# B(A)MC : titles and abstracts of presentations

(A) (B) (C) (D) (E) (F) (G) (H) (I) (J) (K) (L) (M) (N) (O) (P) (Q) (R) (S) (T) (U) (V) (W) (X) (Y) (Z)

"Instability of Vortex Leapfrogging"

Abstract
In this paper I present some computer simulations which give a new twist to a 100-year old problem in the theory of vortex motion. While Love showed, in 1894, that two vortex pairs in an inviscid fluid can 'leapfrog' repeatedly, it turns out that this motion can be unstable in a most peculiar way.

"Propagating reaction fronts in zirconia tubes"

Abstract
Solid oxide fuel cells (SOFC) are a clean and efficient method of energy production. A mathematical model is presented of the flow, mass fraction and temperature profiles in the zirconia tube of the SOFC, when the tube is initially heated from one end. Experimental results show that a reaction front propagates down the zirconia tube from the point of initial heating. The model takes into account diffusion, advection and reaction but does not include any electrochemistry. It is assumed that the reaction front propagates as a wavefront with constant speed along the tube. Matched asymptotic expansions are used to find solutions for the mass fractions, temperature and flow profiles in different asymptotic regions. Numerical and, where possible, analytical results are given. It is found that the speed of this travelling wave decreases as the activation temperature of the reaction increases.
Joint work with: John Billingham and Andy King

"Higher power residue codes and unimodular lattices"

Abstract
We will introduce a construction of lattices by using the higher power residue codes. The lattices obtained by this construction are all invariant under an action of the group SL(2,p) on the vector space R^{3(p+1)}. The centralizer C of this action is a ring acting on R^{3(p+1)}. Invertible elements in C act on the set of SL(2,p)-invariant lattices. Certain even unimodular lattices are shown to be constructed by this method. This includes some Niemeier lattices especially the Leech lattice. Using elements of the centralizer of this action we can define lifting for both higher power residue codes and quadratic residue codes. Certain self dual codes are obtained by this procedure.

"The non-integrable coupled nonlinear Schroedinger equations"

Abstract
An investigation into the homoclinic structures of the perturbed non-integrable coupled nonlinear Schroedinger equations which arise in nonlinear optics and other fields as a model for modulational instabilities in optical fibres and exhibit much richer dynamics than the perturbed single nonlinear Schroedinger equation. Melnikov methods and Hamiltonian techniques are used together with numerical methods to investigate the homoclinic structure of the perturbed equations.
Joint work with: VM Rothos and AR Champneys

"hp-Edge finite element for approximation of solutions of Maxwell's equations"

Abstract
If a standard finite element method is used to approximate the time harmonic Maxwell's equations, then in many cases of practical interest, the sequence of approximations may converge at an optimal rate to a function that is not a solution of the Maxwell equations. Equally well, if one is trying to compute resonant frequencies, then a standard method results in a discrete spectrum that is polluted with spurious, non-physical modes that are not readily distinguished from the physical values. We discuss these phenomena and describe a modified approach that does not suffer from this type of problem.

"Bifurcations in the flow between two infinite parallel planes"

Abstract
We consider the impact of symmetry assumptions on the bifurcation sequence for the flow of a viscous incompressible fluid between infinite, parallel, porous planes. We assume that the flow possesses a self-similarity for which the full Navier-Stokes equations reduce exactly to a set of nonlinear, ordinary-differential equations. The existence of a bifurcation sequence for the two-dimensional flow in an infinite-length channel with uniform transpiration at the walls has been described by Cox (1991). However this sequence is not present if one instead assumes an axisymmetric flow. We discuss the two problems and show that in fact there exists the possibility of a bifurcation to swirl (from a non-swirling state) and non-axisymmetry (supercritical and subcritical respectively) at the same Reynolds number.

"Buoyant convection in cavities in a strong magnetic field"

Abstract
Three-dimensional buoyant convection in a vertical cylindrical cavity with horizontal temperature gradient in a strong, uniform, vertical magnetic field is considered. An asymptotic solution of the problem in the inertialess approximation is obtained for high values of the Hartmann number, Ha. Symmetry of the flow is investigated for small Peclet number, Pe = O(1/Ha). Even for such a small Peclet number, high-velocity jets in the parallel layer lead to significant convective heat transfer. If undisturbed temperature distribution is an even function of the vertical co-ordinate, it creates an asymmetric heat flux at the boundary of the core region. This results in an asymmetric temperature distribution of O(Pe.Ha^1/2) in the core. The addition is small, but the symmetry properties of the problem are such that an odd temperature component with respect to the vertical co-ordinate produces an even core velocity of the same magnitude as that of the odd one. The effect is most expressed for a fixed external heat flux.
Joint work with: S Molokov

"Horizontal girders"

Abstract
Effects of varying width and shape of girders.

"Invariant curves for planar piecewise isometries"

Abstract
Piecewise isometric maps that are continuous on a partition into convex polygons show dynamics that is highly analogous to smooth area-preserving maps. In particular, for typical maps of this type one can find elliptic periodic points surrounded by islands of invariant curves sitting in a "sea" of aperiodically coded states. This poster presents some examples some examples of piecewise isometries that have aperiodically coded non-smooth invariant curves. These include non-trivial interval exchange transformations on the curves. Joint work with:
Joint work with: Arek Goetz

"A low-frequency analogue of the Saint-Venant principle for an elastic semi-strip"

Abstract
Plain harmonic motion of an elastic semi-strip is considered. Asymptotic conditions are derived to provide decay of low-frequency vibrations induced by prescribed end data. These represent a perturbation of the well-known static formulae expressing the Saint-Venant principle. The methodology developed is of great importance for formulating boundary conditions in 2D plate theories. In particular it enables to analyse plate vibrations under self-equilibrium (statically)loads.
Joint work with: J Kaplunov

"On the matrix type of a ring "

Abstract
The matrix type of a ring R is defined to be a multiplicative congruence relation\theta=\theta(R) on N:given by m\theta n if M_m(R)\cong M_n(R). This is a multiplicative congruence on N. It is well-known that there are rings for which this relation is not the equality, but if it is then we say that R has the Invariant Matrix Number (IMN) property.
In this talk we will investigate properties of the matrix type, IMN, its relationship to the Invariant Basis Number (IBN) and allied properties.
Joint work with: Peter Vamos

"Quasi-2d decaying turbulence in channel flow"

Abstract
A DNS of quasi-2d decaying turbulence in channel flow was performed in order to verify the theory (Nikitin and Chernyshenko [1]) according to which near-wall coherent structures result from an instability caused by anisotropy of normal Reynolds stresses. Incompressible flow of constant mean pressure gradient in a plane channel flow was calculated. The initial conditions and, hence, the entire solution is independent of the longitudinal coordinates. We supply an initial random velocity field which had no organized structures and was divergence-free. The cross-flow eventually decays but before it comes to rest coherent structures in the form of vortices in v and w (the velocity components) and their by-products streaks in u near the walls are expected to emerge. The work is in progress and it shows favorable evidence in favour of the theory. [1] Nikitin, N.V and Chernyshenko, S.I., 1997, Fluid Dynamics, Vol. 32, No.1, pp. 18-23.

"Modelling ion and solute transport in the eye lens"

Abstract
The cells of eye lens have no vascular supply, since this would scatter in-coming light rays. Instead the lens cells rely upon an intricate network of transport processes that channels nutrients and waste products between the lens surface and the lens nucleus. In this talk I will outline some basic lens physiology and present results of simulations using a finite element implementation of a simple lens model.

"Entropy jumps in the presence of a spectral gap"

Abstract
Let X be a square-integrable random variable with density f satisfying a spectral gap inequality with constant c for the f-Laplacian, and Y be an independent copy of X. Let E_1= -integral(f log f) be the entropy of X, E_2 the entropy of the normalised sum (X+Y)/sqrt(2) and E_infty the entropy of the Gaussian with the same variance. Then
E_2 - E_1 >= c/4 (E_infty - E_1)
so that a fixed proportion of the entropy gap from the Gaussian is captured with each convolution.
The proof depends upon a new formula for the entropy of a marginal which can be regarded as an inverse to the classical Brunn-Minkowski inequality.

"Bifurcations in Differential-Algebraic Equations"

Abstract
The semi-explicit, differential-algebraic boundary-value problem
Lx=\lambda f(x,y),g(x,y)=0,x(0)=x(1)=0,
is considered, where L is an elliptic operator, and conditions are given under which a bifurcating branch emanates from a trivial solution at a critical parameter value. Moreover, it is shown that there are countably many branches which bifurcate at this parameter value, and using a uniqueness theorem for non-Lipshitz differential equations, we show that the problem also possesses a continuum of bifurcation points.
It is shown by example that solution branches need not be unbounded, in contrast to the well- known results of Krasnoselskii, Crandall, Rabinowitz et al, and consequences for numerical discretisations of the problem will be discussed.
Joint work with: R Laister

"Homoclinic orbits of invertible maps"

Abstract
A numerical method for locating and identifying all homoclinic orbits of invertible maps in any (finite) dimension.

"On the mechanisms of propagation failure in biological excitable systems"

Abstract
Hodgkin and Huxley (1952, Nobel Prize 1963) have proposed a mathematical model of the electric action of the giant squid axon, the first to describe such an exclusively biological phenomenon as excitability. This spawned a large family of models describing other biophysically related phenomena, e.g. excitability of heart muscle. All these models are rather complicated, do not admit analytical solution, and mostly treated numerically. FitzHugh (1961) has suggested that the key properties of the Hodgkin-Huxley system of equations can be qualitatively simulated by an appropriate modification of the van der Pol's model of an electronic circuit with an active element. Nagumo et al. (1963) have proposed a spatially-modified version of FitzHugh's system, to describe the propagation of excitation. Throughout the last four decades, the FitzHugh-Nagumo (FHN) system and its numerous modifications have served well as very simple but qualitatively reasonable models of the complicated processes of excitation and propagation in nerve fibre, heart muscle and other biological spatially-extended excitable systems.
It appears that while successfully describing successful propagation of excitation, FHN-type systems fail to adequately describe propagation failure. Yet it is propagation failure that is most important for some applications of these models, e.g. in relation with onset of fibrillation of heart muscle and sudden cardiac death. In this talk, I will demonstrate a particular kind of propagation failure which is IMPOSSIBLE in FHN-type systems. I will suggest an explanation of this impossibility and identify the key feature responsible for this difference between the biophysical models and their FHN caricature. I will also present a simple model, that retains this key feature of the biophysical models, can reproduce this type of propagation failure and still admits analytical treatment.

"Recognizing the unknot"

Abstract
Every fisherman understands the problem one faces in untangling a tangled ball of string. To state it as a problem which is mathematically precise, mathematicians make the string infinitely long, so that the ends cannot be "passed through". (Alternatively, one considers knots in circular strings). One can then ask the question: "Is the string unknotted? How can we tell?" This simple question turns out to be a basic question in that part of 3-manifold topology which relates to complexity theory. In this talk we will survey what is known.

"Intermittency in weak turbulence"

Abstract
To be supplied

"Micro-fluidics: there's plenty of room at the bottom(1) for fluid dynamicists!"

Abstract
Laboratory on chip technology, bacterial locomotion, gamete transport, mucous transport, suspensions, red blood cell motion, ink-jet printing are all areas of study in micro-hydrodynamics. At these small micron scales the presence of boundaries, surface forces, contact lines, Brownian motion, non-Newtonian fluid behaviour are aspects of the flow that might also be important. From the micro-hydrodynamic viewpoint, transport and mixing, the very low Reynolds number equivalent of turbulence, are of great significance.
This lecture will consider two features of this research area - mixing and feeding processes that are associated with micro-organisms and potential advances to 'laboratory on chip' technology by taking advice from nature on the movement of fluids of differing viscosities in narrow channels.
(1) From R. P. Feyman, the "Father of Nanotechnology" lecture to the American Physical Society, 29th December, 1959 on 'There's plenty of room at the bottom'.
Joint work with: SR Otto

"Chaotic flow in a pulsating pipe"

Abstract
The unsteady flow in a circular pipe driven by time-periodic motion of the wall is investigated. The pipe is taken to be of infinite extent and the fluid is assumed incompressible. A similar problem involving pulsating flow in a channel was studied by Hall & Papageorgiou (JFM V. 393, 1999). For the axisymmetric problem, it is found that sufficiently small driving frequencies, with a fixed amplitude, produce a flow which is in synch with the wall motion. However, when the driving frequency is increased, eventually a Hopf bifurcation is encountered and the two motions become asynchronous and, sometimes, chaotic. The problem is tackled primarily by numerical means, although, in a steady-streaming limit, some progress can be made using high Reynolds number asymptotics.
Joint work with: P Hall and DT Papageorgiou

Poster for Micromechanics of solids:

"General Becker & Doering equations: effect of dimer interactions"

Abstract
Becker & Doering presented an enduring model of nucleation in 1935; clusters form by the addition, or subtraction, of single particles (monomers) with no interaction between larger clusters. Such larger clusters evolve by maintaining a dynamic balance of monomer aggregation and fragmentation. The effect of dimers (a two-particle cluster) interactions on the Becker-Doering model of nucleation is investigated. We consider the problem with size-independent aggregation and fragmentation coefficients and initially a constant monomer concentration. Either an equilibrium, or a steady-state, solution is found; the former when fragmentation is stronger than aggregation, the latter otherwise. By employing asymptotic techniques, the manner in which the system reaches these states is examined. The dimer interaction is found to accelerate the system toward the equilibrium solution, but has little impact on the relaxation time to the steady-state solution. In cases where aggregation is dominant, the steady-state cluster size distribution can only be determined consistently when the manner of approach to steady-state is also known. In the terminology of asymptotic methods, one needs to know the first correction term in order to deduce the leading-order solution. We indicate how this can be derived and so at steady-state find a flux of matter to larger aggregation numbers due to monomer interactions, with a small and decreasing reverse flux due to dimer interactions. We then consider the case of constant density, that is allowing the monomer concentration to vary, and investigate the effect of a strong dimer interaction on the convergence to equilibrium. Two timescales are present and each are investigated. We determine the intermediate meta-stable state, the final state and the timescales over which the system relaxes into these states.

Poster for Mathematical biology 2:

"The matrix effect in vesicle formation; a theoretical approach"

Abstract
Self-reproducing molecular systems have been investigated experimentally over recent years since they exhibit many of the fundamental properties of a living system, and so have the potential to model life in pre-biotic conditions. One such self-reproducing system which has received attention is that of spherical lipid bi-layers: vesicles. For large enough concentrations of surfactant these also have the attractive property of spontaneously forming structures analogous to a biological cell. More recent experiments start with a stock of monomers and a small amount of pre-added vesicles of a particular diameter. The final distribution of vesicle size is found to be strongly biased towards the diameter of the pre-added vesicles; a size-templating matrix'' effect (for example see Berclaz, J. Phys. Chem. B 105, 1056 (2001)) . We develop a microscopic model of vesicle formation, based on a novel generalisation of the Becker-Doeoring equations and solve this numerically. Furthermore we reduce this model to a low dimensional system of ODE's in the macroscopic limit and solve it using a combination of matched asymptotics and phase plane analysis; concluding that the model correctly captures the experimentally observed behaviour.

"Modelling fungal growth in patchy environments: Derivation, calibration and solution"

Abstract
Fungi form a vital component in nearly all natural vegetative ecosystems. Moreover, certain fungi have great potential for use as biocontrol agents in agriculture and as bioremediation agents in e.g. the cleansing of polluted landscapes. The study of this important class organisms is very difficult using traditional experimental means alone, particularly when the growth environment is heterogeneous as is most often the case in their natural habitats. In this talk, a two-dimensional partial differential equation model describing the branching-type architecture of mycelial fungi will be discussed by careful consideration of events occurring at the micro-scale. The model equations are of mixed hyperbolic-parabolic type and it is essential in their solution that both mass is conserved and positivity is maintained. An efficient numerical scheme, using flux limiters in the spatial discretization and time splitting in the integration proceedure, will be presented which satisfies these important demands. We will show that on calibrating the model using known properties for the species Rhizoctonia solani information is provided on the growth behaviour which is in very good qualitative and quantitative agreement with experimental data. A discrete probabilistic model derived from the continuum model will also be discussed and preliminary results of its application to a soil-like environment will be presented.
Joint work with: Helen Jacobs, Fordyce Davidson, Geoff Gadd and Karl Ritz

"The rupture of a thin film on a horizontal plate"

Abstract
The rupture of a thin film on a horizontal plate with an insoluble surfactant is considered. Marangoni and surface viscous stresses are included. Lubrication theory yields equations for the film surface, the surface velocity and the surface concentration. Rupture occurs for all values of the surface viscosity and the rupture time compared to the clean case may be postponed by more than a factor of 4 for some initial conditions. Similarity behavior is also explored.
Joint work with: C DeBisschop, SA Snow and LP Cook

"Mathematical modelling of surfactant driven flows"

Abstract
We discuss several situations where the standard mathematical model for the motion of an aqueous solution of soluble surfactant is insufficient to describe experimentally observed behaviour. We focus on situations where we have an interface which is not in thermodynamic equilibrium with the solution. In this case, there is no functional relationship between the concentration of surfactant residing in the free surface and the concentration just below. Further, the surface tension may vary due to changes in both the surface and subsurface surfactant concentrations.
We present a revised model which addresses these points, and we discuss the model's solution.

"Fractals and Morse Code"

Abstract
Recently, working with J.P.Huke and M.R.Muldoon, we have been studying the use of iterated function systems (IFS) as models of digital communications channels. This approach allows the use of delay embedding methods for analysing the channel output, and so allows signal processing techniques to be developed which do not rely on assuming that the channel is linear.
In practice, a channel might be a piece of wire or an optical fibre (it may be something much more complicated) and so it would be good to be able to derive IFS channel models from appropriate partial differential equations which describe the basic physics of the channel in question.
This talk will describe some results we have obtained for the case of the (distortionless) telegrapher's equation driven by random sequences of short pulses. We show how to constuct for this system, an IFS on a suitable Banach space and show that this has a finite-dimensional compact attractor. We derive an upper bound for the box- counting dimension of this set and show that it is proportional to the rate at which symbols are input to the channel.
Joint work with: AG Brown

"Receptivity of boundary-layers to vortical disturbances"

Abstract
We consider the boundary-layer receptivity at the leading edge of a streamlined solid body with finite downstream thickness. The mean flow is taken to be two-dimensional and incompressible. Theory has been developed recently for the interaction of acoustic waves with Rankine bodies and generalisations of Rankine bodies. In this paper we consider the effect of small amplitude vortical disturbances in the mean flow on the boundary-layer receptivity.
The development of the unsteady disturbances in the boundary-layer are forced by the unsteady slip velocity obtained from the outer potential solution. In order to solve the outer velocity field it is necessary to split the flow into two parts: a vortical part which is a known function of the external disturbance and a potential part to be found by solving a linear inhomogeneous wave equation.
Since the body has a stagnation point at its leading edge the vortical part develops a singularity along the entire body surface. Hence, it is necessary to further split the flow in order to determine the outer flow such that this singular behaviour is cancelled at the body. We apply this method to the above example and introduce a numerical scheme and discuss the limitations and problems that arise in this case. Numerical Results will be presented.
Joint work with: PW Hammerton

"Visions of maths and science"

Abstract
One of the best ways to teach and learn mathematics is through hands on workshops As part of a public understanding of science project in collaboration with the EPSRC and the Royal Institution, a team from the University of Bath has been giving workshops to young people around Bath and Bristol. These have been on magic, chaos, liquid Nitrogen, ... . Each workshop has been professionally filmed and extra graphis included. The resulting videos form (we hope) a valuable teaching resource. I will describe this project in more detail and show some clips from the videos.

