**The first one ties in with my work on
stochastic dynamics. If a stochastic
nonlinear process is formulated as a functional integral and then analysed by
obtaining the optimal paths, this is the beginning of a systematic
approximation scheme based on an expansion in the noise strength. The zeroth
order term is simple to find, but the first order term involves calculating a
functional determinant. In addition, a zero mode may be present (by Goldstone's
theorem, if a continuous symmetry is broken) and the determinant has to be
calculated with the zero mode extracted. Remarkably, formulae can be derived
for these determinants which are simple and elegant. Partial results were
obtained some time ago (see J. Phys. A28, 6931-6942 (1995) or this
pdf file), but with Klaus Kirsten, I have
completed a much more comprehensive study of the evaluation of these
determinants. Initially we investigated simple second order operators with
no negative eigenvalues. The method used to derive the results is easily
accessible since it uses only contour integration methods and elementary facts
concerning the solution of differential equations. We studied general linear
boundary conditions which included all the most well-known types (Dirichlet,
Neumann, Robin and periodic) and found a simple expression for the
determinant which covered all these cases. This result only depends on the
behaviour of the zero mode at the boundaries and therefore does not require
any information about the other modes, or even the details of the nature of
the zero mode away from the boundaries
(see Ann. Phys. 308, 502-527 (2003)). In a second paper
(J. Phys. A37, 4649-4670 (2004)) we generalised the
treatment to cover operators of the general Sturm-Liouville type, those with
negative eigenvalues and systems of second order operators. Again the same
general result was found.**

**The second project involves describing the dynamics
of vesicle growth and the instability which leads to vesicles changing shape.
Vesicles are small cell-like structures in which the membrane separating the
contents of the vesicle from the environment takes the form of a lipid bilayer.
Part of their appeal comes from the fact that living cells are essentially
very complex vesicles --- with the membrane containing mixtures of different
lipids and other components, a cytoskeleton and complex surface structures.
This has led to the use of vesicles as the basic component of models of
protocells. On the other hand simple vesicles, without any additional
structure, have many fascinating properties when observed in the laboratory.
Their self-assembly, their growth, their shape and the fact that they divide
to produce daughter vesicles have many aspects which are little-understood.
The latter property of replication is especially interesting in the context
of models of protocells. One can ask: how much of simple protocell dynamics
can be explained using the statistical thermodynamics of vesicles, without
the introduction of more complex processes or of genetic material? This
question was the motivation for some work recently carried out with Duccio
Fanelli from the University of Florence. This work has only just begun; an
account which discusses our approach is published in
Phys. Rev. E78, 051406 (2008).**