Level: MSci, MSc
Title: Curve shortening flow in the plane
Supervisor:

Dr Reto Buzano
Research Area: Geometry and Analysis
Description:

This project focuses on a different curve shortening process, namely the curve shortening flow, a "geometric heat equation" which deforms a closed curve (in the plane) in such a way that it becomes more and more round (while at the same time shrinking it towards a point). The goal of this project is to study some properties of this flow and reprove the result that any closed curve without self-intersections becomes eventually convex before vanishing to a (round) point. This project is quite a bit harder than the project on Birkhoff's process described above, it is therefore only recommended for particularly strong students. You will have to read actual research papers, follow and understand the computations and restate the results in your own words.
Further Reading:

Literature suggestions will be provided depending on the level of the student and the precise expected outcome (which can be discussed before the start of the project).
Key Modules:

MTH5109 Geometry II: Knots and Surfaces

MTH5104 Convergence and Continuity
Other Information:

A rather solid background in Analysis is required – you will have to understand and possibly reformulate lengthy abstract computations!
Current Availability: Yes