Past

We calculate an analytic value for the correlation coefficient between a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.

Galois groups of p-class towers of number fields have long been a mystery,

but recent calculations have led to glimpses of a rich theory behind them,

involving Galois actions on trees, families of groups whose derived series

have finite index, families of deficiency zero p-groups approximated by

p-adic analytic groups, and so on.

We consider semilinear Sturm-Liouville and elliptic problems with jumping

nonlinearities. We show how `half-eigenvalues' can be used to describe the

solvability of such problems and consider the structure of the set of

half-eigenvalues. It will be seen that for Sturm-Liouville problems the

structure of this set can be considerably more complicated for periodic than

for separated boundary conditions, while for elliptic partial differential

operators only partial results are known about the structure in general.

We construct spaces of manifolds of various dimensions following

Vassiliev's approach to the theory of knots. These are infinite-dimensional

spaces with hypersurface, corresponding to manifolds with Morse singularities.

Connected components of the complement to this discriminant are homotopy

equivalent to the covering spaces of BDiff(M). These spaces appear to be a

natural base over which one can consider parametrised versions of Floer and

Seiberg-Witten theories.

The question whether the measure of a Levy process starting from x>0 and "conditioned to stay positive" converges to the corresponding obiect for x=0 when x tends to 0 is rather delicate. I will describe work with Loic Chaumont which settles this question, essentially in all cases of interest. As an application, I will show how to use this result and excursion theory to give simpler proofs of some recent results about the exit problem for reflected processe derived from spectrally one-sided Levy processes due to Avram. Kyprianou and Pistorius.

The formation of steady patterns in one space dimension is generically

governed, at small amplitude, by the Ginzburg-Landau equation.

But in systems with a conserved quantity, there is a large-scale neutral

mode that must be included in the asymptotic analysis for pattern

formation near onset. The usual Ginzburg-Landau equation for the amplitude

of the pattern is then coupled to an equation for the large-scale mode.

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These amplitude equations show that for certain parameters all regular

periodic patterns are unstable. Beyond the stability boundary, there

exist stable stationary solutions in the form of spatially modulated

patterns or localised patterns. Many more exotic localised states are

found for patterns in two dimensions.

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Applications of the theory include convection in a magnetic field,

providing an understanding of localised states seen in numerical

simulations.