Past

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.

We reconsider Merton's problem under proportional transaction costs.

Beginning with Davis and Norman (1990) such utility maximization problems are usually solved using stochastic control theory.

Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process.

In this study we show that this dual approach can actually be used for both deriving a candidate solution and verification.

In particular, the shadow price process is determined explicitly.

Collaborators from Industry will speak to us about their proposed projects for the MSc in Math Modelling and Scientific Computation. Potential supervisors should attend. All others welcome too.

After a short introduction to the section conjecture, I plan to present a "minimalistic" form of the birational p-adic section conjecture. The result is related to both: Koenigsmann's proof of the birational p-adic section conjecture, and a "minimalistic" Galois characterisation of formally p-adic valuations.

Whispering gallery modes in optical resonators have received a lot of attention as a mechanism for constructing small, directional lasers. They are also potentially important as passive optical components in schemes for coupling and filtering signals in optical fibres, in sensing devices and in other applications. In this talk it is argued that the evanescent field outside resonators that are very slightly deformed from circular or spherical is surprising in a couple of respects. First, even very small deformations seem to be capable of leading to highly directional emission patterns. Second, even though the undelying ray families are very regular and hardly differ from the integrable circular or spherical limit inside the resonator, a calculation of the evanescent field outside it is not straightforward.

This is because even very slight nonintegrability has a profound effect on the complexified ray families which guide the external wave to asymptopia. An approach to describing the emitted wave is described which is based on canonical perturbation theory applied to the ray families and extended to comeplx phase space.