My research is focused on applications of the mathematical language of geometric algebra. This is described in detail on the Cambridge group's website.
Geometric algebra can be applied to any subject in mathematics, physics or engineering which is in some part rooted in geometry. Recent highlights in this research are described here.
A black hole has a spectrum of gravitational bound states in much the same manner as the Hydrogen atom. These have only recently been determined, and are contained in the paper Bound States and Decay Times of Fermions in a Schwarzschild Black Hole Background.
Conformal geometric algebra provides an ideal setting for a range of problems in computational geometry. An introduction to this idea is contained in the paper Conformal Geometry, Euclidean Space and Geometric Algebra.
This method extends projective geometry to treat circles and spheres on the same footing as points, lines and planes. Simple algebraic operations are then used to intersect two spheres, or a line and a sphere. This idea has a range of applications in computer vision and graphics.
A new method for analysing the gravitational field inside and outside an axisymmetric rotation source is developed in the paper New Techniques for Analysing Axisymmetric Gravitational Systems. 1. Vacuum Fields. This raises the possibility of solving one of the long-standing problems in general relativity, that of finding the fields outside a rotating start.
The scattering cross-section for a black hole can be calculated in an
analogous manner to the Mott formula in QED. This calculation was first
described in detail and performed in the paper Perturbation
Theory Calculation of the Black Hole Elastic Scattering Cross Section.