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MA953 - Topics in Partial Differential Equations

Lecturer: Felix Schulze

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Delivery: Via Teams. The lectures will be Monday 13-14, Tuesday 11-12, Thursday 11-12. The first lecture will be on 11 January 2021. If you have signed up for the course you will receive an invitation via Teams. If you have not signed up for the course, or are from outside Warwick, please contact the lecturer directly.

Course page and updated course materials: These can be found on the lecturers personal homepage.

Prerequisites: Fundamentals on the geometry of hypersurfaces are needed, as taught in MA6C0 Differential Geometry. It will be helpful to have some background on elliptic and parabolic PDE, see for example MA6A2 Advanced PDEs.

Content: Mean curvature flow is the prototype of an extrinsic curvature flow, with its intrinsic cousin being the Ricci Flow. This course will be introduction to mean curvature flow, trying to give students a direct route to current developments in the field. I will try to cover the by now classic results on curve shortening flow and convex mean curvature flow, some of the surgery techniques for mean convex mean curvature flow as well as an introduction to weak mean curvature flow and its existence theory. The course is partially based on a course I gave 3 years ago at the LSGNT, with added material. The contents below are subject to change:

  • Maximum principles for scalar functions and tensors
  • Huisken's monotonicity formula
  • Evolution of closed curves in the plane
  • Evolution of closed, convex hypersurfaces
  • Weak compactness of submanifolds
  • Brakke flows
  • Ilmanen's compactness theorem for integral Brakke flows
  • Existence via elliptic regularisation
  • Stratification

References:

My lecture notes for the LSGNT course can be found here. Other resources are

  • B. White, Topics in mean curvature flow, lecture notes by O. Chodosh. Available here.
  • K. Ecker, Regularity theory for Mean Curvature Flow, Birkhäuser
  • R. Haslhofer, Lectures on curve shortening flow. Available here.
  • R. Haslhofer, Lectures on mean curvature flow. Available here.