"A Product of Dirichlet Spaces"

Abstract
We present a new construction for the product of Dirichlet spaces. These result can be applied also for a constructing product of a large class differential operators and Markov processes. We investigate after that properties of capacity and negligible sets on the product space.

"Classical and quantum nonlinear Schroedinger equations"

Abstract
The classical nonlinear Schroedinger (NLS) models are nonlinear pdes in complex fields, \psi(x,t), appearing in the two types repulsive' (c>0, self-defocussing) and attractive (c<0, self-focussing). The real coupling constant c controls the nonlinearity in these NLS equations---which are of course -i\psi_t=\nabla^2\psi-2c\psi^*\psi^2, with x\in R^d and t\in R. For d=1, only, the translationally invariant NLS equations are nonlinear completely integrable Hamiltonian systems in the usual Liouville-Arnold sense (extended to infinite dimensional dynamical systems) for any c\ne0. For d=2 or d=3 the systems are not completely integrable, there being an insufficient number of constants commuting under the Poisson bracket {.,.} [1]. Under translational invariance the quantum NLS models show the same features: for d=1 there is quantum complete integrability while for d=2 or d=3 the systems (c\ne0) are not quantum integrable. These features become relevant to recent experiments on Bose-Einstein condensed metal vapours, e.g. ^{87}Rb vapour, held in magnetic traps at circa 300 nanoKelvin [2]. Here c>0 and d=3, and moreover the magnetic trap, modelled as a paraboloidal potential, breaks translational invariance and linear momentum is not a constant (not a good quantum number'). Even so we have successfully calculated [3] the 2-point quantum correlation functions of the form <\hat\psi^\dag(x,\tau)\hat\psi(x',\tau')> for the quantum fields \hat\psi, \hat\psi^\dag ([\hat\psi,\hat\psi^\dag]=\hbar\delta(x-x') etc. at equal times, [.,.] is the Lie bracket and \hbar=h/2\pi is Planck's constant and we set \hbar=1). Here \tau is a thermal time for thermal equilibrium (which will be explained). So far a tantalising measure of agreement is obtained with the experiments [2] and we expect still better agreement from further experiments still to be done with particular reference to the breakdown of translational invariance. For c<0 and d=3 still other experiments [4] show instability and (literally!) an actual blow-up.' But for the classical NLS model for c<0 and d=3 [5] we show blow-up' and collapse' in the sense of Zakharov [5] in which the blow-up displays itself as a \delta-function singularity in the density \psi^*\psi centred at the origin of the trap x=0.
[1] Lecture Integrable Turbulence,' VE Zakharov at the III Potsdam-V Kiev Workshop,' Clarkson University, Potsdam NY, USA, August 1--11, 1991.
[2] I Bloch et al, Nature 403, 166 (2000).
[3] NM Bogoliubov et al, Europhysics Letters 55(6), 755--761 (2001).
[4] SL Cornish et a, Phys. Rev. Lett. 85 1795 (2000) and subsequent experiments.
[5] A Rybin et al, Phys. Rev. Lett. E62, 6224 (2000)

"Commuting Involution Graphs"

Abstract
Given a group G and a subset X of G, a commuting graph has vertex set X and edges {x, y} whenever x and y in X commute. The authors consider the cases when G is a symmetric group and X is a conjugacy class of involutions of G. They determine when such graphs are connected and the diameter of graph in the connected cases. They also give some information about the orbits of vertex stabilizers on the graph.
Joint work with: C Bates, S Perkins and P Rowley

"Relativistic localisation and causality for unsharp quantum measurements"

Abstract
The conflict between relativistic causality and localizability is analyzed in the light of the existence of unsharp localization observables. A theorem due to S. Schlieder is generalized, showing that the assumption of local commutativity implies the localization observable in question to be unsharp in a strong sense. Furthermore, a recent generalization of a theorem of Luders is applied to demonstrate that local commutativity is a necessary consequence of Einstein causality even in the case of unsharp observables if they admit local measurements. These findings seem to corroborate the result known for sharp localization observables, namely, that they cannot be measured by means of local operations.

"Linearity of pro-p groups"

Abstract
This talk will consider recent work with Marcus du Sautoy on the linearity of pro-p groups analytic over pro-p rings.

"A Model of the Development of cytochrome c oxidase negative regions in muscle fibres"

Abstract
Skeletal muscle fibres are large multi-nucleated cells. The primary source of energy to the muscle fibres is supplied by their mitochondria and mutations in the mitochondrial DNA (mtDNA) can cause serious skeletal myopathies. One signature of these myopathies is a existance of cytochrome c oxidase (COX) deficient regions along the length of the individual fibres, associated with the presence of high levels of mutated mtDNA in those sections of the fibre. Serial muscle biopsies show that these regions often take the form of long sections of COX deficient fibre, with the rest of the fibre having normal COX levels. These experiments give us a snapshot in time of the COX deficiency, but how do these regions form over time? Using a stochastic simulation, we model an individual muscle fibre as a series of connected compartments, with each compartment representing the cytoplasm, including the mitochondria, supported by a single nucleus. Within each compartment we use a model of cellular mtDNA populations. By allowing the motion of mtDNA along the length of the muscle fibre, modelled as diffusion between compartments, we are able to investigate the formation and development of the COX deficient regions.
Joint work with: David C Samuels, Patrick F Chinnery and Joanna L Elson

Poster:

"A model of the nuclear control of mitochondrial DNA replication"

Abstract
Mitochondria are the semi-autonomous organelles that are responsible for generating the majority of the energy required by mammalian cells under normal conditions. They are only semi-autonomous because the replication, transcription and translation of the DNA molecules within the mitochondrion, mtDNA, are ultimately controlled by the cell nucleus. We present a series of three models of the nuclear control of mtDNA replication, with an increasing complexity in the role of mtDNA mutations in the models. We solve these deterministic models exactly, and compare these solutions to the results of stochastic simulations of the same systems. We use the steady states of the deterministic model to explain behaviors that are often seen in the cells of patients affected by mitochondrial diseases, and that also occur with age. The parameters of these models illustrate the dual control of mitochondria by both the nuclear and mitochondrial DNA.

"Spatial pattern formation in a model of vegetation growth"

Abstract
We consider a spatial version of Watson and Lovelock's (1983) model daisyworld'. Two plant types, black daisies and white daisies compete on a hyperthetical planet, stabilizing the global temperature via an albedo feedback. Numerical solutions show a striped pattern of black and white daisies. A stability analysis shows that there are two mechanisms involved in the pattern formation. A Turing-like process causes the uniform equilibrium state to be unstable to non-constant perturbation.

"A Renormalization approach for the quantum Frenkel-Kontorova model"

Abstract
I will show a generalization of the classical Transition by Breaking of Analiticity in the Frenkel-Kontorova model (a discrete sine-Gordon) to finite Planck's constant and temperature. This analysis is based on the study of a renormalization operator for the case of irrational mean spacing using Feynman's functional integral approach.

"Non-local bifurcation of solitary waves for coupled nonlinear Schrodinger-type equations"

Abstract
This talk concerns models which arise in optics and other areas of physics that can be written as a non-integrable system of two coupled nonlinear Schrodinger-type equations which are coupled through their nonlinearity. Examples include models for birefrigence and satuarable nonlinearity. We will focus on a new global bifurcation of solitary waves that can be predicted by a local analysis. Specifically we consider a mechanism where a non-local event, namely the bifurcation at infinity' of a vector soliton from a plane polarised one, can be predicted from the linearisation about the the plane wave. These bifurcations are shown to be a natural counterpart to the truely local bifurcation of a wave and daugther wave'. It also explains the earlier numerical findings of Kivshar and Ostrovskaya on a model with saturable nonlinearity.
Joint work with: Jianke Yang,

"Subcritical transition in channel flows"

Abstract
Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. A resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as R^\gamma say, with \gamma < 0), so that small but finite perturbations lead to transition. Numerical experiments by Lundbladh, Henningson & Reddy (1993) indicated that for streamwise initial perturbations \gamma = -1 and -7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations \gamma = -5/4 and -7/4.
The small domain of attraction is the result of the non-normality of the Orr-Sommerfeld operator, which leads to initial conditions exhibiting large transient growth in the solution, which may then undergo a secondary instability. These ideas will be illustrated by simple toy models, and then, through a formal asymptotic analysis of the Navier-Stokes equations, it will be shown that for streamwise initial perturbations \gamma = -1 and -3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations \gamma = -1 and -5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

"Semi-prime Noetherian rings of injective dimension one"

Abstract
It is not always easy to tell whether a semi-prime Noetherian ring has injective dimension one. We shall describe how to answer this question in some special situations and discuss some of the consequences when the dimension is one. Idealiser rings provide one special context in which it is possible to make considerable progress.

"An application of anti-integrability theory to billiards"

Abstract
Two examples of scattering billiard systems which exhibit anti-integrability are shown. Under some non-degeneracy conditions, we proved that all anti-integrable orbits can be continued from "delta-billiards" to the usual billiards and that any periodic orbit has infinitely many homoclinic orbits as well as heteroclinic orbits to any others. There exists a Cantor set such that the billiard map restricted to it is conjugate to a subshift of finite type with an arbitrarily given number of symbols.

"New developments on non-local homogenised constitutive relations for periodic composite media"

Abstract
Following our talk at BAMC2K we study further a linear non-uniformly elliptic periodic problem set on a conducting composite with highly anisotropic fibres, which was earlier shown to exhibit non-local behaviour in the homogenised limit [3].
We present a new high-contrast Poincare-type inequality that allows to justify rigorously the limiting procedure in the above problem without the "damping term" that was introduced in the earlier formulation.
The homogenisation is performed using the method of two-scale asymptotic expansions. This includes an explicit procedure for finding higher-order correctors to the homogenised solution and gives the usual error estimates. Thus the method of asymptotic expansions complements the two-scale convergence technique, which is briefly reviewed and compared to the former.
We present the corresponding non-local homogenised constitutive relation between electric field and current, and outline application of the above mathematical tools in linearised elasticity.
Finally, using the model setting of anti-plane shear of a linear elastic periodic composite, we describe the interrelations between the above non-local behaviour and higher-gradient asymptotics studied by the first two authors in [1,2].
[1] Smyshlyaev, V.P., Cherednichenko, K.D. On derivation of strain gradient'' effects in the overall behaviour of periodic heterogeneous media. J. Mech. Phys. Solids, 48, 1325--1357, 2000.
[2] Cherednichenko, K.D., Smyshlyaev, V.P. On full asymptotic expansion of the solutions of nonlinear periodic rapidly oscillating problems, 1999. Isaac Newton Institute for Mathematical Sciences. Preprint NI99028-SMM.
[3] Cherednichenko, K.D., Smyshlyaev, V.P., Zhikov, V.V. Non-local'' homogenised limits for periodic composite media, preprint, 2000.
Joint work with: VP Smyshlyaev and VV Zhikov

"Calculating spatial statistics of biased and correlated random walks"

Abstract
The motion of swimming microorganisms that have a preferred direction of travel, such as single-celled algae moving upwards (gravitaxis) or towards a light source (phototaxis), has been modelled as the continuous limit of a correlated and biased random walk by Hill and Hader (1997).
Othmer et al. (1988) originally derived a generalized equation to describe velocity jump processes and calculated spatial statistics from it. In their work, they assumed that the turn angle distribution was the sum of a symmetric probability distribution and a bias term. We have extended this theory to use turning angle distributions which implicitly include bias, as measured by Hill and Hader (1997) in experiments. Thus we derive equations for the mean swimming velocity, diffusivity and other spatial statistics that are required in continuum models for the flow of suspensions of such swimming microorganisms.
We present a computer algorithm for simulating correlated and biased random walks of swimming microorganisms and compare results from simulations to expected values given by the derived equations.
Joint work with: NA Hill

"Non-commutative localization"

Abstract
The basic problem is to embed a ring in a skew field, when possible, or more generally to study homomorphisms to a skew field. The key feature is to invert matrices rather than elements, and a closer examination leads to a criterion for embeddability.

"Nonlinear models of rotating spherical convection"

Abstract
The problem of rotating spherical convection has been of great interest due to its astrophysical and geophysical applications. Soward (1977) identified that the previous local' theories, which determine a critical Rayleigh number, R_L, could not be acceptably embedded into the global' theory (ie a WKBJ type solution), due to the phenomenon of phase mixing. Yano (1992) revisited the spherical problem but neglected certain terms in the governing equations based on a small inclination of the boundary. This had the advantage of yielding an algebraic dispersion relation. Solving this system, with the appropriate global' conditions, determined the true critical Rayleigh number, R_c, which was found to be an O(1) amount greater than R_L . Encouragingly, this agreed well with the study of the full equations by Jones, Soward, Mussa (2000).
We have now extended Yano's linear model to include nonlinear terms and have numerically traced the nonlinear solutions. It is seen that the subcritical or supercritical behaviour of the system is dependent on the Ekman number although, notably, Ekman numbers as low as 3\times 10^{-8} are required to fully understand the complexity of the problem. A time dependent code has been developed to investigate stability.

"Computing Violent Water-Wave Impact"

Abstract
How can a water wave exert a big transient pressure when it impacts a vertical seawall? A wave of breaking-height h_b, meeting a wall, can exert pressures measured to be about 10\rho gh_b, for a few milliseconds. Wave impact is so quick that it is difficult to resolve photographically what the water surface is doing when these high pressures occur. Recent computations of unsteady, 2D, free-surface flow, [1], will be shown, which accord with published cine-film images. Impact pressures are not necessarily due to the direct collision of water onto the wall, but instead can be related to a focussing of the forward face of the wave towards a small zone on the wall. Focussing culminates in a narrow, ascending jet at the wall, which can accelerate at 100,000g. The computed pressures are as intense as in wave-tank measurements.
[1] Cooker, M.J. & Peregrine, D.H. 2002 Computations of violent water wave impact on a vertical wall and flip-through. In preparation.

"Evolution of Benjamin-Ono auto-solitons"

Abstract
In the presence of long wavelength instability and short wavelength dissipation, a nonlinear dispersive wave system can have auto-soliton solutions. In particular, for nonlinear magneto acoustic waves this regime can be reached in the presence of active non-adiabaticity, associated radiative or thermal instability and viscous, resistive or thermal conduction dissipation. Taking these phenomena into account, nonlinear magneto acoustic modes of a magnetic flux slab are described by the extended Benjamin-Ono (eBO) equation. We studied the interaction of localized solutions of the extended Benjamin-Ono equation with the use of multi-soliton perturbation theory, developed for the BO equation by Matsuno (1994). It is always possible to distinguish between two stages of the evolution. The initial stage of the soliton interaction corresponds qualitatively to the exact two-soliton solution of the BO equation. However, when the soliton amplitudes approach the auto-soliton amplitude and, consequently, become almost equal, the two-soliton peaks were found to repel. Thus, we conclude that two interacting auto-solitons always repel, even though the parameters associated with them tend to the same position. A similar behavior was observed in three-soliton and four-soliton systems.
Joint work with: VM Nakariakov

"The close interaction of underwater seismic airgun bubbles"

Abstract
In underwater seismic surveying it is important to have a signal that is well known in the far field (when it impacts against the Earth's crust), and that is sharp and powerful to allow easier analysis of the reflections from the layers in the Earth's crust. One of the current methods used is an airgun cluster, consisting of 2 to 3 airguns of varying volumes with separations of ~1m. An airgun contains air compressed to about 140atm. When fired this air is released explosively into the sea at typical depths of 3-10m. The air bubble produced expands rapidly, past equilibrium due to momentum built up in the surrounding water and then collapses down to near starting volume. It repeats this cycle oscillating with a period of ~100ms and also rising due to buoyancy towards the sea surface. In an airgun cluster, the bubbles produced interact with each other giving a longer period of oscillation and greater damping in the first collapse which produces a better signal. This talk explains why this happens and shows what the boundary integral method reveals about the importance of deformation and close interactions in airgun generated bubbles. In particular, jet formation as a source of damping, and the phenomena of frequency locking are investigated. Underwater films are presented backing up some of the predictions.
Joint work with: John R Blake and Antony Pearson

"Instability and localisation of patterns under the influence of a conserved quantity"

Abstract
The presence of a conserved quantity in a pattern-forming system can significantly affect the stability of patterns, and can even lead to strong localisation of the pattern. Near their onset, the evolution of simple, regular patterns is governed by coupled Ginzburg--Landau equations for the amplitudes of the various pattern modes involved, coupled to an equation governing the modulation of a large-scale mode related to the presence of a conserved quantity. For instance, in a fluid layer with a free surface the conserved quantity is the mass of fluid, and the corresponding large-scale mode is the deflection of the fluid surface. We describe the influence of a conserved quantity on various regular patterns (rolls/stripes, squares and hexagons) near the stationary onset of pattern formation, and show how strong localisation can occur right at onset.

"Representations of algebras, vector bundles with parabolic structure and monodromy"

Abstract
We explain the connections between the following topics, and discuss some progress. (1) The representation theory of a certain class of finite-dimensional associative algebras, the quasi-tilted algebras. (2) The existence of indecomposable parabolic vector bundles on the projective line. (3) A problem of Deligne and Simpson concerning the existence of matrices in given conjugacy classes whose product is the identity.

"Exact solutions for two bubbles in a 4-roller mill"

Abstract
Using techniques from complex analysis, the free boundary problem for the determination of the steady shapes of two interacting inviscid bubbles in the slow viscous flow-field of a 4-roller mill is addressed. Closed form solutions for the bubbles shapes and the associated flow field are found. The results generalize the work of Antanovskii (JFM, vol 327, 1996) who considered the case of a single bubble to the case of two interacting bubbles. Potential applications of the results are discussed.

