Past

We shall discuss a simple low order numerical integration scheme for ODEs and DAEs. The scheme has a parameter that allows for regularization of Jacobian of stiff problems and for numerically elucidating multi-scale response, if any, in some problems.

We discuss homological finiteness Bredon types FPm with respect to the class of finite subgroups and seperately with respect to the class of virtually cyclic subgroups. We will concentrate to the case of solubles groups and if the time allows to the case of generalized R. Thompson groups of type F. The results announced are joint work with Brita Nucinkis

(Southampton) and Conchita Martinez Perez (Zaragoza) and will appear in papers in Bulletin of LMS and Israel Journal of Mathematics.

I will show that given smooth embedded Lagrangian L in a Calabi-Yau, one can find a perturbation of L which lies in the same hamiltonian isotopy class and such that the correspondent solution to mean curvature flow develops a finite time singularity. This shows in particular that a simplified version of the Thomas-Yau conjecture does not hold.

We revisit the problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time sorting algorithm achieving this bound up to a constant factor. They established a crucial link between the entropy of the input partial order and the information-theoretic lower bound. However, their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it unpractical. We develop efficient algorithms for sorting under partial information, derived from approximation and exact algorithms for computing the partial order entropy.

This is joint work with S. Fiorini, G. Joret, R. Jungers, and I. Munro.

We explore two different threading approaches on a graphics processing
unit (GPU) exploiting two different characteristics of the current GPU
architecture. The fat thread approach tries to minimise data access time
by relying on shared memory and registers potentially sacrificing
parallelism. The thin thread approach maximises parallelism and tries to
hide access latencies. We apply these two approaches to the parallel
stochastic simulation of chemical reaction systems using the stochastic
simulation algorithm (SSA) by Gillespie. In these cases, the proposed
thin thread approach shows comparable performance while eliminating the
limitation of the reaction system's size.

Link to paper: 

http://people.maths.ox.ac.uk/erban/papers/paperCUDA.pdf

Much effort has been devoted to the study of weak scale particles, e.g. supersymmetric neutralinos, which have a relic abundance from thermal equilibrium in the early universe of order what is inferred for dark matter. This does not however provide any connection to the comparable abundance of baryonic matter, which must have a non-thermal origin. However "dark baryons" of mass ~5 GeV from a new strongly interacting sector would naturally provide dark matter and are consistent with recent putative signals in experiments such as CoGeNT and DAMA. Such particles would accrete in the Sun and affect heat transport in the interior so as to affect low energy neutrino fluxes and can possibly resolve the current conflict between helioseismological data and the Standard Solar Model.

We develop a theory based on relative entropy to show stabilityand uniqueness of extremal entropic Rankine-Hugoniot discontinuities forsystems of conservation laws (typically 1-shocks, n-shocks, 1-contactdiscontinuities and n-contact discontinuities of big amplitude), amongbounded entropic weak solutions having an additional strong traceproperty. The existence of a convex entropy is needed. No BV estimateis needed on the weak solutions considered. The theory holds withoutsmallness condition. The assumptions are quite general. For instance, thestrict hyperbolicity is not needed globally. For fluid mechanics, thetheory handles solutions with vacuum.

Abstract: We will explain a certain natural way to project elements of

the mapping class to simple closed curves on subsurfaces. Generalizing

a coordinate system on hyperbolic space, we will use these projections

to describe a way to characterize elements of the mapping class group

in terms of these projections. This point of view is useful in several

applications; time permitting we shall discuss how we have used this

to prove the Rapid Decay property for the mapping class group. This

talk will include joint work with Kleiner, Minksy, and Mosher.

We consider a dilute stationary system of N particles uniformly distributed in space and interacting pairwise according to a compactly supported potential, which is repellent at short distances and attractive at moderate distances. We are interested in the large-N behaviour of the system. We show that at a certain scale there are phase transitions in the temperature parameter and describe the energy and ground states explicitly in terms of a variational problem

 Rough path theory, invented by T. Lyons, is a successful and general method for solving ordinary or stochastic differential equations driven by irregular H\"older paths, relying on the definition of a finite number of substitutes of iterated integrals satisfying definite algebraic and regularity properties.

