Past

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.

There are two main statistical mechanical models of ferromagnetism: the simpler and better-understood Ising model, and the more realistic and more challenging classical Heisenberg model, where the spins are in the 2-sphere instead of in {-1,+1}. In dimensions one and two, the classical Heisenberg model with nearest-neighbor interactions has no phase transition, but in three dimensions it has been intractable.

To shed some light on the qualitative behavior of the 3D Heisenberg model, we use the versatile tools of mean-field theory and Stein's method in recent work with Elizabeth Meckes, studying the Heisenberg model on a complete graph with the number of vertices going to infinity. Our results include detailed descriptions of the magnetization, the empirical spin distribution, the free energy, and a second-order phase transition.

A map betweem metric spaces is a bilipschitz homeomorphism if it

is Lipschitz and has a Lipschitz inverse; a map is a bilipschitz embedding

if it is a bilipschitz homeomorphism onto its image. Given metric spaces

X and Y, one may ask if there is a bilipschitz embedding X--->Y, and if

so, one may try to find an embedding with minimal distortion, or at least

estimate the best bilipschitz constant. Such bilipschitz embedding

problems arise in various areas of mathematics, including geometric group

theory, Banach space geometry, and geometric analysis; in the last 10

years they have also attracted a lot of attention in theoretical computer

science.

The lecture will be a survey bilipschitz embedding in Banach spaces from

the viewpoint of geometric analysis.

Abstract: In this talk, I will discuss the proportionality between tree amplitudes and the ultraviolet divergences in their one-loop corrections in Yang-Mills and (N < 4) Super Yang-Mills theories in four-dimensions. From the point of view of local perturbative quantum field theory, i.e. Feynman diagrams, this proportionality is straightforward: ultraviolet divergences at loop-level are absorbed into coefficients of local operators/interaction vertices in the original tree-amplitude. Ultraviolet divergences in loop amplitudes are also calculable through on-shell methods. These methods ensure manifest gauge-invariance, even at loop-level (no ghosts), at the expense of manifest locality. From an on-shell perspective, the proportionality between the ultraviolet divergences the tree amplitudes is thus not guaranteed. I describe systematic structures which ensure proportionality, and their possible connections to other recent developments in the field.

• Savina Joseph - Current generation in solar cells
• Shengxin Zhu - Spectral distribution, smoothing effects and smoothness matching for radial basis functions
• Ingrid von Glehn - Solving surface PDEs with the closest point method

The section conjecture of Grothendieck's anabelian geometry speculates about a description of the set of rational

points of a hyperbolic curve over a number field entirely in terms of profinite groups and Galois theory.

In the talk we will discuss local to global aspects of the conjecture, and what can be achieved when sections with

additional group theoretic properties are considered.

Complex networks have been used to model almost any

real-world complex systems. An especially important

issue regards how to related their structure and dynamics,

which contributes not only for the better understanding of

such systems, but also to the prediction of important

dynamical properties from specific topological features.

In this talk I revise related research developed recently

in my group. Particularly attention is given to the concept

of accessibility, a new measurement integrating topology

and dynamics, and the relationship between frequency of

visits and node degree in directed modular complex

networks. Analytical results are provided that allow accurate

prediction of correlations between structure and dynamics

in systems underlain by directed diffusion. The methodology

is illustrated with respect to the macaque cortical network.

Uncertainty quantification can begin by specifying the initial state of a system as a probability measure. Part of the state (the 'parameters') might not evolve, and might not be directly observable. Many inverse problems are generalisations of uncertainty quantification such that one modifies the probability measure to be consistent with measurements, a forward model and the initial measure. The inverse problem, interpreted as computing the posterior probability measure of the states, including the parameters and the variables, from a sequence of noise-corrupted observations, is reviewed in the talk. Bayesian statistics provides a natural framework for a solution but leads to very challenging computational problems, particularly when the dimension of the state space is very large, as when arising from the discretisation of a partial differential equation theory.

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In this talk we show how the Bayesian framework leads to a new algorithm - the 'Variational Smoothing Filter' - that unifies the leading techniques in use today. In particular the framework provides an interpretation and generalisation of Tikhonov regularisation, a method of forecast verification and a way of quantifying and managing uncertainty. To deal with the problem that a good initial prior may not be Gaussian, as with a general prior intended to describe, for example a geological structure, a Gaussian mixture prior is used. This has many desirable properties, including ease of sampling to make 'numerical rocks' or 'numerical weather' for visualisation purposes and statistical summaries, and in principle can approximate any probability density. Robustness is sought by combining a variational update with this full mixture representation of the conditional posterior density.