric heat equations, where some geometric quantity evolves over time and - in the best case - approaches an equilibrium. A simple example is the curve shortening flow where a curve in the plane evolves to minimize its length, but other examples include the Ricci flow and the mean curvature flow. All of these flows behave like the classical heat equation for a short amount of time, but they are nonlinear and these nonlinearities dominate over longer time intervals leading to many new phenomena.

I will next give an introduction to mean curvature flow of hypersurfaces. Mean curvature flow is a nonlinear heat equation where the hypersurface evolves to minimize its surface area. The major problem is to understand the possible singularities of the flow and the behavior of the flow near a singularity.

Many physical phenomena lead to tracking moving fronts whose speed depends on the curvature. The level set method has been tremendously succesful for this, but the solutions are typically only continuous. We will discuss results that show that the level set flow has twice differentiable solutions. This is optimal.

These analytical questions crucially rely on understanding the underlying geometry. The proofs draws inspiration from real algebraic geometry and the theory of analytical functions. Further developing these geometric techniques gives solutions to other analytical questions like Rene Thom's gradient conjecture for degenerate equations.

Finally, the techniques should have applications to other geometric flows. If time permits, then we will discuss results about this.

This is joint work with Bill Minicozzi.