Past

We classify certain linear representations of finite groups with a large orbit. This is motivated by a question on the number of conjugacy classes of a finite group.

There are many theorems concerning cycles in graphs for which it is natural to seek analogous results for directed graphs. I will survey

recent progress on certain questions of this type. New results include

(i) a solution to a question of Thomassen on an analogue of Dirac’s theorem

for oriented graphs,

(ii) a theorem on packing cyclic triangles in tournaments that “almost” answers a question of Cuckler and Yuster, and

(iii) a bound for the smallest feedback arc set in a digraph with no short directed cycles, which is optimal up to a constant factor and extends a result of Chudnovsky, Seymour and Sullivan.

These are joint work respectively with (i) Kuhn and Osthus, (ii) Sudakov, and (iii) Fox and Sudakov.

We consider problems of elastic plastic deformation with isotropic and  kinematic hardening.

A dual formulation with stresses as principal variables is used. 

We obtain several results on Sobolev space regularity of the stresses  and strains.

In particular, we obtain the existence of a full derivative of the  stress tensor up to the boundary of the basic domain.

Finally, we present an outlook for obtaining further regularity  results in connection with general nonlinear evolution problems.

The goal of this talk is to give sufficient conditions for the existence of points on certain varieties defned over finite fields.

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.

We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.

This talk will mention methods of testing whether a given integer is prime. Included topics are Carmichael numbers, Fermat and Euler pseudo-primes and results contingent on the Generalised Riemann Hypothesis.

The notion of a sutured 3-manifold was introduced by Gabai. It is a powerful tool in 3-dimensional topology. A few years ago, Andras Juhasz defined an invariant of sutured manifolds called sutured Floer homology.

I'll discuss a simpler invariant obtained by taking the Euler characteristic of this theory. This invariant turns out to have many properties in common with the Alexander polynomial. Joint work with Stefan Friedl and Andras Juhasz.

This talk is based on a joint work with Zhan Shi: We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou (2005). Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn (1988). Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

Let $\mathbb{E}$ be an elliptic curve defined over a number field $k$,

and let $a\in\mathbb{E}(\mathbb{C})$ be a complex point. Among the

possible choices of sequences of division points of $a$, $(a_n)_n$

such that $a_1 = a$ and $na_{nm} = a_m$, we can pick out those which

converge in the complex topology to the identity. We show that the

algebraic content of this effect of the complex topology is very

small, in the sense that any set of division sequences which shares

certain obvious algebraic properties with the set of those which

converge to the identity is conjugated to it by a field automorphism

of $\mathbb{C}$ over $k$.

As stated, this is a result of algebra and number theory. However, in

proving it we are led ineluctably to use model theoretic techniques -

specifically the concept of "excellence" introduced by Shelah for the

analysis of $L_{\omega_1,\omega}$ categoricity, which reduces the

question to that of proving certain unusual versions of the theorems

of Mordell-Weil and Kummer-Bashmakov. I will discuss this and other

aspects of the proof, without assuming any model- or number-theoretic

knowledge on the part of my audience.

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k≥1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and coloured in space.

Brownian Molecular Motors are crucial for cell motility, muscle contraction or any other mechanical task carried out by proteins. After a short introduction to protein motors, I will talk about a numerical appraoch I worked on during the last months, which should enable us to deduct properties for a broad range of protein motors. A special focus should lie on the calculation of the eigenvalue spectrum, which gives insight to motors' stability.

My talk will be about optimal investment in Kabanov's model of currency exchange with transaction costs. The model is general enough to allow a random, discontinuous bid-offer spread. The investor wishes to maximize their "direct" utility of consumption, which is measured in terms of consumption assets linked to some (but not necessarily all) of the traded currencies. The analysis will centre on two conditions under which the existence of a dual minimiser leads to the existence of an optimal terminal wealth. The first condition is a well known, but rather unintuitive, condition on the utility function. The second weaker, and more natural condition is that of "asymptotic satiability" of the value function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer. This is joint work with Luciano Campi.

About a third of us will have a cancer during our lives, and we all know someone with the disease. Despite enormous progress in recent years, so that being diagnosed with cancer is not the death sentence that it once was, treatment can be aggressive, leading to short and long term reductions in quality of life. Cancer and its treatment is still feared, and rightly so - it is a major health concern. Destroying cancer non-invasively using protons or charged light ions such as carbon (Particle Therapy Cancer Research or PTCR) offers advantages over conventional radiotherapy using x-rays, since far lower radiation dose is delivered to healthy normal tissues. PT is also an alternative to radical cancer surgery. Most radiotherapy uses a small electron linear accelerator to accelerate an electron beams to a few million volts and then to generate hard x-rays, whereas CPT uses cyclotrons or synchrotrons to accelerate protons to a few hundred million volts, which themselves sterilise the tumour. More recently, a new concept in accelerators – the “non-scaling Fixed Field Alternating Gradient” accelerator – has been advanced, which offers the prospect of developing relatively compact, high acceleration rate accelerators for a variety of purposes, from neutrino factories and muon acceleration to cancer therapy. However, there are formidable technical challenges to be overcome, including resonance crossing. We have recently been awarded funding in the UK to construct a demonstrator non-scaling FFAG at the Daresbury laboratory (EMMA, the Electron Model with Many Applications), and to design a prototype machine for proton and carbon ion cancer therapy (PAMELA, the Particle Accelerator for MEdicaL Applications). I will describe some of the motivations for developing this new type of accelerator. Finally, although the physics of CPT says that it should be qualitatively and quantitatively better than conventional radiotherapy, the robust clinical analyses (for example, randomised control trials) have not been done, and the meta-analyses seem to suffer from large sample biases. The Particle Therapy Cancer Research Institute (part of the James Martin 21st Century School in Oxford) will study the clinical effectiveness of charged particle therapy to treat cancer, promoting its use in the UK and elsewhere on the basis of robust clinical evidence and analysis.

I will provide an overview of theoretical models aimed at understanding how self-excited oscillations arise when flow is driven through a finite-length flexible tube or channel. This problem is approached using a hierarchy of models, from one to three spatial dimensions, combining both computational and asymptotic techniques. I will explain how recent work is starting to shed light on the relationship between local and global instabilities, energy balances and the role of intrinsic hydrodynamic instabilities. This is collaborative work with Peter Stewart, Robert Whittaker, Jonathan Boyle, Matthias Heil and Sarah Waters.