"Elastic deformations of helically wound composite cables"

Abstract
There has been a great deal of interest in the problems of modelling cables and ropes. A recent review by Cardou and Jolicoeur [1] considers the modelling of a cable which consists of a central core surrounded by one or several helically-wound wire layers and cites 107 papers. Other authors have adopted a continuum approach regarding each layer as a transversely isotropic material whose principal direction is along a helix surrounding the central axis of the cable. In each layer the helix angle is constant so that, when referred to cylindrical polar co-ordinates, the cylinder has a constant stiffness matrix in each layer. The intention of this presentation is to use the continuum approach and describe the analytical solutions that govern some elastic deformations of an anisotropic elastic cylinder consisting of a single material of this type. The extension of this work to a composite cylinder consisting of several concentric layers, surrounding a central core, which are either bonded together or make frictionless contact, will be briefly described if time allows.
[1] A. Cardou and C. Jolicoeur, "Mechanical models of helical strands", Appl Mech Rev, Vol.50(1), 1997, pp.1-14
Joint work with: AH England and AJM Spencer

"The effect of ureteric stents on urine flow"

Abstract
A ureteric stent is a plastic tube placed into the ureter to relieve or prevent obstruction. The ureter is a muscular tube which transports urine from the kidney to the bladder.
Whilst the stent may be organ- or life-saving in some cases, it presents patient and surgeon with a new set of problems. The stent can crust up' with crystals of calcium oxalate (present in solution in urine), and since it holds open the junction between the bladder and the ureter it can lead to reflux' of bacteria-rich bladder urine back up towards the kidney when bladder pressure increases.
We present a simple mathematical model of the flow in a stented ureter, and examine solutions in certain physiologically-relevant asymptotic regimes. It is hoped that a better understanding of the flow dynamics may lead to improved stent design and performance.
Joint work with: Stuart Graham, Sarah Waters and Jonathan Wattis

"Using monomial modular representations to construct pre-images of sporadic groups"

Abstract
Let N be a finite group, and let H be a subgroup of N of index n. Further let \rho: H --> Z_p^*, the non-zero integers modulo the prime p, be a linear representation of H, so H is mapped onto a cyclic subgroup C_m of Z _p^*, where m | p-1. Inducing \rho up to N yields \rho^N, an n-dimensional monomial representation of N whose non-zero entries are mth roots of unity in Z_p^*.
Now let p^{*n} denote a free product of n copies of the cyclic group of order p; thus
E = < t_1, t_2, ... , t_n | t_i^p = 1, forall i > \cong p^{*n}.
For x \in N, the matrix \rho^N(x) describes how x can act (by conjugation) as an automorphism of E as follows:
If the {ij}th entry of \rho^N(x) is r, then t_i^x = t_j^r.
Thus x permutes the cyclic subgroups generated by the t_i, but may map t_i to some non-trivial power of t_j. This defines a semi-direct product of the form
P = E : N = p^{*n}:N,
which we call a progenitor. Finite homomorphic images of such progenitors are investigated, and it turns out that many sporadic simple groups are obtained in a revealing manner.
The simplest case, when p = 2 and the monomial representation just yields the permutation representation, is particularly fruitful.
Recent results involving some of the larger sporadic groups will be described.

"The amenability of measure algebras"

Abstract
The process of differentiation' is, of course, of fundamental importance in our subject; the abstract version of this is to look when there is a derivation on an algebra. A Banach algebra A is said to be amenable if every continuous derivation from A into a dual Banach A-bimodule is inner. I will explain what this means and why it seems to be a centrally important notion.
I shall recall some classical theorems and examples on amenability, and also describe some recent results. However, I shall point out that some basic questions remain open.
The term amenable' for Banach algebras comes from B E Johnson's famous theorem that the group algebra L^1(G) is amenable if and only if the locally compact group G is amenable. A related algebra is M(G), the measure algebra on G. I shall describe our recent solution to the question when M(G) is amenable: this latter is joint work with F Ghahramani and A Ya Helemskii.

"Renormalisation for the Harper Equation for quadratic irrationals"

Abstract
In this paper we construct a renormalisation fixed point corresponding to the strong coupling limit of the Harper equation, which is an important quantum mechanical model, for quadratic rotation numbers with continued fraction expansion [a,a,a,a,...] for a \in N. We use renormalisation methods to study self-similarity in the fluctuations \eta_i in the strong-coupling limit \lambda --> \infty, E =~ 2\lambda, where E is the eigenvalue corresponding to the eigenfunction \psi_i defined on the one-dimensional integer lattice indexed by i\in Z, for the case of \omega = (\sqrt{a^2 + 4} - a)/2, a \in N, with the phase \phi = 0.
Joint work with: BD Mestel

"Effect of Volumetric heat-loss on triple flame propapagation"

Abstract
We study the effect of volumetric heat-loss on the propagation of triple flames in the counterflow configuration. Analytical results are derived in the weak-strain asymptotic limit for the flame propagation speed and its shape. These are compared to, and complemented by numerical results covering a wide range of values of the strain rate up to near-extinction values. In particular, a monotonic variation of the propagation speed from positive to negative values, similar to that obtained in the adiabatic case, is found provided that the heat-loss is sufficiently weak. For stronger heat-loss, however, a non-monotonic dependence is obtained associated with the existence of different burning regimes. The different regimes observed are delimited in a two-dimensional plane in terms of the heat-loss intensity and the strain rate.
Joint work with: J Daou and J Dold

"Wavelets and their applications"

Abstract
Wavelets are a relatively new approach used in the analysis of sounds and images, as well as in many other applications. The wavelet transform provides a mathematical analog to a music score: just as the score tells a musician which notes to play when, the wavelet analysis of a sound takes things apart into elementary units with a well defined frequency (which note?) and at a well defined time (when?). For images wavelets allow one to first describe the coarse features with a broad brush, and then later to fill in details, similar to zooming in with a camera. For this reason, the wavelet transform is sometimes called a "mathematical microscope".
Wavelets are used by many scientists for many different applications. Outside science as well, wavelets are finding their uses: the FBI has Been using a wavelet scheme for the compression of its vast library of fingerprint data for several years, and wavelets are incorporated in the next generation image compression standard.
The talk will start by explaining the basic principles of wavelets, which are very simple. Then they will be illustrated with some examples, including pictures of the wavelet scheme used by the FBI. Throughout the talk we will see how wavelets emerged as a synthesis of ideas from many different directions.

"Spectral properties of random non-selfadjoint matrices and operators"

Abstract
We discuss the difference between two methods of calculating the spectrum of a random non-self-adjoint operator, one by taking the limit of a finite chain as its length tends to infinity, and the other by considering the inifinite chain directly. We describe the results obtained using the second method in some detail.

"Spiralling viscous jets"

Abstract
The trajectory and stability of a spiralling viscous liquid jet arising from a rotating container is discussed. Due to the rotation, surface tension and gravity, the trajectory of the jet is curved in space, and the path of this jet is determined using an asymptotic method. The stability of this jet is examined using a multiple scales methodology, and this gives rise to a prediction for the break up length of the jet. Experimental results will also be described as well as the industrial motivation for this problem.
Joint work with: AC King, M Simmons, IM Wallwork, D Wong and E Parau

"From Set Theory to Braids via Self-Distributive Algebra"

Abstract
We show how studying large cardinals in set theory (whose existence is, and will remain, an unprovable assumption) has led to new examples of algebraic systems satisfying the left self-distributivity law, and, from there, quite naturally, to the discovery of some canonical linear ordering of braids. The latter has now received a number of geometrical or topological constructions, and it has led to efficient new braid algorithms, with possible cryptographical applications.

"F-manifold structure on the base space of the miniversal deformation of a function on space curve"

Abstract
It is well known that the base space of the miniversal unfolding of a function with an isolated singular point can be equipped with a multiplicative structure on the tangent bundle and a flat metric, making it into a Frobenius manifold. If f is a function on a space curve, then the base space of its miniversal deformation can also be equipped with a multiplication on the sheaf of vector fields tangent to the discriminant of the family of curves. There is also a candidate for flat metric.

"Lefschetz formulae, trace formulae, and zeta functions"

Abstract
For non-discrete dynamical systems Lefschaetz formulae take a different shape and become identities of distributions. For geodesic flows of locally symmetric spaces they can be derived from Selberg type trace formulae. This then gives rise to a theory of generalized Selberg zeta functions. There are applications like the prime geodesic theorem which in turn can be applied to compute asymptotics of class numbers. Attempts to transfer this theory to an adelic setting have led to a spectral interpretation of the "good" zeros of the Riemann zeta function and conjectural adelic trace formulae.

"K-theory, regulators and L-functions for curves over number fields"

Abstract
There is a classical relation between the residue at s=1 of the zeta function of a number field, and the regulator of the units of its ring of integers. There is a similar relation between the values at n>=2 of the zeta function, and a regulator of the (2n-1)-st K-group of the number field, which was proved by Borel. Bloch proved a similar statement relating K_2 of an elliptic curve over the rationals with complex multiplication to the value of its L-function at 2, and Beilinson conjectured a sweeping generalization of this to smooth projective varieties over number fields. After reviewing some of the more classical cases, we discuss the case of curves, in particular for K_2 and K_4. We also discuss a p-adic regulator on K_2 and K_4, which for curves involves p-adic integration.

"Multi-symplectic systems, symmetry and stability"

Abstract
Multi-symplectic systems are PDEs which have a (pre)-symplectic form associated with both temporal and all spatial evolutions. Many evolutionary Hamiltonian PDEs can be written as multi-symplectic systems. Symmetries in multi-symplectic systems occur often and lead naturally to a generalisation of relative equilibria. Such relative equilibria can correspond to solitary waves or periodic and quasi-periodic patterns. In this talk, a short introduction to multi-symplectic systems will be given and it will be indicated how this structure can be used to derive results about the instability of solitary waves.

"Recent developments in periodic orbit theory"

Abstract
Periodic orbits play an important role in chaotic systems. In low dimensional Axiom A systems periodic orbit expansions for long time properties of chaotic systems converge faster than exponentially. I will describe attempts to extend the theory to cases of nonuniform hyperbolicity, stochastic perturbations and spatiotemporal chaos.

"The Search for Randomness"

Abstract
I will take a careful look at some of our most primitive images of random phenomena; tossing a coin, rolling a roulette ball, shuffling cards. In each case, mathematics coupled with practical analysis shows that while randomness can be approached, usually, it just ain't so.

"Diophantine approximation on manifolds"

Abstract
In classical Diophantine approximation there are two standards forms of approximation; the first is to approximate by rational points and the second by rational hyperplanes. The usual questions which are asked concern the measure or Hausdorff dimension of the sets of points which get "close" to infinitely many such objects and in the classical case these have been almost completely solved. When the same questions are asked of sets of points restricted to lying on a manifold the problems become much harder and in the first case (that of approximating by rational points) very little is known except for specific manifolds such as the circle or the parabola. The main problem for this case is to obtain good estimates of how many rational points with denominators in a certain range lie on (or "very close") to the manifold. For the second case the picture is far more complete although there are still some questions remaining. In this talk there will be a discussion of the two cases and of what is and is not known.

"Energy dissipation in body-forced turbulence"

Abstract
Bounds on the bulk rate of energy dissipation in body-force driven steady state turbulence are derived directly from the incompressible Navier-Stokes equations. We consider flows in 3 spatial dimensions in the absence of boundaries and derive rigorous {\it a priori} estimates for the time averaged energy dissipation rate per unit mass, \epsilon, without making any further assumptions on the flows or turbulent fluctuations. We prove
\epsilon \le c_1 \nu \frac{U^2}{\ell^2} + c_2 \frac{U^3}{\ell}
where \nu is the kinematic viscosity, U is the root mean square (space and time averaged) velocity, and \ell is the longest length scale in the applied forcing function. The prefactors c_1 and c_2 depend only on the functional shape of the body-force and not on its magnitude or any other length scales in the force, the domain or the flow.
We also derive a new lower bound on \epsilon in terms of the magnitude of the driving force F. For large Grashof number Gr = F\ell^3/\nu^2, we find
c_3 \frac{\nu F \ell}{\lambda^2} \le \epsilon
where \lambda = \sqrt{\nu U^2/\epsilon} is the Taylor microscale in the flow and the coefficient c_3 depends only on the shape of the body-force. This estimate is seen to be sharp for particular forcing functions producing steady flows with \frac{\lambda}{\ell} \sim {\cal O}(1) as Gr goes to infinity. We interpret both the upper and lower bounds on \epsilon in terms of the conventional scaling theory of turbulence---where they are seen to be saturated---and discuss them in the context of experiments and direct numerical simulations.

"Polar factorisation: existence and applications"

Abstract
An integrable vector-valued function is said to have a polar factorisation if it can be written as the composition of the gradient of a convex function with a measure-preserving (that is "size-preserving") mapping. This concept was introduced by Y. Brenier, and has been used in diverse applications. We describe one of these, identification of the trajectory mapping for a model of weather front formation. Recent work with G.R. Burton has settled the question of when the factorisation is unique; however existence is still open. There are integrable functions which do not have polar factorisations; we present a class of examples. Finally we examine the case when the integrable vector-valued function has additional regularity, for example when it is Lipschitz.

"The deep quantum structure of spacetime"

Abstract
One approach to solving the problem of finding a theory that reconciles quantum mechanics and gravity is based on the causal set hypothesis, which states that the deep structure of spacetime is discrete and is what is known as a partial order' or poset', a kind of extended family tree'. This talk will describe ten reasons to be optimistic that the approach is on the right track, mentioning some of the new mathematical challenges that are being thrown up along the way.

"Embedded trapped modes near an indentation in a strip wave-guide"

Abstract
We investigate the existence of embedded trapped modes near an indentation in a strip wave-guide, which corresponds to a channel of uniform water depth in water waves and a two dimensional wave- guide in acoustics. Modes are sought which are either symmetric about the centreline of the guide and below the first nonzero cut-off for symmetric wave propagation or anti-symmetric and between the first and second cut-off for anti-symmetric wave propagation. In spectral theory, this means that the eigenvalues associated with the trapped modes are embedded in the continuous spectrum of the relevant operator.
An eigenfunction expansion for the trapped mode potential is obtained. A crude approximation is first obtained by drastically truncating the eigenfunction expansion and a transcendental equation for the trapped mode frequency is obtained. A full numerical solution is then obtained by applying Galerkin approach. The known form of singularity in the velocity at a corner is exploited by expanding the velocity as a series of ultra-spherical Gegenbauer polynomials multiplied by a suitable singular function. Results show that the approximate solutions are very close to the full solutions. For a given depth of indentation, embedded trapped modes can be found for a series of discrete values of the length of indentation and the wave frequency.
Joint work with: Maureen McIver

"Preconditioning for numerical solution of BE problems"

Abstract
In this talk, we consider the solution of large dense systems of equations that arise from the discretization of boundary element methods in the solution of electromagnetic scattering problems. In particular, we will examine techniques for preconditioning the matrices so that standard iterative methods like GMRES converge quickly. We have found that variants of sparse approximate inverse techniques are the most robust on our target applications and we discuss ways of making them more robust including employing imaginary shifts. Since, for very large systems, iterative methods can only really be viable in a fast multipole framework, we concentrate on the implementation of our sparse approximate inverse preconditioners in this context and discuss their scalability on problems of up to over a million degrees of freedom. We consider accelerating the convergence by nesting our preconditioned GMRES within outer FGMRES iterations and show that this can give substantial performance improvement and greater robustness. Finally, we examine the use of low rank corrections implemented as an additional additive preconditioner and show that this can greatly accelerate the convergence particularly when the first preconditioner has done a good job of leaving only a few eigenvalues close to zero.
Joint work with: Bruno Carpentieri and Luc Giraud

"About isotopy classes of non-singular vector fields on the three-sphere"

Abstract
Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are isotopic if and only if the linking number of those links is zero. If we have time, we will show how use this criterion to give a new proof of a result of Yano: every non-singular vector field on the three-sphere is isotopic to a non-singular Morse-Smale vector field.

"Hyperbolic Hamiltonian Monodromy"

Abstract
We consider a geodesic flow with 3 degrees of freedom that is Liouville integrable but has positive topological entropy. We show that its Hamiltonian and Quantum monodromy is described by a hyperbolic matrix from SL(2,Z). The hyperbolicity is responsible for a rigid spectrum with unbounded degeneracies, related to the number of solutions of a quadratic diophantine equation. The example can be generalised with any matrix from SL(2,Z).
Joint work with: A Bolsinov and AP Veselov

"A proof of the Poincare conjecture"

Abstract
We give a proof of the Poincare conjecture. The proof is inspired by the beautiful algorithm to recognise the unknot given by Hyam Rubistein and the proof of this using David Gabai's "thin position" given by Abigail Thompsom.

"Dynamics of monopolar vortices on the beta plane"

Abstract
The ability to predict the direction of motion of monopolar vortices moving on a beta plane is of fundamental importance in weather predictions. Despite the large body of work on the theoretical work on monopolar vortex dynamics, the models do not generally provide any real connection between what is observed experimentally or in the atmosphere. Moreover, most models do not predict a unique vortex trajectory. Drawing on new detailed experimental observations on monopolar vortices moving on a topographic beta plane, we extend Rossby's original inviscid body of monopolar dynamics and show that it provides reasonable agreement with experimental measurements and a clearer insight into the controlling physical processes.
Joint work with: J.B. Flor

"Application of double Fourier series to high order differential equations in MHD"

Abstract
The governing equations for fluid flow in a rotating sphere using a toroidal-poloidal decomposition are sixth order in the meridional co-ordinate. We show here that we can make use of Fourier series as opposed to spherical harmonics with a view to numerically modelling non-linear systems. It is well documented that spherical harmonics, while the obvious choice for linear analysis, are not suitable for large scale non-linear models due to the time spent transforming between spectral space and physical space. Hence, with the option of the FFT, Fourier series are the natural alternative. We will show that the use of Fourier series does not compromise accuracy in the attempt to achieve greater numerical efficiency in modelling the high order equations that govern MHD and convection in spherical systems.
Joint work with: Keke Zhang

"Stochastic Solitons in Integrable Systems"

Abstract
We consider the thermodynamic type limit of the finite-gap potentials on a special band/gap scaling of the hyperelliptic Riemann surface. This limit has a natural description in stochastic terms and can be associated with a one-dimensional soliton gas (homogeneous soliton turbulence). We also propose a scenario of transition to the disordered soliton structure from a deterministic nondecaying initial distribution.

