I would be particularly interested in receiving PhD applications from people in under represented (in mathematics) social groups and under represented world regions.

If you are thinking about studying for a PhD and would like to know the sort of research I do, here is a general idea. Take a look at my papers to get a more precise idea.

 Geometric, spectral and topological index theory lie at an intriguing point of interaction of a number of
areas of pure mathematics and theoretical physics. Topological index theory over a closed manifold M is
essentially concerned with the identification of certain fundamental topological and holomorphic invariants
of bundles over M as indices of naturally associated elliptic differential operaotors. The Atiyah-Singer
Index Theorem gives a precise statement of this relation, generalizing classical integrality theorems such as
the Riemann-Roch Theorem and the Gauss-Bonnet Theorem. Perhaps most importantly, these are
computable invariants, and hence form very useful tools in analysis, topology and theoretical physics. In
particular, many fundamental `quantum numbers' coincide exactly with the type of topological invariant
described above.

Geometric and spectral index theory are concerned with more refined questions concerned with
constructing `local' differential form and spectral invariants which may provide canonical representatives
for their topological partners. These invariants often require a delicate analysis of the heat kernel of the Dirac Laplacian, which describes how the heat flow over the manifold of a point like source of heat connects initial (small time) local geometric information with global topological information.

I am particularly interested in determinants of Dirac operators and the closely related Index Theory for Families of Dirac operators. This refers to the 'determinant' in the usual sense of linear algebra, but for linear operators on infinite dimensional spaces (of functions) whose eigenvalues diverge. For example, the operator D = id/dx + a, where a is in (0,1), acting on functions over the circle has a discrete spectrum consisting of eiegenvalues
                                     Spec(D) = { n+a : n an integer}.
Hence its determinant, which is formally the product of its eigenvalues, is undefined. We therefore look for natural procedures for defining 'regularized' determinant, which roughly speaking involves a canonical method for throwing away the 'infinite part'.  This leads to deep ideas in geometric index theory, and the closely relate theory of anomalies in QFT.

In this sense, the determinant may be regarded as a refinement of the index, but one which requires a regularization procedure to make sense of it as a number. This is essentially well-understood for closed manifolds. However, if we consider a manifold with non-empty boundary then matters are more complicated since one has to impose an elliptic boundary condition, which may introduce new non-local invariants. On the other hand, the boundary condition provides a new degree of freedom which makes the determinant of an elliptic boundary value problem a more computable invariant than the determinant over a closed manifold. The question is then, can we exploit this computability to compute the determinant of a Dirac operator over a closed manifold via sewing rules for the determinant? More generally, can the geometry of the determinant bundle for a family of Dirac operators over a closed manifold be computed via splitting the underlying manifold into two halves and then sewing together the answers from the two sides? The answer is a cautious yes.

For an idea of what the precise answer should be, we turn first to an intuitive theory of the sewing rules
for the determinant arising from sewing rules for the corresponding path integral in QFT. This says that
the determinant on the closed manifold is obtained by averaging away the choice of boundary condition.
Formally this requires integration over the infinite-dimensional space of elliptic boundary conditions. Such
an integral has (so far) no rigourous formulation (indeed the abscence of a general infinite-dimensional
integration theory is the essential mathematical problem with quantum field 'theory'). However, there is a
rigourous analogue of the averaging process defined through taking `adiabatic limits', by blowing up the
geometry of the manifold around the separating hypersurface.

There is also a rigourous algebraic sewing formula for the determinant as a 'Topological' QFT, which
abstracts into an algebraic framework the properties the mythical path integral would satisfy if it existed.
This is closely related to the construction of irreducible representations of gauge groups, e.g. of loop
groups and Virasoro algebras.  In a nutshell, the sewing formula for the determinant is essentially an
infinite-dimensional version of the Weyl orthogonality relations for the characters of these representations.