Past

Speakers include:

* David Abrahams (Manchester, UK); * Stuart Antman (Maryland, USA); * Martine Ben Amar (Ecole Normale Supérieure, France); * Mary Boyce (MIT, USA); * John Hutchinson (Harvard, USA); * Nadia Lapusta (Caltech, USA); * John Maddocks (Lausanne, Switzerland); * Stefan Mueller (Bonn, Germany); * Christoph Ortner (Oxford, UK); * Ares Rosakis (Caltech, USA); * Hanus Seiner (Academy of Sciences, Czech Republic); * Eran Sharon (Hebrew University, Israel); * Lev Truskinovsky (Lab de Mécanique des Solids, France); * John Willis (Cambridge, UK).

• “Two Problems Relating to Sand Dune Formation” by Andrew Ellis

• “Interface Sharpening with a Lattice Boltzmann Equation” by Tim Reis

• “A Dual Porosity Model for the Uptake of Nutrients by Root Hairs” by Kostas Zygalakis

Atomistic computer simulation models are constructed to study a range of materials in which

the atoms appear in novel environments. Two key research areas are considered:

• The Growth and Structure Inorganic Nanotubes. A range of materials have been

observed to form nanotubular structures (inorganic nanotubes - INTs) analogous to those

well known for carbon. These INTs, which may have unique low-dimensional morphologies

not simply related to known bulk polymorphs, potentially offer unique mechanical and electronic properties. A useful synthetic pathway is to use carbon nanotubes as templates using

molten salts. Atomistic simulation models, in which the atom interactions are treated utilizing relatively simple potential energy functions, are developed and applied to understand

the INT formation and stability. INT morphologies are classified by reference to folding

two dimensional sheets. The respective roles of thermodynamics and kinetics in determining

INT morphology are outlined and the atomistic results used to develop an analytic model to

predict INT diameters.

• Ordering on Multiple Length-Scales in Network-forming Liquids. Intermediate-range order (IRO), in which systems exhibit structural ordering on length-scales beyond

the nearest-neighbour (short-range), has been identified in a wide range of materials and is

characterised by the appearance of the so-called first sharp diffraction peak (FSDP) at low

scattering angles. The precise structural origin of such ordering remains contentious and a full

understanding of the factors underlying this order is vital if such materials (many of which are

technologically significant) are to be produced in a controlled manner. Simulation models,

in which the ion-ion interactions are represented by relatively simple potential functions

which incorporate (many-body) polarisation and which are parameterised by reference to

well-directed electronic structure calculations, have been shown to reproduce such IRO and

allow the precise structural origin of the IRO to be identified. Furthermore, the use of

relatively simple (and hence computationally tractable) models allows for the study of the

relatively long length- and time-scales required. The underlying structures are analysed with

reference to both recent (neutron scattering) experimental results and high level electronic

structure calculations. The role of key structural units (corner and edge sharing polyhedra)

in determining the network topology is investigated in terms of the underlying dynamics and

the relationship to the glass transition considered.

We will begin by reviewing the construction of the symplectic quotient and the definition of the Kirwan map. Then we will give an overview of Kirwan's original proof of the surjectivity of this map and some generalizations of this result. Finally we will talk about the techniques that are being developed to construct right inverses for the Kirwan map.

Speakers include:

* Graeme Ackland (School of Physics and Astronomy, Edinburgh) * Andrea Braides (Rome II) * Thierry Bodineau (École Normale Supérieure, Paris) * Matthew Dobson (Minneapolis) * Laurent Dupuy (CEA, Saclay) * Ryan Elliott (Minneapolis) * Roman Kotecky (Warwick) * Carlos Mora-Corral (BCAM, Bilbao) * Stefano Olla (CEREMADE, Paris-Dauphine) * Bernd Schmidt (TU Munich) * Lev Truskinovsky (École Polytechnique, Palaiseau) * Min Zhou (Georgia Tech, Atlanta)

We study a stochastic control problem in the context of utility maximization under model uncertainty. The problem is formulated as /max min/ problem : /max /over strategies and consumption and /min/ over the set of models (measures).

