Oxford-Man Institute

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges.

Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.

Based on joint work with Oliver Riordan.

We give a necessary and sufficient condition for a map defined on a compact, quasiconvex and simply-connected space to factor through a tree. This condition can be checked using currents. In particular if the target is some Euclidean space and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number. Moreover, this shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.

The idea of A-free group, where A is a discrete ordered abelian group, has been introduced by Myasnikov, Remeslennikov and Serbin. It generalises the construction of free groups. A proof will be outlined that a group is A-free for some A if and only if it acts freely and without inversions on a \lambda-tree, where \lambda is an arbitrary ordered abelian group.

The Taylor expansion of a controlled differential equation suggests that the solution at time 1 depends on the driving path only through the latter's iterated integrals up to time 1, if the vector field is infinitely differentiable. Hambly and Lyons proved that this remains true for Lipschitz vector fields if the driving path has bounded total variation. We extend the Hambly-Lyons result for weakly geometric rough paths in finite dimension. Joint work with X. Geng, T. Lyons and D. Yang.    

We consider the numerical approximation of Kolmogorov equations arising in the context of option pricing under L\'evy models and beyond in a multivariate setting. The existence and uniqueness of variational solutions of the partial integro-differential equations (PIDEs) is established in Sobolev spaces of fractional or variable order.

Most discretization methods for the considered multivariate models suffer from the curse of dimension which impedes an efficient solution of the arising systems. We tackle this problem by the use of sparse discretization methods such as classical sparse grids or tensor train techniques. Numerical examples in multiple space dimensions confirm the efficiency of the described methods.

This work is joint with Peter Sarnak.

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

We study infinite (random) systems of interacting particles living in a Euclidean space X and possessing internal parameter (spin) in R¹. Such systems are described by Gibbs measures on the space Γ(X,R¹) of marked configurations in X (with marks in R¹). For a class of pair interactions, we show the occurrence of phase transition, i.e. non-uniqueness of the corresponding Gibbs measure, in both 'quenched' and 'annealed' counterparts of the model.

A model of a financial market is complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. It is well known that numerous models, for example stochastic volatility models, are however incomplete. We present conditions, which, in a general diffusion framework, guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. From a purely mathematical point of view we prove an integral representation theorem which guarantees that every local Q-martingale can be represented as a stochastic integral with respect to the vector of primitive assets and derivative contracts.

Abstract: We introduce a new framework for defining integration against rough path. This framework generalizes rough integral, and gives a natural explanation of some of the regularity requirements in rough path theory.