Past

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.

The computation of flows of compressible fluids will be

discussed, exploiting the symmetric form of the equations describing

compressible flow.

We shall review recent progress in the understanding of

isoperimetric inequalities for product probability measures (a very tight

description of the concentration of measure phenomeonon). Several extensions

of the classical result for the Gaussian measure were recently derived by

functional analytic methods.

The Hopfield model took his name and its popularity within the theory

of formal neural networks. It was introduced in 1982 to describe and

implement associative memories. In fact, the mathematical model was

already defined, and studied in a simple form by Pastur and Figotin in

an attempt to describe spin-glasses, which are magnetic materials with

singular behaviour at low temperature. This model indeed shows a very

complex structure if considered in a slightly different regime than

the one they studied. In the present talk we will focus on the

fluctuations of the free energy in the high-temperature phase. No

prior knowledge of Statistical mechanics is required to follow the

talk.

The first seminar will be given with the new students in

mind. It will begin with a brief overview of quantifier elimination and its

relation to the back-and-forth property.I shall then discuss complexity issues

with particular reference to algebraically closed fields.In particular,how much

does the height and degree of polynomials in a formula increase when a

quantifier is eliminated? The precise answer here gave rise to the definition

of a `generic' transcendental entire function,which will also be

discussed.

A cell is a wonderously complex object. In this talk I will

give an overview of some of the mathematical frameworks that are needed

in order to make progress to understanding the complex dynamics of a

cell. The talk will consist of a directed random walk through discrete

Markov processes, stochastic differential equations, anomalous diffusion

and fractional differential equations.

Joe Doob, who died recently aged 94, was the last survivor of the

founding fathers of probability. Doob was best known for his work on

martingales, and for his classic book, Stochastic Processes (1953).

The talk will combine an appreciation of Doob's work and legacy with

reminiscences of Doob the man. (I was fortunate to be a colleague of

Doob from 1975-6, and to get to know him well during that year.)

Following Doob's passing, the mantle of greatest living probabilist

descends on the shoulders of Kiyosi Ito (b. 1915), alas now a sick

man.

We are interested in a microscopic stochastic description of a

population of discrete individuals characterized by one adaptive

trait. The population is modeled as a stochastic point process whose

generator captures the probabilistic dynamics over continuous time of

birth, mutation and death, as influenced by each individual's trait

values, and interactions between individuals. An offspring usually

inherits the trait values of her progenitor, except when a mutation

causes the offspring to take an instantaneous mutation step at birth

to new trait values. Once this point process is in place, the quest

for tractable approximations can follow different mathematical paths,

which differ in the normalization they assume (taking limit on

population size , rescaling time) and in the nature of the

corresponding approximation models: integro or integro-differential

equations, superprocesses. In particular cases, we consider the long

time behaviour for the stochastic or deterministic models.

The role of electron emission (either thermionic, secondary or

photoelectric) in charging an object immersed in a plasma is

investigated, both theoretically and numerically.

In fact, recent work [1] has shown how electron emission can

fundamentally affect the shielding potential around the object. In

particular, depending on the physical parameters of the system (that

were chosen such to correspond to common experimental conditions), the

shielding potential can develop an attractive potential well.

The conditions for the formation of the well will be reviewed, based

on a theoretical model of electron emission from the

grain. Furthermore, simulations will be presented regarding specific

laboratory, space and astrophysical applications.

[1] G.L. Delzanno, G. Lapenta, M. Rosenberg, Phys. Rev.

Lett., 92, 035002 (2004).