My main research is in the areas of Dynamical Systems, particularly the application of complex methods to exponential asymptotics arising in the study of codimension-two bifurcations and renormalisation, and Stochastic Analysis, particularly questions of existence, uniqueness and approximation of solutions of Stochastic Differential Equations.
I also have some interest in applications of analysis to PDE, complexity of matrix multiplication, the BMV conjecture (see below) and applications of mathematics to biology, particularly protein folding.
Uniqueness of solutions of stochastic differential equations, IMRN 2007 No. 24, Art. ID rnm124. pdf
Differential equations driven by rough paths: an approach via discrete approximation, AMRX 2007 No. 2, Art. ID abm009. pdf
The Bessis-Moussa-Villani conjecture originated in Mathematical Physics and can be stated in several equivalent forms. One is that, if A and B are positive semi-definite matrices of the same size, and k and l are positive integers, and if S is the sum of all products of k A's and l B's, then the trace of S is non-negative (e.g. if k=2 and l=1 then S=AAB+ABA+BAA).
This is easy to prove for 2 by 2 matrices, and also for any size if either k or l is less than 3. Recently it has been proved if either k or l is less than 5, and for a few other values, but the general case seems hard. Progress on the conjecture has used techniques from Fourier-Laplace transforms, variational methods, semidefinite programming and combinatorial algebra.
Some notes and references on this conjecture (updated 21/5/09) can be found here