The Curve Shortening Flow
Suitable for: Students with a basic mathematical background, particularly vector calculus and differential equations. Students with greater knowledge of differential geometry will find ample opportunity for deeper investigation leading towards open research problems.
Project description: The Curve Shortening Flow (CSF) is a particular geometric evolution equation that serves as an excellent model for the study of more complex flows and with inherent interest in its own right with applications in geometry and in applied fields. Geometric evolution equations in general have proved an enormously powerful tool in proving geometric and topological theorems resistant to other forms of attack including the striking solution of the Poincare conjecture. A study of the CSF reveals many techniques pertinent to geometric analysis such as used in the study of the Ricci Flow, submanifold flows such as the Mean Curvature Flow and the Inverse Mean Curvature Flow used in the proof of the Penrose inequality, as well as techniques applicable to minimal surface theory and the calculus of variations. At the same time, the CSF is tractable at a fairly elementary level.
Expected outcomes: The student will gain knowledge of standard and novel techniques used in geometric analysis and PDE, particularly how careful use of the maximum principle leads to elegant solutions of a range of problems. There is also scope for numerical analysis for students so inclined who will learn modelling techniques for concrete, nonlinear geometric problems.
Project duration: 6 weeks
Primary Supervisor: Dr Paul Bryan Further information: firstname.lastname@example.org