Geometric aspects of partial differential equations

The key aim of Arjun's project is to investigate a selection of tentative conjectures concerning the behaviour of partial differential equations (PDE) that arise specifically in geometry. Such equations appear often to have properties that are much stronger than one would expect from generic PDE theory. One key objective within this research is to understand the well-posedness of mean curvature flow with very rough initial data. A completely novel methodology is being pursued in the lowest possible dimension, involving establishing sharp decay results for the length of curves undergoing curve shortening flow. Analogous results in Ricci flow over recent years have had profound implications in the study of differential geometry. The project fits into both the `Mathematical Analysis' and `Geometry and Topology' EPSRC research areas.