"Scattering of sound waves by an infinite grating composed of equally spaced rigid walls"

Abstract
There are numerous interesting physical problems, in the fields of acoustics, electromagnetism, elasticity, etc., which, when modelled mathematically, are reduced to Wiener-Hopf equations defined in some region of complex plane. In simple models this equation is scalar. However, for complex boundary value problems, this procedure often leads to a matrix Wiener-Hopf equation. The key step in the solution of such an equation is to decompose the kernel into a product of two factors with certain analyticity properties. Although it is possible to exactly decompose scalar kernels with the use of Cauchy's Integral Theorem, no procedure has yet been devised to exactly factorise general matrix kernels.
This research discusses the scattering of plane sound waves by an infinite grating composed of equally spaced rigid walls by using a class of matrix kernels, K, which are meromorphic and have the additional property
K = K^{-1}(Q/\nabla)
where Q has entire elements and \nabla is the determinant of Q (The simplest such example is the scattering of sound waves in a duct by a rigid barrier partially obstructing the duct [1]). Amongst authors who have considered the present problem are Dalrymple and Martin [2] who considered only normal incidence and Porter and Evans [3] who considered oblique incidence and unequal spacing. These latter authors used a suitable eigenfunction expansion and after having the solution employed Galerkin approximation to get numerical results accurate up to the three decimal points. They also extended the problem to consider two identical parallel arrays. The approach used here makes use of the Khrapkov type matrices and decomposes the kernel to two factors which are "almost" commutative and have appropriate algebraic behaviour in respective domains of complex plane. After the exact decomposition is achieved, the analytical solution is found by using the well-known procedure of Wiener-Hopf problems. Some numerical results are presented for re ection and transmission coefficients. The work in unequal spacing is already in progress and this method can be applied to many more problems which will be mentioned in the talk.
References
[1] Erbas, B., Abrahams, I.D., Scattering of Acoustic Waves by a Rigid Barrier Across a Duct, BAMC 2001, University of Reading.
[2] Dalrymple, R., Martin, P.A., Wave Diffraction Through Offshore Break- waters, J. Waterway, Port, Coastal, and Oc. Eng. 116, 727-741 (1990)
[3] Porter, R., Evans, D.V., Wave Scattering by Periodic Arrays of Breakwa- ters, Wave Motion 23(2), 95-120 (1996)
Joint work with: I David Abrahams

"Extensions of extensions in module categories"

Abstract
Let R be an associative algebra over an algebraically closed field k, let F be a family of left R-modules, which are orthogonal points in the notation of Ringel, and consider the category Mod(F): The finite length category built on F. The ultimate goal is to construct the objects of Mod(F), when the family F is given. We introduce some new methods for this, using Laudal's non-commutative deformations of modules. We also give an elementary proof of a known criterion for the category Mod(F) to be uniserial. Since our methods are constructive, we can construct all indecomposable objects in Mod(F) in this case. There are some applications to the first Weyl algebra when char(k)=0. We believe that our methods will be suited to study more difficult cases, where the obstructions come into play in a more essential way.

"Random polynomials: an overview"

Abstract
There are many known results concerning the expected number of real zeros of polynomials with random coefficients. Several types of polynomials are studied; among them algebraic polynomials are known most. For these types of polynomials, with sufficiently large degree, we present the asymptotic value for the expected number of real zeros, and discuss interesting variations that occur for this asymptotic value for the different assumptions on the distribution of the coefficients. There are significant differences for the behaviour of this expected number of real zeros for different types of polynomials. We give some of these differences for the cases of random trigonometric and random hyperbolic polynomials. However, there are several points of interest shared between purely mathematical developments and the physical and applied properties of random polynomials. In particular the distribution of complex roots of random algebraic polynomials is of joint interest. The results obtained for random fields could be applied to random polynomials to give a formula for the expected density of the complex roots. We will discuss a select few works recently published in this direction.

"The dual reciprocity boundary element method for inverse problems of the Poisson equation"

Abstract
This study relates to a particular class of problems governed by the Poisson equation in which the source distribution, i.e. the nonhomogenous part of the Poisson equation, has to be determined from data observed over the boundary of the solution domain. This corresponds to inversely solving the Poisson equation. The Dual Reciprocity Method (DRM) has been chosen as the solution procedure since it provides a technique for considering the whole equation with the nonhomogenous terms expressed in terms of boundary integrals. The present DRM approach to the solution of the inverse problem is tested on several test examples that have analytical solutions, using both exact (error-free) and noisy data. The overdetermined system, obtained with the inverse problem, is solved using the least square method. Due to the ill-condition nature of the resulting system of equations, a regularization procedure, namely the Tikhonov regularization combined with the discrepancy principle, is also employed.

"Memory effect in turbulent dynamo"

Abstract
We are concerned with large scale magnetic field dynamo generation and propagation of magnetic fronts in turbulent electrically conducting fluids. An effective equation for the large scale magnetic field is developed here that takes into account the finite correlation times of the turbulent flow. We find that the memory effects can drastically change the dynamo growth rate, in particular, non-local turbulent transport might increase the growth rate several times compared to the conventional gradient transport expression.
Joint work with: A Ivanov and A Zubarev

"Critical modelling issues for sea ice rheology"

Abstract
A brief overview of sea ice dynamics models used for climate simulation is presented and areas requiring further work are described. A novel approach to modelling the effect of flaws in the ice cover on rheology is outlined through the use of the structure tensor and closure postulates on its higher order moments. We model how the anisotropy of the flaw distribution affects the constitutive law for stress.
Joint work with: Alexander Wilchinsky

"Biracks: the search for quaternionic invariants of braids and knots"

Abstract
Solutions of the Yang Baxter equations can be found using 2X2 matrices with quaternionic entries. It is hoped that some of these will yield better results than the Burau representation of braids and the Alexander polynomial of knots and links. A computer search is taking place and has come up with some results.

"Reciprocal transformations of Hamiltonian operators of hydrodynamic type"

Abstract
Reciprocal transformations of Hamiltonian operators of Dubrovin-Novikov type are investigated. The resulting brackets are generally nonlocal, possessing a number of remarkable differential-geometric properties.
Joint work with: MV Pavlov

"Modelling artifical kidneys"

Abstract
Artificial kidney dialysis machines are used to treat patients with kidney failure, due to age, trauma or disease. These devices allow the removal of low-molecular weight solutes and water to take place through a semi-permeable membrane separating the patient\s blood from an artificial dialysate fluid contained within the machine.
We examine a simple channel model of an artificial dialyser. By making estimates of the model parameters from a clinical dialysis machine, we simplify the model and construct steady-state asymptotic solutions for the fluid velocity and solute concentration profiles. These are used to predict the physiologically important effect on the solute and water removal rates of the membrane permeability properties, the solute and fluid input conditions and the dialyser geometry.
By understanding how solute and water removal rates depend on the dialyser properties, it should be possible in a future study to solve the inverse problem which would allow solute and solvent fluxes to be precisely specified by controlling the dialyser parameters in vivo, potentially improving performance and minimising discomfort in patient dialysis sessions.
Joint work with: SM Cox and HM Byrne

"The mechanics of human eyes"

Abstract
Some aspects of the mathematical modelling of human eyes will be considered. A number of mathematical problems will be posed, including elastic boundary value problems concerned with scleral buckles, PDE problems associated with retinal detachment and electromagnetic problems associated with vortex keratopathy. It will be shown how the solution of these problems lead to some interesting conclusions about common diseases and conditions that occur in human eyes.

"Rayleigh-Lamb waves in a cracked plate"

Abstract
We study the diffraction of Rayleigh-Lamb waves by a surface breaking crack in an elastic isotropic plate. The solution of the problem is obtained by the projection method. The numerical performance of the method is assessed by energy conservation.
Joint work with: R Douglas Gregory

Abstract
We consider the construction of an adaptive controller for a first order plant achieving a non-zero robustness margin as measured in the sense of the gap metric. The result is then generalised to the class of relative degree one, minimum phase systems of known high frequency gain. Further extensions to higher relative degrees may be discussed.

"The Cauchy-Born rule for elastic crystals: a paradigm in the passage from atomic to continuum theory"

Abstract
Advances in engineering down to the nanoscale have not been followed by comparable advances in theoretical and mathematical understanding. On these scales one has to deal with large atomistic systems (ultra-thin films; carbon nanotubes; ...) in a regime between "atomistic" and "continuum" where collective behaviour dominates but discreteness effects still play a major role.
The speaker will describe some emerging mathematical work in this area and in particular discuss an innocent looking model problem, a 2D elastic mass-spring lattice with nearest and second neighbour interaction. (Note: Nearest-neighbour models are incapable of capturing elasticity as they lead to vanishing shear modulus.) Traditionally, passage to a continuum model is achieved by invoking the Cauchy-Born rule. It postulates that in response to a small linear displacement of the boundary of a specimen, all atoms will follow this displacement.
Our analysis shows that depending on the choice of spring constants and spring equilibrium lengths, the CB rule holds respectively fails. In the failure region, minimum energy configurations of the atoms exhibit fine-scale spatial oscillations. In this case, there is still a limiting continuum theory, but it involves more sophisticated averaging.
Proofs involve adaptation of nontrivial concepts from functional analysis and the calculus of variations to atomistic systems.
This is joint work with Florian Theil (Warwick) and will appear in J. Nonl. Sci.

"The degree sequence of the k-core of sparse random graphs"

Abstract
The k-core of a graph is the maximal subgraph of minimum degree at least k. We are interested in the k-core of the random graph G_{n,m}, where m=cn/2, and in particular in the asymptotic behaviour of its degree sequence. We show that for any fixed integer d\geq k, the number of vertices of degree d in the k-core is asymptotically almost surely equal to n \gamma(c,d)+o(n), and we give a precise estimate for \gamma(c,d).

"Correlation function of Characteristic Polynomials for Random Matrices from GUE and chGUE"

Abstract
We reconsider the problem of calculating a general correlation function containing arbitrary number of products and ratios of characteristic polynomials for large random matrices taken form Gaussian Unitary ensemble (GUE) and from Chiral Gaussian Unitary ensemble (chGUE). The method suggested by one of us recently (Preprint arXiv:math-ph/0106006) is shown to be capable of calculation when reinforced with a generalisation of Duistermaat-Heckman localisation theorem to integrals over non-compact homogenious Kahler manifolds.
Joint work with: Eugene Strahov

"Multiple strains: Stationary and oscillatory dynamics"

Abstract
We examine a generalisation of the susceptible-infected-recovered (S-I-R) model for the infection dynamics of four competing disease strains. This model uses a mass-action formulation to derive a system of ODEs for the dynamics. The four strains are present in one community of hosts at the same time. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the lifetime of a host individual is much longer than the typical infectious period. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the bifurcations associated with the presence of oscillatory dynamics observed in previous work by other authors.

"Iterative Methods for PDE Eigenvalue Problems"

Abstract
When steady solutions of complex physical problems are computed numerically it is often crucial to compute their stability in order to, for example, check that the computed solution is physical', or carry out a sensitivity analysis, or help understand complex nonlinear phenomena near a bifurcation point. Usually a stability analysis requires the solution of an eigenvalue problem which may arise in its own right or as an appropriate linearisation. In the case of discretized PDEs the corresponding matrix eigenvalue problem will often be of generalised form: Ax=\lambda Mx with A and M large and sparse. In general A is unsymmetric and M is positive semi-definite.
Only a small number of dangerous' eigenvalues are usually required, often those (possibly complex) eigenvalues nearest the imaginary axis. In this context it is usually necessary to perform shift-invert' iterations, which require repeated solution of systems of the form (A-\sigma M)y = Mx, for some shift \sigma (which may be near a spectral point) and for various right-hand sides x. In large applications these systems have to be solved iteratively, requiring inner iterations'.
In this talk we will describe recent progress in the construction, analysis and implementation of fast algorithms for finding such eigenvalues, utilising domain decomposition techniques for the inner iterations.
In particular we describe the computation of bifurcations in (2D) Navier-Stokes problems discretised by mixed finite elements applied to the velocity-pressure formulation of the problem. We describe the construction of appropriate parallel preconditioners for the corresponding (3 X 3 block) shifted systems. These use additive Schwarz methods and can be applied to any unstructured mesh in 2D (or also in 3D) and for any selected elements. An important part of the preconditioner is the adaptive coarsening strategy. At the heart of this are recent extensions of the Bath domain decomposition code DOUG, carried out by Eero Vainikko.

"F.S. Macaulay - an English schoolmaster and abstract algebra"

Abstract
Macaulay (1862-1938) was a mathematics teacher for many years, but he also went on to become Britain's leading algebraist and a major figure in the theory of polynomial rings. In this talk I shall describe his life and some of his work, looking at its origins in geometry (the Riemann-Roch Theorem), his exchanges with Charlotte Scott, and his response to the work of Lasker that earned him the respect of Emmy Noether and her school in the 1920s.

"Shock waves, dead zones and particle free regions in rapid granular free surface flows"

Abstract
A series of experiments will be demonstrated, that show the formation of shock waves, dead zones and particle free regions during the three-dimensional flow of granular materials past obstacles. These flows are interesting in their own right, but also provide a wealth of constitutive information that can be used to improve mathematical models. A simplified theory is formulated, which captures the key physical effects.

"Global bifurcations in lasers with delay"

Abstract
Delay differential equations (DDEs) have received a lot of attention recently, particularly in the modelling of lasers with feedback. One reason for this is the introduction of continuation software for DDEs. Adding to this, we developed an algorithm for computing unstable manifolds in DDEs. We expalin how these new methods can be used to identify global bifurcations in DDEs. As an example we consider a semiconductor laser with phase-conjugate feedback.
More specifically, we compute 1D unstable manifolds of saddle periodic orbits, to show how a torus breaks up culminating in a sudden onset of chaos due to a boundary crisis.

"The history of representation theory"

Abstract
From its creation by Frobenius in 1896-97 the representation theory of groups has come to permeate the mathematics and physics of the 20th century. Some examples of this process will be described.

"Modelling flux flow in thin film superconductors"

Abstract
We use a vortex density model to study the distribution of current and magnetic flux in thin film superconductors. The purpose of the model is to examine in detail the transition of a thin film from the superconducting state to the normal state using the the framework of vortex creep/flow. Understanding this transition is useful in the design of fault current limiter devices which exploit the properties of superconductors to protect electrical systems from current surges (such as lightning strikes and short circuits).
We consider transport currents, applied magnetic fields and the effects of Joule heating on the system. Below a critical current level our model produces results in agreement with the Bean model. At higher current levels we find interesting nonuniform dissipative current distributions.
The model leads to a boundary value problem for Laplace's equation. We have found some simple analytical solutions in 2D. The general problem is solved numerically, by recasting the problem as a singular integral equation.
Joint work with: SJ Chapman and SD Howison

"Solitary waves generated by an external force"

Abstract
In many physical contexts the generation of solitary waves by an external force can be described by the forced Korteweg-de Vries equation. In this framework, the typical scenario is the generation of a train of solitary waves propagating upstream, o localised steady downstream depression followed by a downstream oscillatory wavetrain. Here, we extend that model by considering the generation of solitary waves in an extended Korteweg-de Vries equation, which contains a cubic term.

"Stress due to a scleral buckle;"

Abstract
Some mathematical problems in the eye are of great medical interest since their complete understanding would be useful to correct or even prevent some eye conditions or diseases.
One such problem concerns scleral buckles. A scleral buckle is an effective method for repairing retinal detachment. It consists of a band placed around the eye placed where the retina is detached. It exerts some stress on the eye and hence produces displacements. Such displacements alter the natural position of the retina-choroid wall and therefore the focal length of the eye. This may result in distorted vision when such displacements are big enough.
The aim of our work is to find the maximum stress exerted by a scleral buckle in the retina-choroid wall, considered as a spherical elastic membrane, for which the corresponding displacements do not affect the optical system of the eye. The stress components can be found by solving \nabla^4\Psi=0 where \Psi is the Airy stress function and applying the proper boundary conditions for the displacements due to the presence of the scleral buckle.

"Conical incidence for electromagnetic and elastic waves in an array of cylindrical fibers"

Abstract
This talk presents analysis of electromagnetic and elastic waves propagating through a doubly periodic array of cylindrical channels in oblique incidence. A new method, based on a multiple scattering approach, has been proposed to reduce these spectral problems for partial differential equations to certain algebraic problems of the Rayleigh type. We obtain the eigenvalue problem formulations that enable us to construct the high-order dispersion curves and to study photonic and phononic bang gap structures in oblique incidence.
Joint work with: Chris Poulton and Alexander Movchan

"Making doughnuts of Cohen reals"

Abstract
The doughnut property is a combinatorial property of sets of real numbers and like for the Ramsey or the Baire property it is easy to show that not every set has the doughnut property. One can show that every set which has the Ramsey or the Baire property also has the doughnut property. On the other hand, one can show that a finite support iteration of Cohen forcing, starting from L, yield a model in which every Sigma-1-2 set has the doughnut property, but there are Sigma-1-2 sets which have neither the Ramsey nor the Baire property.

"The effect of fine structure on the stability of planar vortices"

Abstract
In 2D turbulence at large Reynolds number it is known that coherent vortices often dominate the flow and move under mutual interactions. The evolution of a Gaussian vortex profile subject to a weak impulsive strain field was studied by Bassom and Gilbert. However, few vortices physically have such smooth structure, due to processes such as vortex stripping. Balmforth, Llewellyn Smith and Young developed an asymptotic framework to study the evolution of sharp edged vortices with weak surrounding skirts. With this approach they discovered a quasi mode with decay rate proportional to the vorticity gradient in the critical layer. The critical layer is a result of resonance between the frequency of the mode and the rotation frequency of the vortex.
But what happens to the stability of the vortex, if the vortex has a sharp edge but a weak non-smooth messy' skirt? Whereas Balmforth et al had constant vorticity gradient in the critical layer we allow finer structure inside the critical layer. We use their asymptotic picture but include extra terms in the initial vorticity profile to allow for extra structure in the critical layer. We derive equations which we solve numerically and asymptotically for various different types of fine structure such as bumps, troughs and steps.

"Characteristic polynomials of random permutation matrices"

Abstract
We consider the logarithm of the characteristic polynomial of a random permutation matrix. Using coupling techniques we obtain a central limit theorem for this quantity which is very similar in form to the random unitary case. Our result enables us to prove a central limit theorem for the counting function for the number of eigenvalues in an interval on the unit circle.
Joint work with: P Keevash, N O'Connell and D Stark

"A model for slow granular flow incorporating rotation"

Abstract
We present here a continuum model for the flow and deformation of granular materials. Such materials exhibit a wide variety of behaviour depending upon their environment and we shall suppose the following conditions to hold. The grains are envisaged to be densely packed, with each grain in contact with several neighbours. Individual contacts are also prolonged, giving rise to non-impulsive contact forces between neighbouring grains. Further, the material, although constrained by its environment, is free to flow. We shall assume a theoretical framework of a rate-independent rigid/plastic material which satisfies a pressure dependent yield condition containing the physical parameters of internal friction and cohesion, together with physically based kinematic equations which encompass shear, dilatation and rotation and contain a physical parameter governing the dilatation. The rotation is introduced via a kinematic quantity, the intrinsic angular velocity, in addition to the usual Eulerian velocity field and this requires that the Cauchy stress tensor be non-symmetric and also that the intrinsic angular velocity satisfy an equation governing the balance of angular momentum.
However, we do not introduce couple stresses, and we may call the model a reduced Cosserat model. The equations governing planar flows are hyperbolic and the model is linearly well-posed. In this talk we consider some properties of the model in preparation for considering applications.

