The questions I am interested in mix geometry with partial differential equations.

Geometric evolution equations: here we deform geometric objects in a smooth way to help us understand their shape. This is a very powerful recent technique in geometric analysis, and has been crucial in solving many open geometric questions, including the Poincare conjecture (proved by Perelman).

Eigenvalue estimates: Every geometric object has a set of numbers attached to it, called the spectrum. These are like the resonant frequencies of a drum. I am interested in how the shape of the object affects the spectrum.

Capillary surfaces and the calculus of variations: Consider a meniscus--- the interface between the water in a glass and the air above it. The special shape of the meniscus is due to an interplay between the energy used in the water/glass interface and the air/water interface, and the gravitational energy--- the shape we see will minimise the sum of these energies. It will also dependent on the shape of the glass and the volume of the water. This is a typical problem in the calculus of variations, where minimizing a physical quantity can give rise to interesting geometric shapes.

There are several PhD-suitable projects associated with these topics, please contact me if you are interested.