Past

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.

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.

There is a conjectural relationship due to Yau-Tian-Donaldson between stability of projective manifolds and the existence of canonical Kahler metrics (e.g. Kahler-Einstein metrics). Embedding the projective manifold in a large projective space gives, on one hand, a Geometric Invariant Theory stability problem (by changing coordinates on the projective space) and, on the other, a notion of balanced metric which can be used to approximate the canonical Kahler metric in question. I shall discuss joint work with Richard Thomas that extends this framework to orbifolds with cyclic quotient singularities using embeddings in weighted projective space, and examples that show how several obstructions to constant scalar curvature orbifold metrics can be interpreted in terms of stability.

Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number) Ec is different from zero, an additional time scale of O(Ec^(?1)) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/ f^(3/2) power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes.

A recent innovation in quantum field theory is the locally covariant

framework developed by Brunetti, Fredenhagen and Verch, in which quantum

field theories are regarded as functors from a category of spacetimes to a

category of *-algebras. I will review these ideas and particularly discuss

the extent to which they correspond to the intuitive idea of formulating the

same physics in all spacetimes.

We consider monochromatic planar electro-magnetic waves propagating through a nonlinear dielectric medium in the optical regime.

Travelling waves are particularly simple solutions of this kind. Results on the existence of guided travelling waves will be reviewed. In the case of TE-modes, their stability will be discussed within the context of the paraxial approximation.

The modern approach to integrability proceeds via a Riemann surface, the spectral curve.

In many applications this curve is specified by transcendental constraints in terms of periods. I will highlight some of the problems this leads to in the context of monopoles, problems including integer solutions to systems of quadratic forms, questions of real algebraic geometry and conjectures for elliptic functions. Several new results will be presented including the uniqueness of the tetrahedrally symmetric monopole.

Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces

for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients.

The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator

and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO)

which guarantee that the solution operator associated with this problem is an isomorphism.

Hairsine-Rose (HR) model is the only multi sediment size soil erosion

model. The HR model is modifed by considering the effects of sediment bedload and

bed elevation. A two step composite Liska-Wendroff scheme (LwLf4) which

designed for solving the Shallow Water Equations is employed for solving the

modifed Hairsine-Rose model. The numerical approximations of LwLf4 are

compared with an independent MOL solution to test its validation. They

are also compared against a steady state analytical solution and experiment

data. Buffer strip is an effective way to reduce sediment transportation for

certain region. Modifed HR model is employed for solving a particular buffer

strip problem. The numerical approximations of buffer strip are compared

with some experiment data which shows good matches.