"Three-dimensional airway reopening"

Abstract
Many pulmonary diseases cause the smaller airways of the lung to collapse and become blocked by the liquid that normally coats their interiors. Motivated by the practical problem of reopening such collapsed airways, we consider the steady propagation of an air finger into a buckled elastic tube initially filled with viscous fluid. The resulting three-dimensional fluid-structure-interaction problem is solved numerically by a fully-coupled finite element method. Geometrically non-linear shell theory is used to determine the deformation of the elastic tube, and the free-surface Stokes equations are used to describe the dynamics of the fluid.
The generic behaviour of the system is found to be very similar to that observed in previous two-dimensional models for the steady propagation of an air finger into a flexible channel. In particular, we find a two-branch behaviour in the relationship between propagation speed and bubble pressure, p_b. At high speeds, p_b increases monotonically with propagation speed, whereas at low speeds, a decrease in p_b is required to increase the propagation speed. We present a simple model which explains this behaviour and why it occurs in both two and three dimensions.
Illustrative flow fields and wall shear stress distributions will be presented. We find that an increase in surface tension causes an increase in the bubble pressure required to drive the air finger at a given speed. Furthermore, we demonstrate that higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications.
Joint work with: M Heil

"Existence of Leray's self-similar solutions of the Navier-Stokes equations in D \subset R^3"

Abstract
Leray's self-similar solution of the Navier-Stokes equations is defined by u(x,t) = U (y) / \sqrt{ 2 b (t^*-t)} ~, where y = x / \sqrt{ 2 b (t^*-t)} , b > 0.
Consider the equation for U(y) in a smooth bounded domain D of R^3 with non-zero boundary condition:
- \nu \triangleup U + b U + b y \cdot \nabla U + U \cdot \nabla U + \nabla P = 0, y \in D, \nabla \cdot U = 0 , y \in D , U = G(y), y \in \partial D .
We prove an existence theorem for the Dirichlet problem in Sobolev space W^{1,2}(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t^* with t^* < +\infty , provided the function G(y) is permissible.

"On the effect of nonparallel terms on the absolute and convective instabilities of the rotating disc boundary layer"

Abstract
The flow produced by a rotating disc in an otherwise still fluid can be described by von Karman's similarity solution. This basic flow, in certain resolved directions, is inflexional and subject to an inviscid crossflow instability, which includes a set of growing stationary vortices and also travelling waves. Consideration of the inversion contours of a Fourier-Laplace transform of the initial value problem shows that the invisicd problem is also absolutely unstable. Viscous, Coriolis, streamline curvature and nonparallel terms all enter the linearized disturbance equations at O(R^{-1}). A fourth order Orr-Sommerfeld equation can be derived when only the viscous terms are added to the inviscid terms, allowing predictions to be made at experimental parameters. Adding also the Coriolis and streamline curvature terms leads to a sixth order ODE, and is found to introduce a new mode to the neutral curve for stationary vortices, and to significantly affect the critical Reynolds number for instability. The nonparallel terms, although of the same order, and so in principle of equal importance, are usually neglected because then the disturbance equations are PDEs. We have obtained numerical solutions to the full nonparallel equations and apparently discovered a new branch of stationary vortices. The critical Reynolds number for absolute instability is also found to be changed significantly. At the time of writing, an analysis based on matched asymptotic expansions is underway to discover the origin of this new branch of solutions.

"Mathematics Works? -- integration of graduate skills"

Abstract
Mathematics graduates in industry, commerce and also those entering postgraduate study are expected to possess a range of mathematical abilities that include knowledge and implementation of mathematical and computational techniques and the development of mathematical skills. Typically these are delivered through formal lectures and assessment is normally heavily weighted to formal examinations. It is widely recognised that some variety of learning and assessment experiences helps students to develop subject-specific but also crucially to develop more general graduate skills. Increasingly, students are selecting their choice of degree to meet the flexible demands of a changing workplace and well designed MSOR programmes have the potential to develop a profile of skills abilities alongside the more traditional subject-specific education. A compulsory third-year mathematical modelling module "Vocational Mathematics" has been introduced at Nottingham University. Students work in groups of 3 or 4 on a succession of three projects to help them develop a strong skills base in applying and communicating mathematical problem solving. As a final year exercise, student groups are expected to function with less staff-directed study than is normally associated with an individual project, with greater autonomy, within a shorter timeframe and provide "professional" quality oral and written reports. Within this talk, an overview of the module development and student experiences will be provided alongside examples of implementation and issues associated with identification of graduate skills, group project work, oral presentations and written reports.

"Travelling Waves in a Simple Phytoplankton-Zooplankton System"

Abstract
Spatiotemporal patchiness is manifest in ocean blooms of phytoplankton on all scales of observation. On small scales the physical features of the environment such as thermoclines and eddies correspond to phytoplankton variabliity. Biological interactions are likely to occur at longer scales of the order of 1 to 100 km (Steele 1978). At these scales the physical features are less clear and patchiness is likely to be the result of the interaction between the physical and biological processes.
We use an excitable medium, two-component phytoplankton-zooplankton model due to Truscott & Brindley (1994), and incorporate the dispersion due to ocean currents and eddies into a spatial diffusion term. A scaling for travelling waves solutions is found, based on the the magnitude of the diffusion coefficient which is a small quantity. The analysis is complemented by a numerical study in which wavetrains and secondary convective instabilities are found. The scales and propagation speeds are consistent with observations suggesting that patchiness may be due in part to travelling wave phenomena.
References
Steele, JH 1978 Some comments on plankton patches. In Spatial patterns in plankton communities', ed. H Steele, Plenum Press.
Truscott, JE & Brindley, J 1994 Ocean plankton populations as excitable media. Bull. Math. Biol. 56, 981--998.

Joint work with: LA Plumpton

"Unitary block designs and finite 3-dimensional unitary groups"

Abstract
We apply the resulting knowledge on the representation theory of these groups to investigate certain invariants of unitary block designs. In particular we find the the ranks (over arbitrary fields) of their incidence matrices. Some information on the elementary divisors of these matrices is also given.

"Bifurcation with icosahedral symmetry"

Poster for Nonlinear dynamics:

Abstract
This poster presents an analysis of the steady state bifurcation with icosahedral symmetry. The Equivariant Branching Lemma is used to predict the generic bifurcating solution branches corresponding to each irreducible representation of the icosahedral group. The relevant amplitude equations are deduced from the equivariance condition, and used to investigate the stability of bifurcating solutions. It is found that the bifurcation with icosahedral symmetry can lead to competition between twofold, threefold and fivefold symmetric structures, and between tetrahedral and threefold symmetric structures.

Poster for Granular and particle laden flow:

"The effect of avalanching in a model of aeolian sand ripples"

Abstract
This poster presents a simple two-species ripple model with avalanching. The effect of the avalanching term is investigated numerically, and is found to be crucial in producing realistic ripple profiles.
Joint work with: Anita Mehta

"Quartic and cubic polynomials over Galois fields- some new results"

Abstract
Given q, a power of a prime p, denote by F the Galois field GF(q) and, for n \in N, by E its extension GF(q^n). The Primitive Normal Basis Theorem states that it is possible to find an element \alpha \in E that is simultaneously primitive and free over F. It is natural to ask whether further conditions may be imposed on \alpha. The PFNT-problem asks: given a finite extension E/F of Galois fields, a primitive element b in F and a non-zero element a in F, does there exist a primitive element \alpha \in E, free over F, whose (E,F)-norm and trace equal b and a, respectively? Equivalently, does there exist a primitive, free polynomial M(x)=x^n + M_{n-1}x^{n-1}+ ... + M_0 with M_{n-1}=-a and M_0=(-1)^n b? The PFNT problem was solved for n > 4 by SD Cohen (2000); however his method failed for n=4 and he conjectured that it might be impractical to expect any progress in the case n=3. Here, the PFNT-problem is solved for the n=4 case using a different approach and a proof is outlined for the n=3 case.
Joint work with: SD Cohen

"Uncertainities, inaccuracies and progress in climate models"

Abstract
The deterministic,'bottom-up' method of computational climate prediction starts from well established principles and equations, but requires approximations about initial and boundary conditions, about processes, and in the numerical methods used for discretisation and for speed of computation. Some of these approximations have now been studied in detail are well understood, but others are less so, eg the role of sea ice, or the effects on large scale phenomena of neglecting or smoothing small scale processes, or the effects of variation in solar radiation/particles. The verification and thence improvement of climate models also depends greatly on the ingenious use of good mathematical 'post-processing'; eg relating time series of the computations with past data.To be supplied

"Conservation laws for multisymplectic systems"

Abstract
The multisymplectic formalism provides a powerful approach to partial differential equations that have a Lagrangian or Hamiltonian structure. Conservation laws are particularly easy to obtain; indeed, some of them are encoded within the multisymplectic structure. For example, in geophysical flows described by the shallow-water and semi-geostrophic equations, the multisymplectic structure incorporates conservation of energy and potential vorticity. These conservation laws can be derived geometrically, without using symmetries and Noether's Theorem.
Many multisymplectic systems have conservation laws in addition to those that are encoded in the structure. This talk introduces a method that uses symmetries to obtain such conservation laws directly from the structure. The method is free from the restrictions of Noether's Theorem.

"The space of trees"

Abstract
We consider the topology on the space of weighted trees with given leaves, and various metrics which are consistent with this topology. Computational issues are considered. There are biological applications to evolutionary trees.
Joint work with: David Epstein

"A dimension formula for the Gurevich Pressure"

Abstract
Let f be an expanding Markov map of the interval. D.Ruelle proved that when the Markov partition is finite ,the root of the Bowen equation P(-tlog|f'|)=0 is the Hausdorff dimension of the repeller associated to f. We consider the case when the Markov partition in countable and we use a generalization of the topological pressure to this case. In this setting it might happen that the Bowen equation has no solution. We proved that the Hausdorff dimension of the set of recurrent points of f is equal to inf{t: P(-tlog|f'|) <= 0}. Note that the Hausdorff dimension of the recurrent set can be strictly less than the one of the repeller.

"Resonance statistics in chaotic open billiards"

Abstract
We study resonances for quantum transport through chaotic open billiards attached to leads supporting single open channel. We numerically identify complex poles and investigate their statistics in detail. The distribution functions for real and imaginary parts of the poles are compared with predictions by random matrix theory.

"Cohomology of Racks and Quandles"

Abstract
A rack is a set equipped with an asymmetric binary operation satisfying axioms related to the Reidemeister moves in classical knot theory. There exists a topological space (the rack space') roughly analogous to the classifying space of a group, whose cohomology has been used successfully in the construction of new state sum' invariants for codimension-2 embedded manifolds.
This talk will investigate a homologically algebraic viewpoint of the cohomology of racks (and certain subspecies), and give explanations of low-dimensional cohomology in terms of extensions' and deriviations' of a rack by/to a module'.

"Operator-valued inequalities"

Abstract
Any inequality expressing positivity or norms of matrices can be generalized to an operator-valued version. Particular cases include the classical inequalities of Hilbert and Hardy.

"Initial Stages of Atherosclerosis"

Abstract
Atherosclerosis is the most common arterial disease in modern society. It is characterised by the formation of plaques at the walls of larger arteries which diminish the radius of the lumen and cause significant health problems. Intimal hyperplasia is the initial stage of atherosclerosis, a small bump (fatty streak) appears in the wall of the artery caused by the thickening of the intima. An increased concentration of LDL cholesterol inside the arterial wall is a suspected cause of intimal hyperplasia.
The problem has been modelled in a 3-dimensional cylindrical geometry. LDL cholesterol is transported from the blood through the endothelium, which lines the inner wall of the artery, to the intima. Once in the intima it is advected and diffuses. AQ proportion is converted to oxidised cholesterol which binds to the substrate, and leads to the thickening of the intima and the formation of a fatty streak. The bump in the vessel wall changes the flow of blood. The shear stresses imposed on the wall by the blood flow alter and in turn cause changes in the permeability of the endothelium.
Modelling this process gives information on how intimal hyperplasia progresses and on what factors, such as ageing and geometry, alter the speed of this progression.
Joint work with: Nick Hill

"Numerical and analytical results for a special class of CGL equations with small dissipation"

Abstract
We consider the problem u_t=\delta u_{xx}+i|u|^2u in one spatial dimension with periodic boundary conditions. We describe the behaviour of solutions in the turbulent regime\, which means we take smooth initial data with supremum norm of order one, and the positive real parameter \delta to have order much less than one. The peak values of the C^m-norms of solutions behave as powers of \delta:
c_m \delta^{-m/2}\leq\sup_t|u(t)|_{C^m}\leq C_m\delta^{-m/2}
Here the upper bound can be proved analytically, while the lower has been observed in numerical experiments. These bounds imply restrictions on the Fourier spectra of solutions related to the Kolmogorov--Obukhov law of fluid dynamics. In particular, they imply that the inner scale\' of the turbulence behaves like \delta^{1/2}.

"Quantum computation - some theoretical challenges"

Abstract
To be supplied

"Method of deciding a single-line programming problem with boolean variable"

Abstract
Created method of deciding a single-line programming problem with boolean variable. This method realizes tacit viewing spots - decisions. Proved and experimental confirmed its convergence to optimum deciding always, at the condition positive factors to target functions and restrictions (problem dares on the maximum). Conducted experimental evaluation of number of iterations. For similar problems this number did not exceed values 10n, where n - a number variable problems. Labour content to iterations approximately is or even iterations less a simplex-method in single-line programming problems. When deciding the problems, amongst factors which be and positive, and negative sometimes appeared a recirculation. Way of struggle with the recirculation until revealled. But much similar problems by the author was flown. Noticed relationship between the appearance of recirculation and volume of area of possible decisions. Than area less, that more probable appearance of round-robin process of viewing the possible decisions. Problems were decided On the computer Pentium-200 with number variable before 5000. The Author hopes that will manage to perfect algorithm toward the struggle with the recirculation, or same - to ensure constancy its work when deciding the single-line programming problems with positive and negative factors and boolean variable. The Temporary evaluations tinned when deciding the problems allow to do the first findings on single-line dependencies of time of deciding from the number variables. Dependency from the number of restrictions too exists, but it reveals itself too as single-line, but with vastly smaller factor.

"Mathematical and Algorithmic Challenges from the Molecular Life Sciences"

Abstract
Molecular biology, genetics and genomics seek to understand life at the level of genes and proteins. Specific goals include sequencing and comparing the genomes of different organisms, identifying the genes and determining their functions, understanding the genetic basis of disease, understanding the evolutionary relationships among organisms on the basis of molecular data, understanding how genes and proteins work in concert to control cellular processes, and predicting the three-dimensional structure of proteins. All of these endeavors require algorithms for the analysis of complex data. The speaker will describe several key algorithmic problems and techniques in this field, including sequence assembly, sequence alignment and comparison, construction of Hidden Markov Models of protein families, gene finding, construction of phylogenetic trees, and the analysis of gene expression using clustering and classification techniques.

"Inertial oscillations driven to blow-up by a quadratic density variation"

Abstract
A horizontally quadratic density profile in an infinite slab generates a two-dimensional stagnation-point flow which blows up in finite time. In a rotating frame at high Burger number, the blow-up is slightly delayed. However, at low Burger number, i.e. when Coriolis forces are large compared with buoyancy forces, the system undergoes inertial oscillations of increasing amplitude before the eventual blow-up. The multiple-scales method is used to analyse the amplitude and phase of the growing oscillations, and asymptotic methods can also be used to analyse the final approach to blow-up. These analytical results are compared with numerical computations, the latter being the only way to determine how the time of blow-up and the number of oscillations before blow-up vary with Burger number.

"A finite element approximation of a variational formulation of Bean's model for superconductivity"

Abstract
We introduce a finite element approximation of a variational formulation of Bean's model for the experimental set-up of a infinitely long cylindrical superconductor subject to a transverse magnetic field. We prove an error bound between the exact solution and the approximate solution, with respect to the finite element mesh size.
Numerical simulations for various applied magnetic fields will be shown.

"Multisingularities and cobordisms"

Abstract
The enumeration of singularities of differend kind is a classical subject of enumerative geometry. For a given map of smooth manifolds the points of the source manifold are classified according to the singularities of the germ of the map at these points. Every singularity class defines a locus in the source manifold formed by points at which the singularity of the map belongs to the given class. By a general principal of R.Thom, the cohomology class dual to the closure of this locus is a characteristic class of the manifolds and can be expressed as the universal polynomial in the Chern (or Stiefel-Whitney) classes of the source and target manifolds. Many authors have contributed to the computation of these polynomials for various classes of singularities; computation of Thom polynomials for singularities of Boardman type \Sigma^{1_k} was for several years one of the most challenging compuational problems in Singularity Theory. Recent developments by A. Szucs and R.Rimanyi,in Budapest, and, independently, by M. Kazarian, in Moscow, have led to a substantial increase in the calculability of Thom polynomials for these singularities and many others.
The points of the target manifold are classified according to the multisingularities of the map (corresponding to singularities at different points with the same image). We present a similar apriori description of the characteristic classes dual to the cycles of multisingularities. The classes dual to the multisingularity loci were in the frmework of intersection theory. The most strong result were the formulas due to Kleiman and Katz for the cycles of multiple points valid for maps admitting only singularities of corank 1. We present a new formula valid for all proper finite maps and for multisingularities of any kind in terms of the so called residue polynomials. The motivation for this formula came from topology, namely, from cobordism theory, the domain being rather far from algebraic geometry and intersection theory, perhaps, this was the reason why this formula have not been observed before. This result demonstrates the unity of mathematics: combinig the methods of completely different domains (intersection theory and cobordism theory in this case) one may obtain new results in both of them.

"A comparison of two computer aided assessment systems"

Abstract
The two Computer Aided Assessment (CAA) systems are CalMaeth and AIM:
http://calmaeth.maths.uwa.edu.au
http://allserv.rug.ac.be/~nvdbergh/aim/docs
Both are underpinned by mass-market Computer Algebra (CA) packages, Mathematica for CalMaeth, Maple for AIM. Both deliver questions to students via Web.

"Modelling diseases in a heterogeneous environment. The 2001 Foot-and-Mouth Epidemic"

Abstract
The 2001 Foot-and-Mouth epidemic provided a stern test for epidemiological modelling. This is one of the few occasions when modelling could be used to influence the on-going dynamics of an epidemic - with the aim of limitting its effect. This talk will discuss a detailed spatio-temporal model which included many of the heterogeneities which influenced the dynamics. The effects of the implemented culling policy will be compared to various other control and vaccination options.

"Nonlinear Waves in a Diverging Fluid Channel"

Abstract
Jeffery-Hamel flows are a family of exact solutions of the Navier-Stokes equations for steady 2-dimensional flow of an incompressible viscous fluid from a line source at the intersection of 2 rigid planes. Although they were discovered over 85 years ago by Jeffery (1915) and Hamel (1916) and look relatively simple (the streamfunction is independent of the polar radius!), they are not widely understood because there is a multiplicity of solutions with a richer structure than most other similarity solutions. Additionally, computations by Tutty (1996) in a closely related geometry have suggested that spatially-periodic, steady flows may also exist in an infinitely diverging channel. We confirm this and discuss how these nonlinear waves may be related to the traditional Jeffery-Hamel family of flows.
Joint work with: Owen Tutty and Philip Drazin

"Unified Analytic Geometry using Determinants"

Abstract
The formula for calculating the area of a triangle with given coordinates is very similar to the calculation of the volume of a tetrahedron using its coordinates. This work analyses a variety of formulae in Analytic Geometry, concentrating on expressions using determinants, as these become increasing useful with the development of calculators. It unifies approaches for several dimensions by defining a global terminology applicable to any dimension. The constructed formulae are validated in one, two and three dimensions, leading, in part, to new expressions.

