# Ancient solutions to fully nonlinear curvature flows

We will consider time-dependent hypersurfaces in Euclidean space evolving by fully nonlinear, curvature-driven diffusion equations. These generalise the mean curvature flow. Solutions to such equations are called ancient if they are defined for all negative times, and we expect ancient solutions to satisfy rigidity results, since at any instant one might consider, diffusion has already had an infinite amount of time in which to act. Our focus will be the proof of one such rigidity result, which says that uniformly convex ancient solutions must in fact be shrinking spheres, and emphasis will be placed on the interplay between geometry and analysis which makes such problems interesting.