I am interested in geometry and symmetry (in particular reflection groups) and their applications in the natural sciences.
My PhD considered spatially homogeneous but anisotropic spacetimes, so-called Bianchi models where the homogeneity symmetry of the universe is modelled by a three-dimensional Lie group manifold, and its general relativistic dynamics with different matter content such as vacuum, dust, radiation or a scalar field. I am particularly interested in singularities and how to avoid them - which turns out is rather difficult. As Hawking and Penrose showed, singularities generically exist, and more recently in very general circumstances their analytical structure has been understood better in terms of hyperbolic Coxeter (reflection) groups and Kac-Moody algebra symmetries of the underlying gravitational theories.
The structure of small viruses is essentially a geometric problem, owing to their genomic simplicity. For instance, certain viruses only have one structural gene. Therefore, the protein capsid which protects the viral nucleic acid needs to consist of identical building blocks, which have exactly the right shape and edge interactions such that multiple copies of this building block can form a spherical shell. This problem is therefore related to the Platonic solids and their polyhedral (reflection) symmetries, in particular the symmetry group of the icosahedron. Caspar-Klug theory and viral tiling theory consider icosahedral tesselations of the sphere, which model viral surface structure organisation. We have developed frameworks that could describe not only the surface structure but also the virusinterior, the physical motivation for this via a generalised symmetry principle as well as other applications of this new principle in the context of fullerenes (carbon onions). The relevant mathematical concepts are root systems, Coxeter (reflection) groups, Kac-Moody algebras, affine extensions and (quasi)lattices.
High Energy Physics
I am interested in the symmetry group of the Standard Model of particle physics, Lie Groups, spinors etc as well as possible extensions of the Standard Model such as family symmetries, for which the polyhedral symmetries are contenders. Another interest is Moonshine, which denotes a surprising connection between two very different areas of mathematics, those of modular formsand finite simple groups. The original Moonshine observation was by John McKay in the 1980's in the context of the largest sporadic finite simple group, the Monster group, and a modular form, called the j(τ) function; Mathieu Moonshine is a recent observation (2010) in the context of string theory, relating the sporadic finite simple group Mathieu M24 to the elliptic genus of the K3 manifold.