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After a year of studying pure Maths and research in molecular Virology during my national service, I read Natural Sciences (biological and physical) and Part III Maths at Cambridge. I went to Harvard on a scholarship, studying computer science and high energy physics and doing research in astrophysics. I did a PhD in theoretical Physics at Cavendish Astrophysics (Kavli Institute for Cosmology), followed by postdocs in Mathematical Virology at York and in Particle Theory at Durham Physics and Maths, before returning to York as a lecturer.

I have served on national panels for the Equality Challenge Unit Athena Swan and Race Equality Chartermarks. After having been the York coordinator for the Pint of Science festival twice, I am now the Chapter manager for York, Hull and Bournemouth.

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 virus interior, 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.

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 forms** **and 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 M_{24} to the elliptic genus of the K3 manifold.

Clifford algebras are very natural objects to consider in the context of reflection groups, or more generally on any vector space with an inner product. They allow a very simple way of performing reflections, as well as a simple construction of spinor groups. For real Clifford algebras, objects with complex and quaternionic algebraic relations arise naturally as different geometric objects such as (collections of) planes and volumes, making a complexification or quaternionification of the whole underlying vector space unnecessary, whilst providing new insight into the real geometry. In particular, such an approach allowed me to construct essentially all the exceptional root systems from the 3D Platonic solids: In general, any 3D root system proves the existence of corresponding root system in 4D, which yields all the exceptional 4D root systems. In particular, via an extension of this idea one can construct the root system of the Lie group E_{8} from the (symmetry group of the) icosahedron. These constructions therefore made new links between exceptional objects in Mathematics, akin to Arnold's Trinities and the McKay correspondence, relating 3D root systems, 4D root systems and (affine) Lie algebras.
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Mathematical Physics Research Group

Mathematical Biology and Chemistry

Applications of Group Theory to Virology

Vector Calculus

Principles of Molecular Virology

Science Communication