Past

I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations.

This is joint work with Rob Meyerhoff.

Coefficients of the characteristic polynomial are generators of the ring of polynomial functions on the space of matrices which are invariant under the conjugation. This was generalized by Chevalley to general reductive groups. By looking closely on the centralisers, one is lead to a very natural 2-category attached to Chevalley characteristic morphism. This abstract, but yet elementary, construction helps one to understand the symmetries of the fibres of the Hitchin fibration, as well as those of affine Springer fibers.

We will also explain how these groups of symmetries are related to the notion of endoscopic groups, which was introduced by Langlands in his stabilisation of the trace formula. We will also briefly explain how the symmetry groups help one to acquire a rather good understanding of the cohomology of the Hitchin fibration and eventually the proof of the fundamental lemma in Langlands' program.

A financial market model is considered on which agents (e.g. insurers) are subject to an exogenous financial risk, which they trade by issuing a risk bond. Typical risk sources are climate or weather. Buyers of the bond are able to invest in a market asset correlated with the exogenous risk. We investigate their utility maximization problem, and calculate bond prices using utility indi®erence. This hedging concept is interpreted by means of martingale optimality, and solved with BSDE and Malliavin's calculus tools. Prices are seen to decrease as a result of dynamic hedging. The price increments are interpreted in terms of diversification pressure.

GIBSON BUILDING COMMON ROOM 2ND FLOOR

(Coffee and Cakes in Gibson Meeting Room - opposite common room)

The effects of electric fields on nonlinear free surface flows are investigated. Both inviscid and Stokes flows are considered.

Fully nonlinear solutions are computed by boundary integral equation methods and weakly nonlinear solutions are obtained by using long wave asymptotics and lubrication theory. Effects of electric fields on the stability of the flows are discussed. In addition applications to coating flows are presented.

The "large sieve" was invented by Linnik in order to attack problems involving the distribution of integers subject to certain constraints modulo primes, for which earlier methods of sieve theory were not suitable. Recently, the arithmetic large sieve inequality has been found to be capable of much wider application, and has been used to obtain results involving objects not usually considered as related to sieve theory. A form of the general sieve setting will be presented, together with sample applications; those may involve arithmetic properties of random walks on discrete groups, zeta functions over finite fields, modular forms, or even random groups.

Geometric integration is the numerical integration of a differential equation, while preserving one or more of its geometric/physical properties exactly, i.e. to within round-off error.

Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. The field has tantalizing connections to dynamical systems, as well as to Lie groups.

In this talk we first present a survey of geometric numerical integration methods for differential equations, and then exemplify this by discussing symplectic vs energy-preserving integrators for ODEs as well as for PDEs.

Diffusion process models for evolution of neutral genes have a particle dual coalescent process underlying them. Models are reversible with transition functions having a diagonal expansion in orthogonal polynomial eigenfunctions of dimension greater than one, extending classical one-dimensional diffusion models with Beta stationary distribution and Jacobi polynomial expansions to models with Dirichlet or Poisson Dirichlet stationary distributions. Another form of the transition functions is as a mixture depending on the mutant and non-mutant families represented in the leaves of an infinite-leaf coalescent tree.

The one-dimensional Wright-Fisher diffusion process is important in a characterization of a wider class of continuous time reversible Markov processes with Beta stationary distributions originally studied by Bochner (1954) and Gasper (1972). These processes include the subordinated Wright-Fisher diffusion process.

I will go over my paper (arXiv:0902.2135v1) which explains how semi-flat Calabi-Yau / G$_2$ manifolds can be constructed from minimal 3-submanifolds in a signature (3,3) vector space.