A fully non-linear flow with surgery for hyper surfaces in a Riemannian manifold

The lecture explains joint work with Simon Brendle on the deformation of hypersurfaces in Riemannian manifolds by a fully non-linear, parabolic geometric evolution system. The surfaces are assumed to satisfy a natural curvature condition ("2-convexity") that is weaker than convexity and move with a speed given by a non-linear mean value of their principal curvatures. It is explained how the possible singularities of the flow can be classified and overcome by surgery to construct a long-term solution of the flow that leads to the classification of all 2-convex surfaces in a natural class of Riemannian manifolds.