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).