"Regularity in the complex spectra of random circulant Jacobi matrices"

Abstract
Circulant Jacobi matrices are tridiagonal matrices with additional non-zero entries at the right-top and bottom-left corners. This talk present results, obtained jointly with Ilya Goldheid, which show that such matrices have surprising distribution of eigenvalues in the limit of large matrix dimension. Under some general conditions on the matrix entries, the eigenvalues are distributed along arcs in the complex plane and segments of the real axis, and the **non-real** eigenvalues are regularly spaced. When the matrix entries are taken randomly, these two properties of the eigenvalue distribution hold with probability one.

"Internal waves in a continuously stratified bubbly fluid"

Abstract
The influence of gas bubbles on the properties of internal waves in a continuously stratified fluid is studied in the framework of a two-dimensional model of a diluted monodisperse mixture of an incompressible fluid with gas bubbles. The model takes into account surface tension on the walls of the bubbles and an effective viscosity. We obtain the dispersion relation for linear waves in the Boussinesq approximation for a uniformly stratified fluid, and show that in the presence of a uniform distribution of bubbles, there are two classes of plane waves. One class, the "bubble" wave may propagate with frequencies higher than the buoyancy frequency, while the other class is a modified internal wave, whose frequency is less than the buoyancy frequency, with a finite gap existing at all wavenumbers. The effective viscosity introduces a damping of both modes, and has a greater effect on the "bubble" mode. We also obtain the dispersion relation for waves propagating horizontally in the oceanic waveguide, both for the case when the fluid is uniformly stratified and contains the bubbles, and when the bubbles are in a thin nearly homogeneous upper layer. Solutions involving an exchange of energy between the fluid and the bubbles are constructed. The possibility of the long-short wave resonance is discussed.
Joint work with: RHJ Grimshaw

"Chaotic diffusion coefficients from velocity field information : wavy Taylor vortex flow"

Abstract
In a recent investigation of particle transport in numerically computed wavy Taylor-vortex flow, Rudman estimated an effective axial diffusion coefficient, D_z, to characterize the enhanced mixing due to chaotic advection [AIChE J. 44 (1998) 1015--1026]. We find that D_z is proportional to the product of two measures of symmetry deviation. The first is a measure of the average deviation of the flow from rotational symmetry, and the second is a measure of the average deviation from flexion-free flow (a flow where the curl of the vorticity is zero). Because these quantities are obtained directly from the velocity field, we call them Eulerian symmetry measures. Thus, we show that the macroscopic transport behaviour in a flow can be quantified directly in terms of the velocity field and its gradients, and hence provides a connection between Eulerian and Lagrangian pictures of transport - a problem of fundamental and wide-spread interest.
Joint work with: G Rowlands, M Rudman, K Coughlin and AN Yannacopoulos

"On extension and torsion of a compressible elastic circular cylinder"

Abstract
In this talk we discuss explicit conditions on the strain-energy function of a compressible isotropic elastic solid needed for the material to support combined extension and torsion of a solid circular cylinder with vanishing traction on the lateral surface of the cylinder. The deformation is maintained by the application of a twisting torque and an axial load on the ends of the cylinder. A general formula connecting these two quantities is derived. A special case of the above deformation is pure torsion. Important examples of hyperelastic materials are shown to support this (isochoric) deformation with the vanishing traction on the lateral surface of the cylinder, and some new classes of strain-energy function satisfying the criteria established are constructed.
Joint work with: RW Ogden

"Classical and Quantum Systems from Symplectomorphisms"

Abstract
We present an untraditional scheme of quantisation as an intertwining operator instead of common definition as a Lie homomorphism of Lie algebras of Poisson brackets and commutator.
Definition. Quantisation is a linear operator from a space of function to an algebra of convolutions, which intertwines two actions of the symplectic group: by symplectomorphis of the phase space in classical mechanics and the metaplectic representation in quantum mechanics.
Our approach naturally wraps both classical and quantum systems of particles and fields into a single object. Thereafter classical and quantum descriptions could be obtained from it by means of different representations.

"Chaotic mixing in a combustion model"

Abstract
We consider chaotic mixing within a combustion reaction. We study the evolution of a localized perturbation in our reaction-advection-diffusion (RAD) problem. In a two-dimensional system,a convergent and a divergent direction can be assigned to any point in the flow associated with the eigenvectors corresponding to the negative and positive Lyapunov exponents -lambda and lambda of the chaotic advection. Based on this we separate our original RAD problem along the (Lagrangian) streching and convergent directions. The one-dimensional equations which describe the behaviour in the convergent direction, 'the Lagrangian filament slice model', play a fundamental part in understanding the basic chaotic advection of reactants. They are the equations that determine the mean transverse profiles that propagate along the divergent direction following the unstable foliation. We investigate this one-dimensional model numerically and analitically. The understanding gained from the 'filament slice model' will be used to investigate the two-dimensional RAD.
Joint work with: John H Merkin and Stephen K Scot

"Strain patterns in large-scale plastic deformation"

Abstract
An approach to stress and deformation has been developed that treats elastic loading (i.e. stress) as a change of state in the sense of the First Law of thermodynamics. The theory uses an equation of state for solids, involves a transformation of the thermodynamic theory from scalar form into vector form, and takes into account that a solid consists of bonded matter. The subject of interest is the mechanical interaction of a thermodynamic system with a surrounding within a larger solid; hence two independent force vector fields exist: the external one is controlled by the boundary conditions, the internal one by the material properties; they are brought into equilibrium with one another by the assumption that in the elastic realm no bonds are broken, and the contact is bonded. It has thus been possible to successfully predict the microstructural properties of simple shear.
At the elastic-plastic transition the state of stress decays irreversibly through breaking of bonds. The transition is connected with a bifurcation: the loaded state can only partially relax into one of two possible states the geometric properties of which have opposite skew. Thus linear elements in the solid oriented /Y and perpendicular to the XZ-plane (e.g. fold axes) are mechanically unstable; they will passively rotate from either side into the X-direction, leading to of sheath folds. The phenomenon is physically related to turbulence in viscous flow and the origin of conjugate joint sets in brittle deformation.

"Modelling tissue invasion by bacteria"

Abstract
The bacterium Pseudomonas aeruginosa employs a mechanism termed Quorum sensing to regulate its behaviour based on population density. This mechanism effectively switches on the production of degradative enzymes and toxins across the whole population once a high density has been reached. These enzymes and toxins destroy host tissues releasing nutrients for the bacteria. The bacteria are then able to move into regions of damaged tissue, and via this process will eventually reach the host blood supply in large numbers, almost certainly leading to septicemia. In this talk a model of the tissue invasion process shall be presented and compared to in vitro experiments.

"Initial asymptotics of gravity-driven flow"

Abstract
The work is concerned with the initial asymptotics of the generalized dam-breaking problem. Several important problems (dam-breaking problem, water-on-deck problem, cavity collapse problem, sloshing problem and some others) are treated with the same technique based on Lagrangian description of the flow. The standard Small-Time Expansion Procedure used by Stoker and Pohle, has been modified to account for Separation Effect and for the Run-Up Phenomenon. Finally, the uniformly valid initial asimptotics of the flow is presented and difficulties with numerical solutions of the problem are discussed.

"Asymptotic formula for solutions to elliptic equations near a Lipschitz boundary"

Abstract
We study an asymptotic behaviour of solutions to the Dirichlet problem for arbitrary order elliptic equations in an n-dimensional domain near a boundary point O. It is supposed that the boundary near 0 is Lipschitz, i.e. it can be given by x_n=\phi(x') near 0 with a Lipschitz function \phi. The only assumption on the boundary is a smallness of the Lipschitz constant of \phi. We give an explicit description of asymptotic behaviour of solutions near O. Some applications of this result will be discussed.
Joint work with: V Mazya

"Mapping Class Group Actions on Ordered Sets"

Abstract
The theme of this talk is whether some recent and some not so recent results on braid groups can be generalized to surface mapping class groups. I start by recalling three results on braid groups: (1) The lattice ordering on them, closely related to the word problem and automaticity; (2) The linearity; (3) The relation between 1 and 2. I then present two ideas how to generalize to mapping class groups: an ordered set on which they act (unfortunately not a lattice), and a seemingly new "linear representation" which has in common with the faithful braid group module that it is a second homology group.

"On the number of real roots of a polynomial"

Abstract
The famous result of Schur gives a bound for the number of positive real roots of a polynomial in terms of L_1 norm of its coefficients. We provide similar best possible inequalities in weighted L_2 norms. More generally, we establish an unexpected connection between the multiplicity of the zero at unity and the L_2 norm of discrete orthogonal polynomials. Explicit bounds are obtained on choosing some known families of the discrete orthogonal polynomials. On the other hand, any (e.g. random) construction of a polynomial with high-fold zero at unity yields bounds on the discrete orthogonal polynomials. We indicate a connection of the above problem with the orthogonality on the unit circle.

"Spectral properties of solutions for nonlinear PDEs with small viscosity (analytical and numerical results)"

Abstract
I shall present some analytical and numerical results which describe spectral properties of solutions for nonlinear PDE with small linear dissipation, and discuss their relations to the theory of decaying turbulence.

"Moments' method in an inverse problem for heat equation"

Abstract
We consider an inverse problem of a coefficient reconstruction in a heat/diffusion equation in a bounded domain. The data used is a finite number of Cauchy data on the lateral boundary of solutions to the heat equations. These solutions correspond to the boundary sources which are restrictions on the boundary of the first few harmonic polynomials. We describe a method of an approximate reconstruction of the unknown coefficient using these data, analyse its rate of convergence when the order of the polynomials used tends to infinity and discuss the numerical implementation of the method.

"Finite travelling waves in singular diffusion equations"

Abstract
New results are presented on the existence of finite travelling waves (FTW) in singular bistable equations with degenerate diffusion. These results extend those of Hosono (1985) in the non-singular case. By singular' we mean that the bistable nonlinearity is non-differentiable at the origin. Such equations arise in many applications including signal propagation in nerve axons, population genetics and combustion theory.
In contrast to the non-singular case we show that FTW fronts exist with negative velocity. The existence of such waves gives rise to two interesting phenomena. Firstly, one can construct (weak) travelling waves comprising two distinct FTW moving with different speeds but in the same direction. We term these multi speed wave trains'. Secondly, solutions of the full Cauchy problem with compactly supported initial data become zero after a finite time (finite-time extinction). The latter behaviour is commonly observed in quasilinear heat equations, but not in diffusion equations with sign-changing nonlinearities. Furthermore, initially positive solutions may become zero on a spatial region which expands over time, giving rise to so-called dead cores'. These are common in chemical reactor theory but not in bistable equations.
We also present some generalisations to multistable diffusion equations.
Joint work with: AT Peplow and RE Beardmore

"Geometric invariants of Cohen-Macaulay modules over 2-dimensional rings"

Abstract
I would like to talk about some invariants of Cohen-Macaulay modules over local 2-dimensional algebras. These invariants are some real (possibly even rational) numbers. For rings of invariants of finite groups they can be computed using the Atiyah-Singer-Segal equivariant Riemann-Roch formula. For other rings they seem to be one of the few known non-trivial invariants that behave additively on the Auslander-Reiten quiver. The main reason to their study are geometric applications to linear systems on normal surfaces, singularities of plane curves, hyperbolicity of algebraic surfaces etc. To be supplied

"Nonexistence of global solutions for higher-order evolution partial differential equations"

Abstract
We investigate nonexistence of global (nontrivial) solutions for some semilinear evolution partial differential equations and inequalities in exterior and cone-like domains. Our proofs are based on nonlinear test-function method and we do not use the properties of correspoding linear problems. Nevertheless most of our critical exponents are sharp for parabolic inequality (Fujita-type exponents), and (partially) for hyperbolic problems (Kato-type exponents). The results were announced in a paper: G.G. Laptev, Some nonexistence results for higher-order evolution inequalities in cone-like domains, Electron. Res. Announc. Amer. Math. Soc. 7 (2001) 87--93.

"Algorithmic theory of Zeta and L-functions over finite fields"

Abstract
A basic problem in algorithmic number theory is: Given a multivariate polynomial f over a finite field determine how many solutions there are to the equation f = 0 over the field, and more generally, over all' extension fields. This information is encoded in the Zeta Function of f. By Dwork's rationality theorem one can compute the Zeta Function. The problem, however, is to compute it quickly' i.e. in polynomial-time in the input parameters. The most famous example of this problem is the case in which f defines an elliptic curve. Here Schoof's algorithm may be used, and this is a crucial part of elliptic curve cryptography. I will describe some recent work with Daqing Wan on this topic. Specifically, we prove polynomial-time computability of the Zeta Function for all polynomials, provided the field characteristic is small'. I will also describe some very fast algorithms for special classes of curves, and discuss the more general problem of computing L-functions for character sums over finite fields.

"L^2 cohomology of Artin groups"

Abstract
For each Artin group we compute the L^2 cohomology of the associated Salvetti complex, which is conjecturally an Eilenberg-Mac Lane space. The answer is surprising for two reasons:
(1) it turns out to be far easier than computing the ordinary rational cohomology (which is not known in general);
(2) many of the groups are non-trivial (whereas most previous calculations using L^2 cohomology have depended on vanishing theorems).
In my talk I shall give an introduction to L^2 cohomology and state our result.
Joint work with: Mike Davis

"Adjustment structures - sets, mappings, calculus"

Abstract
Newtonian methods (tangency) even when extended to (Nonlinear) Riemann methods (multiple tangencies) cannot model all Dynamical effects faithfully - often not at all. Pure Nonlinear methods (Axiality, polarisation) also fail universality tests. But adjustment (Abstract analogue of Physical Interaction and Interference) models definitively all Dynamical effects. This is established and then the forumulations and calculus forms for this new universal dynamical characterisation are explicitly presented.

"Inverse inclusion problems"

Abstract
An inverse problem is considered to identify the geometry of discontinuities in a conductive material \Omega\subset R^d with conductivity (I+(K-I) \chi_D) from Cauchy data measurements taken on the boundary \partial\Omega, where D\subset\Omega, K is a symmetric and positive definite tensor not equal to identity and \chi_D is the characteristic function of the domain D. As an example this models the determination of the shape, size and location of the anisotropic inner core of the Earth from measurements taken at its mantle. There are also other applications in electrical impedance tomography (EIT). The previous results of Ikehata (1998) for estimating the size of the inclusion D are proved and applied to several examples. Further, we develop an integral representation of the solution and we propose an efficient boundary element method (BEM) in conjunction with a least-squares constrained minimization procedure to detect an anisotropic inclusion D, such as a circle, by a single boundary measurement. Numerical results are discussed confirming the previous theoretical estimates of the size of the inclusion and giving an insight into the unresolved uniqueness issues of detecting ellipses.

"Numerical simulation of the flame fronts with two consecutive reactions in enclosed tubes"

Abstract
Slow combustion involving a two-step consecutive reaction A->B->C is studied. The conditions of separation of flame fronts and evolution of their shapes are investigated. A low Mach number limit of the Navier-Stokes system was used. The system was discretized and solved with an explicit, central finite-difference scheme on a staggered grid. Fractional time steps have been used for chemical and diffusion-convection phenomena. The standard solver of stiff ordinary differential equations (ODE)-LSODE has been employed in the chemical step. Pressure correction approach was adopted as an alternative to the method of preconditioned iterations. A set of Poisson solvers including Successive Over-Relaxation (SOR), Preconditioned Conjugate Gradient Squared (Pre-BICG), Preconditioned Bi-Conjugate Gradient (Pre-BICGSTAB), Preconditioned Bi-Conjugate Gradient Stabilization (Pre-BICGSTAB), Fast Fourier Transfermation (FFT) was considered and compared.
Joint work with: V Karlin and G Makhviladze

"A point source method for inverse scattering by rough surfaces"

Abstract
In this poster we propose a new method for determining the location and shape of an unbounded surface from measurements of scattered electromagnetic waves, based on the Point Source method of Potthast for inverse scattering by bounded obstacles. We present numerical results for the case of a perfectly conducting surface in TE polarisation, in which case a homogeneous Dirichlet condition applies on the boundary and the surface is located as a minimum of the total field.

"Instability of MHD-modified interfacial gravity waves revisited"

Abstract
We reveal the basic mechanism of instability of the two-layer conductive fluid system carrying a normal current and exposed to a uniform external magnetic field. This process is a reflection of a MHD-modified interfacial gravity wave from the boundary. Due to special boundary conditions, the reflection coefficient turns out to be greater than 1 for some directions of the wave propagation. We consider two cases: reflection of a monochromatic plane wave from the plane boundary and reflection of rotating waves in a circular geometry. We believe that the proposed mechanism gives a new understanding of the instability formation in the system liquid metal-electrolyte' type.
Joint work with: G El and S Molokov

"Ping-Pong ball avalanches"

Abstract
Ping-pong ball avalanche experiments have been carried out for five years in Japan on a ski jump and in chutes. The speed of these flows is controlled by the balance between air drag and gravity, with surface friction playing only a small role. Estimates are made of internal stresses using kinetic theory and shown to be in balance with air drag and gravity. A simple integral model is shown to accurately describe the flows over most of the slope but the flow is self-similar, but mnore complicated models are needed to describe the initial acceleration phase and the final deposition phase.

Poster for Quantum theory:

"Uniform persistence of spectral projections for networks of quantum units"

Abstract
A strategy to prove persistence of spectral projections for networks of quantum units, uniformly in the size of the network, is presented, and applied to experiments on vibrations in molecular crystals such as solid deuterium, platinum chloride ethylenediamine and 4-methyl pyridine.

Poster for Control:

"Safety criteria for aperiodically forced systems"

Abstract
The theory of uniform hyperbolicity is used to obtain sets of forcing functions for which the response of a dynamical systems remains in a prescribed region, asymptotically optimal for small regions. Joint work with Z Bishnani

Poster for Geom, Top & Mech:

"Five geom/top topics in Hamiltonian mechanics"

Abstract
(a) Anosov parameter values for Thurston's triple linkage. Joint work with T Hunt
(b) Geometric interpretation of Weierstrass' theorem on invariant Lagrangian graphs. Joint work with M Bialy
(c) Second species orbits in the circular restricted 3-body problem. Joint work with S Bolotin
(d) Instability of a vortex in a straining field
(e) Hamiltonian slow manifolds. Includes aspects of joint work with J-A Sepulchre, T Ahn, R Battye, P Sutcliffe

"Inverse boundary value problems in linear elasticiticy"

Abstract
In the formulation of the Cauchy problem in linear elasticity the Lame system of equations has to be solved subject to overspecified boundary conditions on both the displacement and traction vectors over a portion of the boundary of the solution domain, with the remaining portion of the boundary being underspecified. This classical Cauchy problem is ill-posed and direct inversion numerical techniques fail to produce a stable solution. Therefore, in this paper several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition are developed and compared.

