Singularities and Surgery in Geometric Evolution Equations

Curvature-driven evolution equations for surfaces arise in models of crystal growth, moving phase boundaries, flame propagation and oil-water boundaries, and are also useful as tools within mathematics. This project will introduce innovative new techniques for investigating the singularities of a broad range of curvature flows for which the current knowledge is very incomplete. We will give a detailed analytic and geometric description of the regions of high curvature, and use this knowledge to artificially excise singularities as they appear. By controlling the resulting topological changes we will deduce new geometric inequalities, and topological classifications of hypersurfaces satisfying various curvature conditions.