For the minimization problem, we have showed in Bordigoni G., Matoussi,A., Schweizer, M. (2007) that there exists a unique optimal measure equivalent to the reference measure. Moreover, in the context of continuous filtration, we characterize the dynamic value process of our stochastic control problem as the unique solution of a generalized backward stochastic differential equation with a quadratic driver. We extend first this result in a discontinuous filtration. Moreover, we obtain a comparison theorem and a regularity properties for the associated generalized BSDE with jumps, which are the key points in our approach, in order to solve the utility maximization problem over terminal wealth and consumption. The talk is based on joint work with M. Jeanblanc and A. Ngoupeyou (2009).

HSBC Currency Trading has collaborated with the Oxford Maths Institute for over six years and is now working with its third DPhil student. In this workshop, we will look at the some of the academic research which has directly benefited the trading operation.

Evolutionary PDEs on stationary and moving surfaces appear in many applications such as the diffusion of surfactants on fluid interfaces, surface pattern formation on growing domains, segmentation on curved surfaces and phase separation on biomembranes and dissolving alloy surfaces.

In this talk I discuss three numerical approaches based on:- (I) Surface Finite Elements and Triangulated Surfaces, (II)Level Set Method and Implicit Surface PDEs and (III) Phase Field Approaches and Diffuse Surfaces.

We consider rational approximations to the Faddeeva or plasma dispersion function, defined

as

$w(z) = e^{-z^{2}} \mbox{erfc} (-iz)$.

With many important applications in physics, good software for

computing the function reliably everywhere in the complex plane is required. In this talk

we shall derive rational approximations to $w(z)$ via quadrature, M\"{o}bius transformations, and best approximation. The various approximations are compared with regard to speed of convergence, numerical stability, and ease of generation of the coefficients of the formula.

In addition, we give preference to methods for which a single expression yields uniformly

high accuracy in the entire complex plane, as well as being able to reproduce exactly the

asymptotic behaviour

$w(z) \sim i/(\sqrt{\pi} z), z \rightarrow \infty$

(in an appropriate sector).

This is Joint work with: Stephan Gessner, St\'efan van der Walt

Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. Using torus localisation, one can then compute topological invariants of M. We consider the case X=S is a toric surface and compute generating functions of Euler characteristics of M. In case of torsion free sheaves, one can study wall-crossing phenomena and in case of pure dimension 1 sheaves one can verify, in examples, a conjecture of Katz relating Donaldson--Thomas invariants and Gopakumar--Vafa invariants.

This talk is about the ordinary representation theory of finite groups of Lie type. I will begin by carefully reviewing algebraic groups and finite groups of Lie type and the construction and properties of (ordinary) Gelfand--Graev characters. I will then introduce generalized Gelfand--Graev characters, which are constructed using the Lie algebra of the ambient algebraic group. Towards the end I hope to give an idea of how generalized Gelfand--Graev characters can and have been used to attack Lusztig's conjecture and the role this plays in the determination of the character tables of finite groups of Lie type.

This talk will be largely speculative. First we consider the formal properties that could be expected of a "topological field theory" in 6+1 dimensions defined by $G_2$ instantons. We explain that this could lead to holomorphic bundles over moduli spaces of Calabi-Yau 3-folds whose ranks are the DT-invariants. We also discuss in more detail the compactness problem for $G_2$ instantons and associative submanifolds.

The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.

This talk will review material, well-known to specialists, on calibrated geometry and Yang-Mills theory over manifolds with holonomy $SU(3)$, $G_2$ or $Spin(7)$. We will also describe extensions of the standard set-up, modelled on Gromov's "taming forms" for almost-complex structures.

The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.

Extremal black holes are of interest as they are expected have simpler quantum descriptions than their non-extremal counterparts.  Any extremal black hole solution admits a well defined notion of a near horizon geometry which solves the same field equations. I will describe recent progress on the general understanding of such near horizon geometries in four and higher dimensions. This will include the proof of near-horizon symmetry enhancement and the explicit classification of near-horizon geometries (in a variety of settings). I will also discuss how one can use such results to prove classification/uniqueness theorems for asymptotically flat extremal vacuum black holes in four and five dimensions.

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.