"Values of quadratic forms at lattice points"

Abstract
Consider the infinite sequence of values of a quadratic form taken at the grid points of a k-dimensional lattice. I will review recent results which suggest that, provided the quadratic form is sufficiently generic, these values behave like a sequence of independent random numbers. Our studies are motivated by quantitative versions of the Oppenheim conjecture, and by a conjecture of Berry and Tabor in the context of quantum chaos, which asserts that the spectral correlations of quantized integrable systems are those of a Poisson process.

"Intersections of automorphism fixed subgroups in a free group"

Abstract
We give some results concerning when the family of subgroups of a finitely generated free group is closed under intersections.

"What every mathematician ought to know about diseases: an introduction to epidemiological modelling"

Abstract
To be supplied

"Exponential averaging for parabolic systems"

Abstract
The influence of rapid time-periodic forcing on a large class of semilinear parabolic partial differential equations is analysed. After a suitable transformation of the phase space the nonautonomous forcing-terms are exponentially small in the period of the forcing. This transformation result gives also a geometric theory to analyse the influence of rapid forcing to hyperbolic equilibria and to homoclinic orbits.

"Almost sure and second mean stabilization for a class of stochastically perturbed systems"

Abstract
Lyapunov exponents, almost sure and in pth mean, characterize stability of stochasticaly perturbed linear systems. They are described by variants of the Furstenberg-Khasminskii formulas.
Here we develop asymptotic formulas for the Lyapunov exponents ( sample and second mean ) of a 2 dimensional linear stochastic differential equation with multiplicative noise, adopting for the calculation of the almost sure bound limit a Stratonovich form of the solution and for the second mean an Ito one.
Using the latter formulas we conclude that if the noise enters in a purely skew-symmetric way and is strong enough, then our system is high-gain stabilisable almost surely and in second mean, even when the zero dynamics are unstable. If the diffusion matrix is not skew-symmetric then we show that almost sure and pth mean growth rates have different asymptotical behaviour.

"On globally periodic solutions of the difference equation"

Abstract
We study the second-order difference equation
x_{n+1} = \frac{f(x_n)}{x_{n-1}}
where f \in C^1([0, \infty), [0, \infty)) and x_n \in (0, \infty) for all n \in Z. For the cases p \le 5, we find necessary and sufficient conditions on f for all solutions to be periodic with (non-prime) period p.

"About Localized Boundary-Domain Integral and Integro-Differential Equations for Problems with Variable Coefficients"

Abstract
Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a Boundary Value Problem with variable coefficients to a Localized Boundary- Domain Integral or Integro-Differential Equation (LBDIE or LBDIDE). After discretization, this results in a sparsely populated system of linear algebraic equations, which can be solved by well- known efficient methods. This makes the method competitive with the Finite Element Method. Some techniques of the parametrix localization are discussed and the corresponding LBDIEs and LBDIDEs are introduced. Both mesh-based and meshless algorithms for the localized equations discretization are described.

"Fractal Escape Time and Chaotic Dynamics"

Abstract
We consider the time required for a Hamiltonian system to escape from an identified region of phase space, where the dynamics is given by an area-preserving chaotic map of the phase plane. Segments of a line of initial conditions escape at various iterates of the map, and we observe sequences of such segments which escape at successive iterates. These sequences, called epistrophes'', decay geometrically at large iterate number but have unpredictable initial behavior. The epistrophes impart a fractal structure to the plot of ionization time, but their unpredictable beginnings break any true asymptotic self-similarity, leaving a weaker epistrophic self-similarity.'' The existence and characterization of epistrophes follow from a general geometric analysis of homoclinic tangles. Application to the ionization of an excited hydrogen atom in parallel electric and magnetic fields is considered.

"Acoustic scatttering a in waveguide with a membrane bounded cavity"

Abstract
This talk is concerned with acoustic scattering in a 2-D waveguide with a membrane bounded cavity. The waveguide is formed by two semi- infinite ducts occupying the regions 0\leq y \leq a, x<-\ell and 0\leq y \leq a, x>\ell, together with a finite duct of height b>a in the gap -\ell\leq x \leq \ell. The structure is closed by two vertical regid surfaces at x=\pm\ell a\leq y \leq b forming a rectangular cavity. The interior region of the duct contains a compressible fluid of sound speed c and density \rho. The fluid in the cavity, however, is separated from the main body of fluid by a horizontal membrane which lies along y=a, -\ell \leq x \leq \ell. A plane wave is incident in the positive x direction through the fluid towards x=-\ell and is scattered at the discontinuity. The under-lying eigensystem for the cavity region of the duct is non Sturm- Liouville, however, the orthogonality relation is known (see [1]). This enables a mode-matching technique to be employed to determine the amplitudes of the reflected and transmitted waves. The resulting system of algebraic equations is solved numerically. Results are presented in the form of a parametric investigation of the power distributed in the membrane and various fluid regions.
[1] Warren, D.P., Lawrie, J.B. & Mohamed, I.M. (2002) Acoustic scattering in waveguides that are discontinuous in geometry and material property. Wave Motion, (to appear).

"Free surface MHD flows"

Abstract
Magnetohydrodynamic (MHD) liquid metal flows with free surfaces are of interest in the context of divertor applications for tokamaks, as well as potential applications to coating flows, laser welding, semiconductor crystal growth, etc.
Flows of an electrically conducting fluid in the presence of a strong, uniform magnetic field are considered. The main driving forces are gravity and surface tension. The Lorentz force, which results from the interaction of the electric currents induced by the fluid motion, considerably slows down the flow.
Particular attention is made to spreading of a drop on a horizontal surface, pendent volume of fluid, film flow on an inclined plane, and rivulet. The problems are analyzed by matched asymptotic expansions at high values of the Hartmann number. The resulting evolution equations for the free surface location have very rich structure. Depending on the type of the flow and the dimensionless parameters of the problem, various forms of the evolution and equilibrium equations result. The features of various flows depending on the orientation of the magnetic field, and the balance between gravity and surface tension are discussed.
Finally, parallels with Hele-Shaw flow, flow in a porous medium, and spreading of thin drops of a non-conducting fluid are drawn.

"Homfly polynomials and quantum sl(N) invariants of decorated Hopf"

Abstract
We give a formula for the quantum invariant of the Hopf link colored by the irreducible sl(N)_q modules V_\lambda and V_\mu corresponding to partitions \lambda and \mu in terms of an N X N minor of the Vandermonde matrix (q^{ij}). This invariant is a specialisation of the 2-variable Homfly polynomial <\lambda,\mu> of the Hopf link decorated by elements Q_\lambda,Q_\mu in the skein of the annulus. We show how to find <\lambda,\mu> from the Schur function s_\mu of an explicit power series depending on \lambda. The Homfly polynomial of any link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle can then be calculated as a linear combination of suitable <\lambda,\mu>.
Joint work with: SG Lukac

"Some surprises when mathematics meets vision"

Abstract
Take a new scientific area -- in our case, the study of the 2D signals presented by imaging systems. Does mathematics contribute some insight? I'll discuss three interactions. In the first, the math turned out to be interesting but unfortunately not really relevant; in the second, we find that "images" are not measurable functions; in the third, we show how fluid flow ideas provide an approach to analyzing "shape".

"An asymptotic model of unsteady airway reopening"

Abstract
Airway reopening is a fundamental process in pulmonary mechanics involving the propagation of a bubble of air into a liquid-filled collapsed airway. The bubble must be blown at a pressure (P_b) large enough to overcome the adhesive effects of viscosity and surface tension, yet sufficiently low to minimize damage to delicate airway walls. Bench-top experiments designed to mimic repoening a collapsed lung airway show that, during the initiation of reopening using a constant flow-rate pump, P_b can transiently overshoot its ultimate steady value. This physiologically important behavior motivates our study.
We consider a theoretical model for unsteady propagation of a long bubble through a liquid-filled channel whose flexible walls are held under tension T and are either confined between plates separated by a distance D or are supported by linearly elastic springs with stiffness E. An asymptotic approximation valid for large T reduces the model to solving a fourth-order evolution equation for the channel width ahead of the bubble tip. The model predicts both a transient peak as well as a monotonic response in P_b. The type of behavior and magnitude of the peak was recorded for a wide range of values of D, E and pump flow-rate, and may suggest a physiologically useful parameter range for avoiding transient overshoot. The model qualitatively agrees with experiment and provides a first step towards understanding this phenomenon in a variety of reopening configurations.
Joint work with: OE Jensen

"Quasi-periodic motion in a plane parallel shear flow"

Abstract
The transition from laminar flow to an early stage of turbulence in a simple parallel shear layer is investigated numerically by using a bifurcation analysis and DNS technique. It is found that the first bifurcation is described by a two-dimensional transverse vortex flow whereas the second bifurcation is characterised by three- dimensional sinusoidal motions of the transverse vortex. As the controling parameter increases these motions become unstable and a time- dependent state with two distinct frequencies appears. Then, chaotic motions follow. We shall show the spatial structure as well as some physical properties of these flows.
Joint work with: T Itano

"Vibration Frequencies of Rotating FGM Cylindrical Shells"

Abstract
In this a vibration frequency analysis of rotating functionally graded material (FGM) circular cylindrical shells is carried out. The Galerkin method is applied for the study and axial modal dependence is approximated by the trigonometric functions for the simply supported boundary condtion.The influence of volume fractions of the constituenct materials of the shell is considered on shell frequency besides those of geomerical parameters.
Joint work with: CB Sharma

"Small-scale turbulent dynamo"

Abstract
We study evolution of magnetic fields advected by Gaussian white noise with emphasis on the moments of its Fourier amplitudes. We derive and compute numerically the growth rates of these moments and their distribution over scales in two asymptotic regimes: initial perfect conductor phase and the final dynamo state. We identify the stochastic deformations which are responsible for the dynamo growth and the typical magnetic coherent structures dominating such turbulence.
Joint work with: Rob West

"A unified theory for ripples, dunes and roll waves"

Abstract
To be supplied

"Key varieties in algebraic geometry"

Abstract
We give some old and new constructions of algebraic varieties using key-ambient varieties.

"Pattern formation on growing domains: Applications to tumour biology"

Abstract
Recently work has been undertaken to examine the reaction diffusion models of Turing (1952) on growing domains. We look to use such a model as a paradigm for tumour growth and examine the behaviour of chemicals reacting and diffusing within such a regime. Thus far we have mainly concerned ourselves with one-dimensional uniform and nonuniform growth, where the growth is driven by the chemicals within the system.
In the uniform case our results mirror those of Crampin et al. (1999) where we see frequency-doubling (as the pattern changes from one to two to four to eight modes etc.) although we show that inclusion of dilution terms (neglected in Crampin's work) can disrupt this mechanism.
For nonuniform growth we see a range of pattern and domain behaviour. We couple together the net cell birth rate (= F = cell birth-cell death) and the domain growth, whereby divergence of the local cell velocity within the domain equals the net cell birth rate. Taking F to be a function of the chemicals present in the domain, e.g. oxygen, glucose or lactic acid, we have domain growth driven by the chemicals present. This gives rise to a range of patterns on both growing and shrinking domains.

"Toward a notion of quantum Banach spaces"

Abstract
We show that the category of Operator spaces is a natural full subcategory of essential Banach K-bimodules (in term of the whether the canonical embedding of the Banach K-bimodule into its K-bimodule bidual'' is isometric). We will also define Haagerup, projective and injective tensor products for general essential Banach K-bimodules.

"Impulsive pressures associated with a water wave impact on a vertical wall"

Abstract
The impact of a sea wave against a vertical wall results in a rising plume of water up the wall. The early growth of this "splash" can be modelled by an ideal fluid attracted by a sink positioned just outside the fluid domain. This violently unsteady flow has an instantaneous pressure, p, which is found from a Maclaurin expansion of the velocity potential with respect to time. The pressure field is found analytically for finite and infinite fluid depths using Green's functions. It is shown that the pressure has a single spatial maximum close to the free surface, which is associated with its violent eruption and extreme force on the wall.

"Non-Markovian random processes and travelling fronts in reaction-transport systems"

Abstract
In this poster we present an analysis of propagation of travelling fronts into an unstable state of a reaction-transport system. Our approach makes use of generalized master equation, hyperbolic scaling, and Hamilton-Jacobi theory. In particular, we consider the case when the waiting-time distribution is different from exponential one that leads to non-markovian dynamics.
Joint work with: Sergei Fedotov

"Homoclinic branch switching"

Abstract
We present a new numerical method for switching between branches of homoclinic orbits, during a numerical numerical continuation process. Specifically, the method constructs an n-homoclinic orbit (having multiple bumps) where n>1, from a 1-homoclinic orbit (consisting of a single bump), which can then be followed in the usual way. This method is implemented in the software package AUTO/HomCont.
Our approach does not require any a priori knowledge or the computation of any additional periodic orbits. The basis of this scheme is the so-called Lin's method. Our method is very robust and more reliable than traditional shooting methods, especially if the unstable manifold has a dimension higher than one.
It allows us to explore routes to chaos in models arising in applications. We use Sandstede's model, a theoretical normal form like system of ordinary differential equations, as a test bed for the algorithm. Furthermore, we reliably find multi-hump solitary waves in applications such as the FitzHugh-Nagumo nerve-axon equations, a 5th order Hamiltonian KdV model for water waves and a semiconductor laser with optical injection.
Joint work with: Alan Champneys and Bernd Krauskopf

"Deep- and shallow-water slamming at zero deadrise angle"

Abstract
We discuss the normal impact of a flat-bottomed wedge on deep- and shallow-water. In the infinite depth case, we show how the impact is characterized by the similarity solutions to a local canonical problem at each corner. No general theory is available for this problem, but an argument based entirely on the location of the contact point may be used to explore the existence of different solutions depending on the size of the wedge angle. We discuss the problems of trying to reconcile the infinite depth flow with the shallow-water impact theory of Korobkin (1999). Finally, we make some preliminary remarks concerning the effect of air cushioning and forward motion on Korobkin's model.
Joint work with: SD Howison and JR Ockendon

"Array imaging, time reversal and communications in random media"

Abstract
I will present an exposition of the mathematical problems that arise in using arrays of transducers for imaging and communications in random media. The key to understanding their performance capabilities is the phenomenon of statistically stable super-resolution in time reversal, which I will explain carefully. Signals that are recorded, time reversed and re-emitted by the array into the medium tend to focus on their source location with much tighter resolution when there is multipathing because of random inhomogeneities. I will explain how this super-resolution enters into array imaging and communications when there is multipathing.

"Computation of steady three dimensional free surface flows"

Abstract
Steady three-dimensional free surface flows are considered. The fluid is assumed to be inviscid and incompressible and the flow is supposed to be irrotational. The free surface flow is generated by a three dimensional distribution of pressure (with support compact) moving at a constant velocity at the surface of a fluid of infinite depth. On the free surface the full nonlinear boundary conditions are applied: the kinematic and the dynamic conditions. The three dimensional problem is formulated as a nonlinear integro-differential equation.
The problem is solved numerically by using finite differences. After the discretization, the resulting algebraic equations are solved by a Newton method. Some computational results are presented.
This problem can be interpreted as an inverse problem for a three dimensional ship moving at a constant velocity at the surface of a fluid. The shape of the ship is not known in advance but is given at the end of the calculations by the shape of the free surface below the support of the pressure distribution.
Joint work with: J-M Vanden-Broeck

"Bethe-Sommerfeld conjecture and periodic differential operators"

Abstract
To be supplied

"Discrete and continuous structures in defective crystals"

Abstract
Motivated by the need to prescribe symmetry properties of energy densities related to defective crystals, I outline work on the geometrical structure of crystals with uniform dislocation density. In particular I discuss the fundamental connection with the theory of Lie groups, and discrete subgroups of Lie groups, as well as Cartan's theory of equivalence (of coframes).

"Boundary value problems and integrability"

Abstract
I intend to give a survey talk on the most recent developments and problems in this area, focussing mainly on integrable evolution equations on the half line but mentioning also problems with a time-dependent boundary.

"Axial impact on a cylindrical container almost full of liquid"

Abstract
Consider a long horizontal cylindrical tank, with closed ends, which is almost full of liquid. The tank receives a horizontal impact which sets the tank in motion (or the tank was in motion and is suddenly brought to rest). After impact the liquid begins to flow relative to the tank. A simple approximation to the flow development comes from using the solution for "filling flows" in Peregrine, D.H. & Kalliadasis, S. (1996) Filling flows, cliff erosion and cleaning flows, J. Fluid Mech. 310, pp.365-374. This predicts very high pressures on the top of the tank, and permits evaluation of some properties of the motion of the system.

