Bath

"We investigate a construction of a Skorokhod embedding due to Root (1969), which has been the subject of recent interest for applications in Mathematical Finance (Dupire, Carr & Lee), where the construction has applications for model-free pricing and hedging of variance derivatives. In this context, there are two related questions: firstly of the construction of the stopping time, which is related to a free boundary problem, and in this direction, we expand on work of Dupire and Carr & Lee; secondly of the optimality of the construction, which is originally due to Rost (1976). In the financial context, optimality is connected to the construction of hedging strategies, and by giving a novel proof of the optimality of the Root construction, we are able to identify model-free hedging strategies for variance derivatives. Finally, we will present some evidence on the numerical performance of such hedges. (Joint work with Jiajie Wang)"

We study upper bounds on topological complexity of sets definable in o-minimal structures over the reals. We suggest a new construction for approximating a large class of definable sets, including the sets defined by arbitrary Boolean combinations of equations and inequalities, by compact sets.

Those compact sets bound from above the homotopies and homologies of the approximated sets.

The construction is applicable to images under definable maps.

Based on this construction we refine the previously known upper bounds on Betti numbers of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae, and prove similar new upper bounds, individual for different Betti numbers, for their images under arbitrary continuous definable maps.

Joint work with A. Gabrielov.

I shall describe joint work with Gritsenko and Hulek in which we study the moduli spaces of polarised holomorphic symplectic manifolds via their periods. There are strong similarities with moduli spaces of K3 surfaces, but also some important differences, notably that global Torelli fails. I shall explain (conjecturally) why and show how the techniques used to obtain general type results for K3 moduli can be modified to give similar, and quite strong, results in this case. Mainly I shall concentrate on the case of deformations of Hilbert schemes of K3 surfaces.

The parabolic Anderson model is the Cauchy problem for the heat equation with random potential.  It offers a case study for the possible effects that a random, or irregular environment can have on a diffusion process.  In this talk I review results obtained for an extreme case of heavy-tailed potentials, among the effects we discuss our intermittency, strong localisation and ageing.

The uniform spanning forest (USF) in a graph

is a random spanning forest obtained as the limit of uniformly chosen spanning

trees on finite subgraphs. The USF is known to have stochastic dimension 4 on

graphs that are "at least 4 dimensional" in a certain sense. In this

talk I will look at more detailed estimates on the geometry of a fixed

component of the USF in the special case of the d-dimensional integer lattice,

d > 4. This is motivated in part by the study of random walk restricted to a

fixed component of the USF.