The role of complex singularities in geometric flows

Overview

A popular topic in differential geometry involves studying the singularity structure of geometric flows. The most well-known example is mean curvature flow. In this example, surfaces evolve according to a flow rule that relates the speed of the surface to its curvature.  Certain surfaces will evolve until singularities occur in finite time, and these singularities can be studied using similarity solutions and asymptotic analysis.

In this project, a different perspective is applied to these problems, namely the use of complex variable theory and singularities in the analytic continuation of the mathematical solutions. This work has applications throughout the study of nonlinear partial differential equations.

Research activities

• Revisit well-known examples of geometric flows and simulate various initial conditions.
• Reproduce the analysis to describe certain singularities and match with numerical simulations.
• Reformulate the problems in terms of complex analysis, determine singularities in the complex plane and track their evolution.
• Relate the complex behaviour with the real behaviour.
• Expand the analysis to hold for higher-order flows that mimic crystal growth.

Outcomes

• New insight into well-known geometric flows.
• New mathematical approaches to study nonlinear partial differential equations.
• New results for higher-order geometric flows.

Skills and experience

• Strong background in MXB322 Partial Differential Equations and MXB325 Modelling with Differential Equations 2.
• Preferably, background in complex analysis.

Keywords

• Nonlinear partial differential equations
• Singularities
• Complex analysis

Contact

Contact the supervisor for more information.