Past

Numerous studies of asset returns reveal excess kurtosis as fat tails, often characterized by power law behaviour. A hybrid of arithmetic and geometric Brownian motion is proposed as a model for short-term asset returns, and its equilibrium and dynamical properties explored. Some exact solutions for the time-dependent behaviour are given, and we demonstrate the existence of a stochastic bifurcation between mean- reverting and momentum-dominated markets. The consequences for risk management will be discussed.

Several dissipative scalar conservation laws share the properties of

$L1$-contraction and maximum principle. Stability issues are naturally

posed in terms of the $L1$-distance. It turns out that constants and

travelling waves are asymptotically stable under zero-mass initial

disturbances. For this to happen, we do not need any assumption

(smallness of the TW, regularity/smallness of the disturbance, tail

asymptotics, non characteristicity, ...) The counterpart is the lack of

a decay rate.

Lord Desai will discuss how the use of mathematics in economics is as much a result of formalism as of limited knowledge of mathematics. This will relate to his experience as a teacher and researcher and also speak to the current financial meltdown.

A key birational invariant of a compact complex manifold is its "canonical ring."

The ring of modular forms in one or more variables is an example of a canonical ring. Recent developments in higher dimensional algebraic geometry imply that the canonical ring is always finitely generated:this is a long-awaited major foundational result in algebraic geometry.

In this talk I define all the terms and discuss the result, some applications, and a recent remarkable direct proof by Lazic.

Traditional arbitrage pricing theory is based on martingale measures. Recent studies show that some form of arbitrage may exist in real markets implying that then there does not exist an equivalent martingale measure and so the question arises: what can one do with pricing and hedging in this situation? We mention here two approaches to this effect that have appeared in the literature, namely the ``Fernholz-Karatzas" approach and Platen's "Benchmark approach" and discuss their relationships both in models where all relevant quantities are fully observable as well as in models where this is not the case and, furthermore, not all observables are also investment instruments.

[The talk is based on joint work with former student Giorgia Galesso]

Thin films of low refractive index nanoparticles are being developed for use as anti-reflection coatings for solar cells and displays. Although these films are deposited as a single layer, the comparison between a simple theoretical model and the experimental data shows that the coating cannot be treated as a such, but rather as a layer with an unknown refractive index gradient. Approaches to modelling the reflectance from such coatings are sought. Such approaches would allow model refractive index gradients to be fitted to the experimental data and would allow better understanding of how the structure of the films develops during fabrication.

Modelling phase change in the presence of a flowing thin liquid film

There are numerous physical phenomena that involve a melting solid

surrounded by a thin layer of liquid, or alternatively a solid

forming from a thin liquid layer. This talk will involve two such

problems, namely contact melting and the Leidenfrost phenomenon.

Contact melting occurs, for example, when a solid is placed on a

surface that is maintained at a temperature above the solid melting

temperature. Consequently the solid melts, while the melt layer is

squeezed out from under the solid due to its weight. This process

has applications in metallurgy, geology and nuclear technology, and

also describes a piece of ice melting on a table. Leidenfrost is

similar, but involves a liquid droplet evaporating after being

placed on a hot substrate. This has applications in cooling systems

and combustion of fuel or a drop of water on a hot frying pan.

The talk will begin with a brief introduction into one-dimensional

Stefan problems before moving on to the problem of melting coupled

to flow. Mathematical models will be developed, analysed and

compared with experimental results. Along the way the Heat Balance

Integral Method (HBIM) will be introduced. This is a well-known

method primarily used by engineers to approximate the solution of

thermal problems. However, it has not proved so popular with

mathematicians, due to the arbitrary choice of approximating

function and a lack of accuracy. The method will be demonstrated on

a simple example, then it will be shown how it may be modified to

significantly improve the accuracy. In fact, in the large Stefan

number limit the modified method can be shown to be more accurate

than the asymptotic solution to second order.

Invariant subspaces are a well-established tool in the theory of linear eigenvalue problems. They are also computationally more stable objects than single eigenvectors if one is interested in a group of closely clustered eigenvalues. A generalization of invariant subspaces to matrix polynomials can be given by using invariant pairs.

We investigate some basic properties of invariant pairs and give perturbation results, which show that invariant pairs have similarly favorable properties for matrix polynomials than do invariant subspaces have for linear eigenvalue problems. In the second part of the talk we discuss computational aspects, namely how to extract invariant pairs from linearizations of matrix polynomials and how to do efficient iterative refinement on them. Numerical examples are shown using the NLEVP collection of nonlinear eigenvalue test problems.

This talk is joint work with Daniel Kressner from ETH Zuerich.

We investigate calibrating financial models using a rigorous Bayesian framework. Non-parametric approaches in particular are studied and the local volatility model is used as an example. By incorporating calibration error into our method we design optimal hedges that minimise expected loss statistics based on different Bayesian loss functions determined by an agent's preferences. Comparisons made with the standard hedge strategies show the Bayesian hedges to outperform traditional methods.

We will present a self-contained introduction to gauge theory, self-duality and instanton moduli spaces. We will analyze in detail the situation of charge 1 instantons for the 4-sphere when the gauge group is SU(2). Time permitting, we will also mention the ADHM construction for k-instantons.

I will discuss recent results on dispersive estimates for linear PDE's with time dependent coefficients. Then I will discuss how such

estimates can be used to study stability of nonlinear solitary waves and resonance phenomena.

Discussing Christoph Zenger’s paper.

Let G be a permutation group on a set S. A base for G is a subset B of S such that the pointwise stabilizer of B in G is trivial. We write b(G) for the minimal size of a base for G.

Bases for finite permutation groups have been studied since the early days of group theory in the nineteenth century. More recently, strong bounds on b(G) have been obtained in the case where G is a finite simple group, culminating in the recent proof, using probabilistic methods, of a conjecture of Cameron.

In this talk, I will report on some recent joint work with Bob Guralnick and Jan Saxl on base sizes for algebraic groups. Let G be a simple algebraic group over an algebraically closed field and let S = G/H be a transitive G-variety, where H is a maximal closed subgroup of G. Our goal is to determine b(G) exactly, and to obtain similar results for some additional base-related measures which arise naturally in the algebraic group context. I will explain the key ideas and present some of the results we have obtained thus far. I will also describe some connections with the corresponding finite groups of Lie type.

1. "Approximate approximations" and accurate computation of high dimensional potentials.

2. Iteration procedures for ill-posed boundary value problems with preservation of the differential equation.

3. Asymptotic treatment of singularities of solutions generated by edges and vertices at the boundary.

4. Compound asymptotic expansions for solutions to boundary value problems for domains with singularly perturbed boundaries.

5. Boundary value problems in perforated domains without homogenization.