Past

Anderson localisation is an important phenomenon describing a

transition between insulation and conductivity. The problem is to analyse

the spectrum of a Schroedinger operator with a random potential in the

Euclidean space or on a lattice. We say that the system exhibits

(exponential) localisation if with probability one the spectrum is pure

point and the corresponding eigen-functions decay exponentially fast.

So far in the literature one considered a single-particle model where the

potential at different sites is IID or has a controlled decay of

correlations. The present talk aims at $N$-particle systems (bosons or

fermions) where the potential sums over different sites, and the traditional

approach needs serious modifications. The main result is that if the

`randomness' is strong enough, the $N$-particle system exhibits

localisation.

The proof exploits the muli-scale analysis scheme going back to Froehlich,

Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No

preliminary knowledge of the related material will be assumed from the

audience, apart from basic facts.

This is a joint work with V Chulaevsky (University of Reims, France)

Sparse grids yield numerical solutions to PDEs with a

significantly reduced number of degrees of freedom. The relative

benefit increases with the dimensionality of the problem, which makes

multi-factor models for financial derivatives computationally tractable.

An outline of a convergence proof for the so called combination

technique will be given for a finite difference discretisation of the

heat equation, for which sharp error bounds can be shown.

Numerical examples demonstrate that by an adaptive (heuristic)

choice of the subspaces European and American options with up to thirty

(and most likely many more) independent variables can be priced with

high accuracy.

/notices/events/abstracts/Workshop/Falush.shtml

I will present a new formula for diffusion processes which involving

Ito integral for the transition probability functions. The nature of

the formula I discovered is very close to the Kac formula, but its

form is similar to the Cameron-Martin formula. In some sense it is the

Cameron-Martin formula for pinned diffusions.

I will consider a stochastic process ( \xi_u; u \in

\Gamma_\infty ) where \Gamma_\infty is the set of vertices of an

infinite binary tree which satisfies some recursion relation

\xi_u= \phi(\xi_{u0},\xi_{u1}, \epsilon_u) \text { for each } u \in \Gamma_\infty.

Here u0 and u1 denote the two immediate daughters of the vertex u.

The random variables ( \epsilon_u; u\in \Gamma_\infty), which

are to be thought of as innovations, are supposed independent and

identically distributed. This type of structure is ubiquitous in models

coming from applied proability. A recent paper of Aldous and Bandyopadhyay

has drawn attention to the issue of endogeny: that is whether the process

( \xi_u; u \in \Gamma_\infty) is measurable with respect to the

innovations process. I will explain how this question is related to the

existence of certain dynamics and use this idea to develop a necessary and

sufficient condition [ at least if S is finite!] for endogeny in terms of

the coupling rate for a Markov chain on S^2 for which the diagonal is

absorbing.

Much is now known about exp-log series, and their connections with o-

minimality and Hardy fields. However applied mathematicians who work with

differential equations, almost invariably want series involving

trigonometric functions which those theories exclude. The seminar looks at

one idea for incorporating oscillating functions into the framework of

Hardy fields.

When modelling a physical or biological system, it has to be decided

what framework best captures the underlying properties of the system

under investigation. Usually, either a continuous or a discrete

approach is adopted and the evolution of the system variables can then

be described by ordinary or partial differential equations or

difference equations, as appropriate. It is sometimes the case,

however, that the model variables evolve in space or time in a way

which involves both discrete and continuous elements. This is best

illustrated by a simple example. Suppose that the life span of a

species of insect is one time unit and at the end of its life span,

the insect mates, lays eggs and then dies. Suppose the eggs lie

dormant for a further 1 time unit before hatching. The `time-scale' on

which the insect population evolves is therefore best represented by a

set of continuous intervals separated by discrete gaps. This concept

of `time-scale' (or measure chain as it is referred to in a slightly

wider context) can be extended to sets consisting of almost arbitrary

combinations of intervals, discrete points and accumulation points,

and `time-scale analysis' defines a calculus, on such sets. The

standard `continuous' and `discrete' calculus then simply form special

cases of this more general time scale calculus.

In this talk, we will outline some of the basic properties of time

scales and time scale calculus before discussing some if the

technical problems that arise in deriving and analysing boundary

value problems on time scales.