Past

Respiration is a redox reaction in which oxidation of a substrate (often organic) is coupled to the reduction of a terminal electron acceptor (TEA) such as oxygen. Iron oxides in various mineral forms are abundant in sediments and sedimentary rocks, and many subsurface microbes have the ability to respire using Fe(III) as the TEA in anoxic conditions. This process is environmentally important in the degradation of organic substrates and in the redox-cycling of iron. But low mineral solubility limits the bioavailability of Fe(III), which microbes access primarily through reductive dissolution. For aqueous nutrients, expressions for microbial growth and nutrient uptake rates are standardly based on Monod kinetics. We address the question of what equivalent description is appropriate when solid phase Fe(III) is the electron acceptor.

Given a diffusion in R^n, we prove a small-noise expansion for its density. Our proof relies on the Laplace method on Wiener space and stochastic Taylor expansions in the spirit of Benarous-Bismut. Our result applies (i) to small-time asymptotics and (ii) to the tails of the distribution and (iii) to small volatility of volatility.

We shall study applications of this result to stochastic volatility models, recovering the Berestycki- Busca-Florent formula (using (i)), the Gulisashvili-Stein expansion (from (ii)) and Lewis' expansions (using (iii)).

This is a joint work with J.D. Deuschel (TU Berlin), P. Friz (TU Berlin) and S. Violante (Imperial College London).

A pseudofinite field is a perfect pseudo-algebraically closed (PAC) field which

has $\hat{\mathbb{Z}}$ as absolute Galois group. Pseudofinite fields exists and they can

be realised as ultraproducts of finite fields. A group $G$ is geometrically

represented in a theory $T$ if there are modles $M_0\prec M$ of $T$,

substructures $A,B$ of $M$, $B\subset acl(A)$, such that $M_0\le A\le B\le M$

and $Aut(B/A)$ is isomorphic to $G$. Let $T$ be a complete theory of

pseudofinite fields. We show that, geometric representation of a group whose order

is divisibly by $p$ in $T$ heavily depends on the presence of $p^n$'th roots of unity

in models of $T$. As a consequence of this, we show that, for almost all

completions of the theory of pseudofinite fields, over a substructure $A$, algebraic

closure agrees with definable closure, if $A$ contains the relative algebraic closure

of the prime field. This is joint work with Ehud Hrushovski.

The computational analysis of a mathematical model describing a complex system is often based on the following roadmap: first, an experiment is conceived, in which the measured data are (either directly or indirectly) related to the input data of the model equations; second, such equations are computationally solved to provide iconographic reconstructions of the unknown physical or physiological parameters of the system; third, the reconstructed images are utilized to validate the model or to inspire appropriate improvements. This talk will adopt such framework to investigate three applied problems, respectively in solar physics, neuroscience and physiology. The solar physics problem is concerned with the exploitation of hard X-ray data for the comprehension of energy transport mechanisms in solar flares. The neuroscientific problem is the one to model visual recognition in humans with the help of a magnetocencephalography experiment. Finally, the physiological problem investigates the kinetics of the kidney-bladder system by means of nuclear data.

While a general framework of approximating the solution to Hamilton-Jacobi-Bellman (HJB) equations by difference methods is well established, and efficient numerical algorithms are available for one-dimensional problems, much less is known in the multi-dimensional case. One difficulty is the monotone approximation of cross-derivatives, which guarantees convergence to the viscosity solution. We propose a scheme combining piecewise freezing of the policies in time with a suitable spatial discretisation to establish convergence for a wide class of equations, and give numerical illustrations for a diffusion equation with uncertain parameters. These equations arise, for instance, in the valuation of financial derivatives under model uncertainty.

This is joint work with Peter Forsyth.

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann's principle $S = k\log W$, does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold's `cat' map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.

This is intended as an introductory talk about one of the most

important (and most geometric) aspects of Geometric Group Theory. No

prior knowledge of any maths will be assumed.

We present some recent results concerning domain wall motion in one-dimensional nanowires, including the existence, velocity and stability of travelling-wave and precessing solutions.  We consider the case of unixial anisotropy, characteristic of wires with symmetrical (e.g., circular) cross-section, as opposed to strongly anisotropic geometries (films and strips) that have received greater attention.  This is joint work with Arseni Goussev and Valeriy Slastikov.

Organisers: Hilary Priestley, Drew Moshier and Leo Cabrer.

This will be dedicated principally to extensions of duality theory beyond zero-dimensional structures and to its application in novel settings. Topics that are likely to feature include duality for bilattice-based structures and associated semantics; extensions to compact Hausdorff spaces, bitopological duality, and duality for continuous data; applications to coalgebraic logic. We shall be seeking two-way interaction between those focused on a particular application and those who are seeking to extend the theory. Keynote speakers will be Mike Mislove and Drew Moshier. Samson Abramsky will be away from Oxford fromJune 12, but we are grateful for his offer to give a talk on June 11. We are also pleased to announce that, through the good offices of Georg Gottlob (Oxford Department of Computer Science), we are able to include within W1 a tutorial lecture on the applications of bilattice semantics to computer science; this will be given by Ofer Arieli.

 Nuclear fusion offers the prospect of abundant clean energy production, but the physical and engineering challenges are very great. In nuclear fusion reactors, the fuel is in the form of a plasma (charged gas) which is confined at high temperature and density using a toroidal magnetic field. This configuration is susceptible to turbulence, which transports heat out of the plasma and prevents fusion. It is believed that rotating the plasma suppresses turbulence, but experiments are expensive and even modest numerical simulation requires hundreds of thousands of CPU hours. We present a numerical technique for one of the five phase-space dimensions that both improves the accuracy of the calculation and greatly reduces the resolution required.

This is joint work with Dani Wise and builds on his earlier

work. Let M be a compact oriented irreducible 3-manifold which is neither a

graph manifold nor a hyperbolic manifold. We prove that the fundamental

group of M is virtually special. This means that it virtually embeds in a

right angled Artin group, and is in particular linear over Z.

The Schramm-Loewner evolution (SLE(\kappa)) is a family of random fractal curves that arise in a natural way as scaling limits of interfaces in critical models in statistical physics. The SLE curves are constructed by solving the Loewner differential equation driven by a scaled Brownian motion. We will give an overview of some of the almost sure properties of SLE curves, concentrating in particular on properties that can be derived by studying the the geometry of growing curve locally at the tip. We will discuss a multifractual spectrum of harmonic measure at the tip, regularity in the capacity parameterization, and continuity of the curves as the \kappa-parameter is varied while the driving Brownian motion sample is kept fixed.

This is based on joint work with Greg Lawler, and with Steffen Rohde and Carto Wong.

An intriguing, and largely open, question in mathematical fluid dynamics is whether solutions of the Navier-Stokes equations converge in some sense to a solution of the Euler equations in the zero viscosity limit. In fact this convergence could conceivably fail due to oscillations and concentrations occuring in the sequence.

In the late 1980s, R. DiPerna and A. Majda extended the classical concept of Young measure to obtain a notion of measure-valued solution of the Euler equations, which records precisely these oscillation and concentration effects. In this talk I will present a result recently obtained in joint work with L. Székelyhidi, which states that any such measure-valued solution is generated by a sequence of distributional solutions of the Euler equations.

The result is interesting from two different viewpoints: On the one hand, it emphasizes the huge flexibility of the concept of weak solution for Euler; on the other hand, it provides an example of a characterization theorem for Young measures in the tradition of D. Kinderlehrer and P. Pedregal where the differential constraint on the generating sequence does not satisfy the constant rank condition.