Geometric Flows

Summary

Geometric flows (such as Ricci flow, mean curvature flow, inverse mean curvature flow...) are extremely powerful tools which have been used to solve hard open conjectures such as the Poincaré conjecture. They have also provided beautiful proofs of important results such as the differentiable sphere theorem and the Penrose inequality.

Roughly speaking under such a flow, a geometric structure (e.g. a surface or a metric) is deformed over time according to a geometric partial differential equation, hopefully improving the geometric object considered, but possibly also encountering singularities. The analysis and understanding of such singularities guide the application of a given flow. This is an exciting field to work in and significant further applications are to be expected in diverse areas of geometry and topology.

There are many possible starting points for a project in this area, but the overarching theme of this project will be to investigate and understand the properties of an extrinsic geometric flow such as mean curvature flow, inverse mean curvature flow, Gauss curvature flow or symmetric curvature polynomial flows.

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.