Controlled magnetic confinement fusion offers the possibility of an inexhaustible supply of energy with zero greenhouse gas emissions.

The problem of numerically computing eigenvalues and eigenfunctions of the Laplacian, with Dirichlet (zero) boundary conditions, on a plane domain, is computationally intensive and there is a lot of theory behind finding efficient algorithms.

Investigate the relation between group theory and repeating 'crystal' patterns in 1, 2, and 3 dimensions.

The curve shortening flow is a simple and beautiful example of a geometric heat flow, the family of equations which includes the Ricci flow used by Perelman to prove the Poincare conjecture as well as many other interesting examples. 

Curve shortening flows

Clifford algebras and Clifford modules; Spin structures and Dirac operators, their geometric properties, and some examples, possibly including Witten's proof of the positive mass theorem.

The discontinuous Galerkin (DG) method is now an established method for computing approximate solutions of partial differential equations in many applications.

A possible application area for this project is Stellar Astronomy. The project will involve both computational and mathematics aspects, but the focus will depend on the student's interest.

Edge Localised Modes – linear stability and dynamics

Suppose a domain with smooth boundary is given. It turns out that there is a discrete set of eigenfunctions which can be arranged in a sequence for which there is a nontrivial solution to the eigenfunction equation...

On a (compact) Riemannian manifold there is a natural differential operator on functions, the Laplacian. The eigenvalues of this operator are important invariants of the manifold, and there are many interesting results which relate the eigenvalues to other geometric quantities...

Fusion energy promises baseload electricity generation with zero greenhouse gas emissions, a virtually inexhaustible supply of fuel, and significantly reduced radioactive waste, compared to fission and coal.

Monoidal categories describe the "quantum symmetries" of 2-dimensional topological phases of matter. Recently, it's been realised that enriched monoidal categories provide a useful model for 2-dimensional phases at the boundary of a 3-dimension phase.

In ITER, broken toroidal symmetry is introduced deliberately, through the use of resonant magnetic perturbation (RMP) coils, to suppress large explosive instabilities known as edge localised modes (ELMs). It is crucial to evaluate the equilibrium and stability of magnetic field configurations with RMP for ITER scenario

Evaluation of hydrological models