Past

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.