Analysis (harmonic, functional, and/or stochastic) of partial differential equations.

How many solutions does the equation x2+y2 = z2 have if x, y, z are taken from Z/pZ? How many square-free polynomials of degree n are there with coefficients in Z/pZ? Questions of this kind have deep and seemingly unexpected connections with the arithmetic and geometry of algebraic varieties.

The discovery of strong 17 per cent circular polarisation in the IR in the Orion OMC-1 star forming region has provided a new impetus to the view that the homochirality of natural amino-acids seen in biological systems on Earth may have been triggered by asymmetric photolysis of biogenic amino acids in space.

Combinatorial and Geometric Group Theory

About 50 percent of stars in our Galaxy occur in binary systems. The orbits of such systems evolve with time due to a variety of mechanisms involving mass and angular momentum loss.

Can one write a computer program to decide whether two given rings are isomorphic? Or whether a given group is the trivial group? Or whether a given CW complex is a sphere? These are questions on the inteface of logic, theoretical computer science, and mathematics with fascinating results and open problems.

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

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

Edge Localised Modes – linear stability and dynamics

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

Explorations in Geometry and Mathematical Physics: From Symplectic Geometry to String Theory

As we move into exascale computing, and beyond, the chance of a fault occurring in the system increases. Traditional approaches to building resilience into the system, such as check-pointing, may become too expensive.

It is proposed that the student presents a review of the theory of black hole formation, and a critical evaluation of the arguments used to deduce their existence in the universe.