Although these are known to exist, many questions are still open, in

particular:  (1) "how many" possible choices are there ? (2) how to construct one explicitly ?  (3) what is the connection to "true" iterated integrals obtained by an approximation scheme ?

  In a series of papers, we (1) showed that "formal" rough paths (leaving aside

regularity) were exactly determined by so-called "tree data"; (2) gave several explicit constructions, the most recent ones relying on quantum field renormalization methods; (3) obtained with J. Magnen (Laboratoire de Physique Theorique, Ecole Polytechnique)  a L\'evy area for fractional Brownian motion with Hurst index <1/4 as the limit in law of  iterated integrals of a non-Gaussian interacting process, thus calling for a redefinition of the process itself.  The latter construction belongs to the field of high energy physics, and as such established by using constructive field theory and renormalization; it should extend to a general rough path (work in progress).

9:45 DH common room coffee

The new model is that the universal small scale structure of high Reynolds number turbulence is determined by the dynamics of thin evolving shear layers, with thickness of the order of the Taylor micro scale,within which there are the familiar elongated vortices .Local quasi-linear dynamics shows how the shear layers act as barriers to external eddies and a filter for the transfer of energy to their interiors. The model is consistent with direct numerical simulations by Ishihara and Kaneda analysed in terms of conditional statistics relative to the layers and also with recent 4D measurements of lab turbulence by Wirth and Nickels. The model explains how the transport of energy into the layers leads to the observed inertial range spectrum and to the generation of intense structures, on the scale of the Kolmogorov micro-scale.

But the modelling also explains the important discrepancies between data and the Kolmogorov-Richardson cascade concept ,eg larger amplitudes of the smallest scale motions and of the higher moments ,and why the latter are generally less isotropic than lower order moments, eg in thermal convection. Ref JCRHunt , I Eames, P Davidson,J.Westerweel, J Fernando, S Voropayev, M Braza J Hyd Env Res 2010

Let C be a smooth plane cubic curve over the rationals. The Mordell--Weil Theorem can be restated as follows: there is a finite subset B of rational points such that all rational points can be obtained from this subset by successive tangent and secant constructions. It is conjectured that a minimal such B can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This talk is concerned with the corresponding problem for cubic surfaces.


We extend for the first time the linear discretization theory of Schaback, developed for meshfree methods, to nonlinear operator equations, relying heavily on methods of Böhmer, Vol I. There is no restriction to elliptic problems or to symmetric numerical methods like Galerkin techniques.

Trial spaces can be arbitrary, but have to approximate the solution well, and testing can be weak or strong. We present Galerkin techniques as an example. On the downside, stability is not easy to prove for special applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. These results remain valid for the general case of fully nonlinear elliptic differential equations of second order. Some numerical examples are added for illustration.

The theory of C*-algebras provides a good realisation of noncommutative topology. There is a dictionary relating commutative C*-algebras with locally compact spaces, which can be used to import topological concepts into the C*-world. This philosophy fails in the case of homotopy, where a more sophisticated definition has to be given, leading to the notion of asymptotic morphisms.

As a by-product one obtains a generalisation of Borsuk's shape theory and a universal boundary map for cohomology theories of C*-algebras.

In this talk we will survey some aspects of social choice theory: in particular, various impossibility theorems about voting systems and strategies. We begin with the famous Arrow's impossibility theorem -- proving the non-existence of a 'fair' voting system -- before moving on to later developments, such as the Gibbard–Satterthwaite theorem, which states that all 'reasonable' voting systems are subject to tactical voting.

Given time, we will study extensions of impossibility theorems to micro-economic situations, and common strategies in game theory given the non-existence of optimal solutions.