"An infinite analogue of rings with stable rank one"

Abstract
Replacing invertibility with quasi-invertibility in Bass' first stable range condition, we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one. Several categorical constructions will be mentioned, including the behaviour of this property under extensions and pullbacks. This naturally leads to the analysis of the QB-property in multiplier rings of von Neumann regular rings (since those have stable rank 2 or infinity for a large class).
Joint work with: Pere Ara and Gert K Pedersen

"Coxeter Length"

Abstract
Coxeter groups are of central importance in many branches of mathematics. Historically, Coxeter showed that finite reflection groups are the only ones (up to isomorphism) with a presentation of generators which are involutions, where the only relations are those of the form (rs)^{m_{rs}}=1. Now all groups (finite or not) with such a presentation are named Coxeter groups. Coxeter groups arise in many areas, such as Lie algebras, groups of Lie type, the representation theory of such groups, and crystallography.
The notion of the length of an element in a Coxeter group W is, and has been, of fundamental importance in the study of Coxeter groups. The length of an element w of W is defined to be the shortest length of an expression for w as a product of generating elements. However there is an alternative definition of length in terms of the action of w on the root system \Phi of W. This formulation of the length function has recently been extended to assign a length, called the Coxeter length, to all subsets of W. A number of intriguing results have been obtained, some of which will be described, which suggest that the Coxeter length of subsets will be of value in future investigations into Coxeter groups.
Joint work with: Peter J Rowley

"Tumour angiogenesis and reinforced random walks"

Abstract
Solid tumours depend on their ability to induce angiogenesis, the formation of new blood vessels, for sustained growth and metastasis. Angiogenesis proceeds by the controlled migration and proliferation of endothelial cells (EC) from parent vessels.
Here we present a model for tumour angiogenesis in which cell motion is viewed as a reinforced random walk. The tumour produces angiogenic growth factors, which stimlate the EC of nearby capillaries to secrete proteolytic enzymes. These enzymes degrade the basement membrane, allowing the EC to migrate across the extra-cellular matrix towards the tumour, thus forming new capillaries. The EC also proliferate, increasing the rate of vessel formation. The effect of angiostatin, an angiogenic inhibitor which functions by deactivating the proteolytic enzymes, is also considered.
It is found that, in agreement with experimental observations, EC migration alone, in the absence of proliferation, is not sufficient to achieve vascularisation.
Joint work with: Brian Sleeman

"Asymptotic models for elastic wave propagation in an infinite elastic medium"

Abstract
The problem considered is that of a plane elastic wave propagating in an infinite elastic medium, containing an array of "coated" circular cylindrical inclusions. The "coating" layer surrounding each inclusion is assumed to be thin and soft. The inclusions are taken to be infinitely long and aligned along the z-axis. The objective is to study the effect of coating on the propagation of elastic waves through a finite stack. To analyse the stress and displacement fields within the "coating" layer we use the asymptotic expansion technique and derive the constitutive equations for the interface between the inclusion and the surrounding material.
We use a multipole method to find the reflection and transmission matrices of a single grating, and then employ a recurrence procedure to calculate the overall reflection and transmission properties of the whole stack. Disorder within the array is introduced by varying the cylinder radii and the spacing between them from layer to layer. Results are presented in the form of transmission diagrams, which are compared with those for a stack of perfectly bonded inclusions.
Joint work with: NV Movchan

"Patterns formed in a model system for rotating convection"

Abstract
We investigate the patterns generated by thermal convection in a rotating horizontal layer of fluid, heated uniformly from below. Naturally occurring patterns, such as cloud formations on the Earth, are examples of thermal convection on a large scale. Simple convection patterns in the fluid (for example, rolls) are susceptible to a variety of different instabilities, and these result in the formation of new, more complicated patterns. We show that, in addition to the usual instabilities, new instabilities can arise due to the presence of conserved quantities, for example where thermal convection is influenced by rotation of the fluid layers, or by magnetic fields.
In order to understand the dynamics of thermal convection we model the motion in the fluid layer with a set of coupled partial differential equations (pdes). Weakly non-linear analysis of these equations yields the patterns obtained at the onset of convection; further analysis reveals the various forms of the instabilities to which simple patterns may be susceptible. Although the weakly non-linear theory can predict the stability or otherwise of simple patterns, it cannot in general provide information about the ultimate state of the fluid layer following the onset of an instability. In order to describe the convection beyond the scope of weakly non-linear analysis, we simulate the model pdes and find the patterns that are formed subsequent to the non-linear development of the instability. From these simulations we find that an initial pattern of rolls, once unstable, can be replaced by a variety of interesting patterns.
Joint work with: SM Cox and PC Matthews

"The effect of entrained air on pressure pulses in a thin crack"

Abstract
When a breaking, or near breaking, water wave hits a sea wall or breakwater, massive pressures are exerted on that structure. In stormy conditions much air is often entrained in the water in the form of small bubbles, and compressibility effects become important. A simple model of an air-water mixture will be presented. Numerical solutions of the equations are used to show the behaviour of pressure pulses in a thin crack when the crack is filled with a bubbly mixture. In particular, the effects of resonance in the crack will be discussed.
Joint work with: DH Peregrine

"Internal degrees of freedom of an actuator disc model"

Abstract
Depending on the particular application, modeling of a flow in an axial compressor can be performed at different levels of complexity, from simplest relations for overall compressor characteristics to full unsteady three-dimensional calculations. This work is concerned with an intermediate level when a behavior of the compressor blade row is described with an actuator disk model (ADM). Actuator disk model can have internal degrees of freedom, governed by additional differential equations. Generally, being a system with distributed parameters, flow in the interblade passage has an infinite number of internal degrees of freedom. This work presents an attempt to estimate how many of them can be distinguished as the most important. The response of a blade row to a time-periodic excitations is modeled by actuator disk with internal degrees of freedom and by linearized Navier-Stokes equations. It is found that in the case of subsonic flow one internal degree of freedom can be considered as the most important, both for design and off-design regime. In case of transsonic flow in off-design regime two internal degrees of freedom are more important the the rest. However, for transsonic design regime no internal degrees of freedom could be distinguished as especially significant.

"The incompleteness of Euler zeta function"

Abstract
Euler zeta function formula, states that:
Series: 1+(1/2)^s+(1/3)^s+....= Product_primes[1/(1-(1/p)^s)]
Using an arithmetical approach, the author aim to raise concern over the incomplete results of the formula. New mathematical argument and results are presented, which show that: Expanding the Series into (Prime Products), would create another missing series. The new results would take the form:
1+(1/2)^s+(1/3)^s+...=Product_primes[1/(1-(1/p)^s)] + new series

"Similarity solutions to an averaged model for superconducting vortex motion"

Abstract
Under certain conditions the motion of superconducting vortices is primarily governed by an instability. We shall consider an averaged model, for this phenomenon, describing the motion of large numbers of such vortices. The model equations are parabolic, and, in one spatial dimension x, take the form
H_2_t = d/dx (|H_3 H_2_x - H_2 H_3_x|H_2_x),
H_3_t = d/dx (|H_3 H_2_x - H_2 H_3_x|H_3_x).
where H_2 and H_3 are components of the magnetic field in the y and z directions respectively. These equations have an extremely rich group of symmetries and a correspondingly large class of similarity solutions. In the talk we will examine steady solutions to the model, discuss their stability and then investigate similarity solutions which decay towards a steady solution of the model.

"A rigorous theory of experimental observations in fluid flows"

Abstract
Taking the 2d Navier-Stokes equations as a starting point, this talk describes a rigorous theory that guarantees that a finite number of point observations of the velocity serve to distinguish fully-developed' flows throughout the entire domain. These observations can be distributed in both space and time; taking equally spaced observations in time yields a version of Takens' time-delay embedding theorem, while taking all observations at one time and with a uniform spatial distribution we recover the 2d Kraichnan lengthscale. The theory is also valid in 3d provided that we assume that the 3d Navier-Stokes equations are well-posed.

"Dynamics of adaptive control; normal forms"

Abstract
One key issue in adaptive control designing is the possibility of destabilised limit systems. The situation becomes worst when a destabilised limit system attracts a large subset of all initial conditions. Such situation gives rise to bad behaviour. The main issue of this paper is to calculate the normal form of this bad behaviour in the scalar control system with one unknown parameter.

"Homoclinic solutions for mKdV and Lattice mKdV equations"

Abstract
The hyperbolic structure and homoclinic solutions of integrable mKdV-type equations are constructed through the Backlund trnasformation and Lax pair. Dynamical systems methods are used to establish the existence of homoclinic solution for the near-integrable modified KdV equation and the lattice mKdV (discrete in space and time) under periodic boundary conditions.

"Some geometrical properties of hydrodynamical models"

Abstract
We study complex structures arising in Hamiltonian models of nearly geostrophic flows in hydrodynamics. These approximate models seek to describe flows in which there is a dominant balance between the Coriolis, buoyancy and pressure-gradient forces on fluid particles. Such approximations to Newton's second law are commonly referred to as balanced models'. Semi-geostrophic theory, which has a special importance in dynamical meteorology, and its contact structure and other formal properties are first of all reviewed in the context of the shallow water equations. A number of these properties are remarkably simple and elegant, and mathematically important. We then ask which of those properties might generalize to more accurate hamiltonian models of balanced vortex motion. In many of these models an elliptic Monge-Ampere equation defines the relationship between a balanced velocity field and the materially conserved potential vorticity. Elliptic Monge-Ampere operators define an almost-complex structure and in this talk we show that a natural extension of the so-called geostrophic momentum transformation of semi-geostrophic theory defines a hyper-Kahler structure on phase space. The implications of this result will be discussed.

"There are two 2-twist-spun trefoils"

Abstract
We give a short proof inspired by Carter et al, arxiv:math.GT/9906115, that the 2-twist-spun trefoil is not isotopic to its orientation reverse. The proof uses a computer calculation of the third homology group of the three colour rack. We also give a new proof using the same calculation of the well-known fact that the left and right trefoil knots are not isotopic.

"Comparison of piecewise-linear, longwave and numerical eigenvalue solutions for inviscid stability problems"

Abstract
We are concerned with the inviscid linear stability analysis of prescribed boundary layer type fluid flows. For example, the behaviour of normal mode disturbances can be found by solving the Rayleigh equation and imposed boundary conditions as an eigenvalue problem for phase speed c in terms of real valued wave number \alpha, the imaginary part of c determining the growth or decay of the normal modes.
We have obtained a longwave series solution for c accurate to O(\alpha^2), which includes logarithmic terms. Although this gives an efficient way of computing dispersion relations, they appear to be accurate for only a very restricted range of \alpha. This range can be significantly extended by evaluating certain integrals numerically instead of forming a power series, but their evaluation is no quicker than the numerical solution to the Rayleigh equation.
We also consider the analytical solutions obtained for piecewise linear profiles. Using up to four linear segments to model smooth profiles, we demonstrate how spacing the linear segments appropriately gives some of the qualitative features of the numerical solutions.
Finally, we consider the complete initial value problem (which predicts normal mode disturbances in the far-field). Work is currently in progress to examine how disturbances which eventually decay in the linear theory can undergo transient algebraic growth which could also cause transition to turbulence. We are investigating the usefulness of the analytic theories in this context too.
Joint work with:JJ Healey

"Tracking with Prescribed Transient Behaviour"

Abstract
Universal tracking control is investigated in the context of a class S of dynamical systems modelled by functional differential equations. The class of systems encompasses a wide variety of nonlinear and infinite-dimensional systems and contains, as a prototype subclass, all finite- dimensional linear single-input single-output minimum-phase systems with positive high frequency gain. The control objective is to ensure that, for any admissible reference signal (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error between plant output and reference signal evolves within a prespecified performance envelope or funnel. A simple (non-adaptive and non-dynamic) error feedback control is described which achieves the objective whilst maintaining boundedness of the control and associated gain signals.

"Overall behaviour of two dimensional periodic composites"

Abstract
The overall properties of a binary elastic periodic fibre-reinforced composite are atudied here for a cell periodicity of square type. Exact formulae are obtained for the effective stiffnesses, which give closed-form expressions for composites with isotropic components including ones for empty and rigid fibres. The new formulae are simple and relatively easy to compute. Examples show the dependences of the sitffnesses as a function of fibre volume fraction up to the percolation limit. The equations easily lead to Hill's universal relations. The exact formulae explicitly display Avellaneda and Schwarts's microestructural parameters, which have a physical meaning, and provide formulae for them.

"Asymptotic approximation of eigenvalues of vector equations"

Abstract
A vectorial extension of the Keller-Rubinow method of computing asymptotic approximations of eigenvalues in bounded domains is presented. The method is applied to several problems, including that of a multimode step-profile cylindrical optical fibre, including the effects of polarisation. A comparison of the asymptotic results with the exact eigenvalues is made, and the agreement is shown to be good.

"Stable chaos in coupled lattices of simple maps"

Abstract
We consider a system consisting of coupled discrete maps on a lattice. The individual maps have very simple behaviour in time: the asymptotic behaviour of any initial condition is periodic. The dynamics of a finite coupled system also tends asymptotically to a periodic orbit; however, the length of the transient before the periodic orbit is reached grows exponentially fast with the size of the finite lattice. This implies that in the case of an infinite lattice, the expected periodic orbit is in fact never attained, and the transient dynamics, which is hard to characterise, is all that is observed.

"New opportunities for encouraging higher level mathematical learning by creative use of emerging computer aided assessment"

Abstract
We define the higher level mathematical skill of creating instances of mathematical objects satisfying certain properties. We provide examples of such questions and examine how often, in practice, students are asked to perform these tasks by an analysis of existing university level courses.
Secondly we explain how such questions may be assessed in practice without the imposition on staff of a large marking load. Included are a selection of working examples which have been implemented on a free computer aided assessment system (AIM) with a technical appendix giving source code for these questions and explaining how such questions may be authored.

"Eigenfunctions on arithmetic surfaces"

Abstract
The problem of understanding the behaviour of high frequency eigenstates of the Laplacian on a surface of negative curvature is central one in the theory of quantum chaos. In the special case that the surface is hyperbolic and arithmetic there has been substantial progress recently. We will review the general problem and describe some of the results and methods.

"Mathematics problem in HE: Recent Developments"

Abstract
The presence in HE of increasing numbers of mathematically ill-prepared students now affects a wide range of disciplines across the whole university sector. The report 'Measuring the Mathematics Problem' (Engineering Council 2000) presented evidence of a rapid decline in the basic mathematical skills of students entering degree courses in mathematics, physical sciences and engineering. This decline is now having serious effects on teaching and learning, on recruitment and retention and on standards. The current debacle with AS mathematics will exacerbate an already difficult situation and so the need for action to address the problem is now acute.
This paper will report on a number of recent developments including (i) the setting up of a joint DfES/UUK/HE working group to review the mathematics problem at the interface and make recommendations to facilitate a smoother transition from school to university and (ii) the initiative to set up, with the support of the LTSN's, a UK Mathematics Support Centre.

"Preconditioning for sedimentary basin simulations"

Abstract
The simulation of sedimentary basins aims at reconstructing its historical evolution in order to provide quantitative predictions about phenomena leading to hydrocarbon accumulations. The kernel of this simulation is the numerical solution of a complex system of time dependent, three-dimensional partial differential equations of mixed parabolic-hyperbolic type. A discretisation (Finite Volumes + Implicit Euler) and linearisation (Newton) of this system leads to very ill-conditioned, strongly non-symmetric and large systems of linear equations with three unknowns per mesh element, i.e. pressure, geostatic load, and hydrocarbon saturation.
The preconditioning which we will present for these systems consists in three stages. First of all the equations for pressure and saturation are locally decoupled on each element.The pressure part is then in a second stage preconditioned by AMG. The third step finally consists in "recoupling" the equations. In almost all our numerical tests with the basin simulator TEMIS3D (Institut Francais du Petrole) on real test problems from case studies we observed a considerable reduction of the CPU-time for the linear solver, up to a factor 4.3 with respect to ILU(0) (which is used at the moment in TEMIS3D). Furthermore, the preconditioner shows no dependency on physical parameters or discretisation parameters.

Joint work with: R. Masson and J. Wendebourg (IFP)

"The core chain of circles in the limit set of the Jorgensen's group"

Abstract
In the limit set of Jorgensen's group there is a fractal curve which can be obtain as limit of chains of circles. These circles appear in the limit sets of groups close to the Jorgensen's group (by close it means that it will be considered a limit of Fibonacci numbers tending to the Golden mean). Some pictures will also be shown as well as some conjectures.

"A depth-averaged model for rapid granular flows"

Abstract
Rapid granular flows have applications to many geophysical phenomena, such as snow avalanches and rockfalls. Kinetic theories are available which describe these flows as a continuum, based on an analogy between the motion of molecules in a gas and the motion of the particles in a granular flow. The governing equations arising from these theories are similar in structure to those which describe the conservation of mass and momentum in a compressible Newtonian fluid, but are supplemented by an expression for the conservation of fluctuation energy.
We have developed a shallow layer, depth-averaged model from these equations for exploring the evolution of a rapid granular flow down a slope. In the process of depth-averaging, approximations have to be made about the functional form of certain terms. For example, it is necessary to express the total dissipation of fluctuation energy (due to inelastic collisions) in the layer as a function of mass flux, mean velocity, and mean fluctuation energy. In order to achieve this, we have determined some general relationships from numerical investigation of depth profiles of steady, non-developing flows down an inclined plane.
The predictions of this model for the evolution of a rapid granular flow down a slope will be presented.
Joint work with: AJ Hogg

"Modelling of multi-channel interaction in a packed bed catalytic reformer"

Abstract
The aim of this work is to investigate the effect of a thin conducting wall on chemical processes which occur in two adjacent channels of a packed bed catalytic reformer. We assume that the chemical reactions are combustion in one channel and reforming in the other. To describe the heat flow in the channels we use one-dimensional energy and mole balance equations for both gas and solid phases. The temperature field within the conducting wall satisfies the two-dimensional heat equation. The heat transfer across the wall is analysed using the asymptotic expansions technique which provides the representation for the heat flux from the wall to the gas phase in both channels. The latter is used to describe the coupling between the two channels. The results of numerical calculations are compared with those for the uncoupled case when there is no heat transfer across the wall and each channel therefore can be treated separately.
Joint work with: NV Movchan, AB Movchan, ST Kolacz

"Dissipative Particle Dynamics"

Abstract
A system of stochastic differential equations used in the mesoscopic description of fluids, known as Dissipative Particle Dynamics, is introduced. We discuss numerical methods and erogidicity for this system.

"On the local connectivity of Julia sets of real polynomials"

Abstract
We show that a real polynomial with only real critical points has a locally connected Julia set, provided that the Julia set is connected and the polynomial has no periodic attractor. The proof is based on "complex bounds", constructed by Sebastian van Strien, Edson Vargas and the speaker.

"Fast black-box preconditioners for self-adjoint elliptic PDE problems"

Abstract
Discretization of self-adjoint PDEs using mixed approximation leads to symmetric indefinite linear system of equations. Preconditioners based on the positive definite symmetric Schur complement are common in the literature since they open up the possibility of using Preconditioned Conjugate Gradients as solver. Such methods have an inherent deficiency which is avoided in our alternative approach.
We outline a generic block preconditioning technique for such systems with the property that the eigenvalues of the preconditioned matrices are either contained in intervals that are bounded independently of the mesh size, or else have a small number of outlying eigenvalues that are inversely proportional to the mesh size.
The attractive feature of our technique is that the basis of the preconditioning is a readily available building block; namely, a fast Laplacian solve based on a single multigrid V-cycle. In the talk, we review the theoretical foundation for our approach. We also present numerical results showing it\'s effectiveness in the context of three model problems; anisotropic diffusion equations arising in ground- water flow, the biharmonic equation arising in plate modelling, and the Stokes equations that arise in incompressible flow modelling.

"Coupling between optical fibres and multi-dimensional tunneling"

Abstract
Optical fibres carry electrical signals in the form of pulses of light. When fibres are placed in parallel the signals have a tendancy to slosh back and forth between fibres, a phenomenon known as tunneling, or evanescent coupling.
We introduce two methods, adapted from quantum mechanics, for calculating the rate at which this tunneling occurs. The first establishes an operator equation for the system, which we treat as a matrix equation and calculate the tunneling rates by applying perturbation theory. The second relates the rates to an integration over a curve dividing the two fibres.
Our intention is to demonstrate the advantages of both methods, and illustrate possiblities for further study.

"Optimal computation of radionuclide chain transport"

Abstract
Optimal computation of radionuclide chain transpor Ronald Smith The optimally accurate compact implicit numerical scheme is derived for the concentrations of successive isotopes in a radionuclide chain as they transmute, are sorbed into the rock matrix, are transported and spread out with the groundwater flow. A key step is a non-local change of dependent variables, based on classical work of Bateman (1910). That change of dependent variables can be performed numerically to the requisite high accuracy using the corresponding steady numerical scheme. An exact point release solution is used to demonstrate the accuracy of the numerical scheme with large time steps and low spatial resolution.

"Conspiring with the gods: the mathematics of William Brouncker (c. 1620-1684)"

Abstract
William Brouncker, first President of the Royal Society, was a skilled and intuitive mathematician. He was largely written out of history by Euler, and his achievements are now little recognised, but he worked on some original and unusual mathematics whose full significance emerged only many years after he died. In this lecture I shall explore Brouncker's continued fraction for 4/pi and his work on 'Pell's equation', and contrast him with his closest mathematical colleague, Oxford Savilian Professor John Wallis.

"Moments of characteristic polynomials enumerate lexicographic arrays"

Abstract
A combinatorial interpretation is provided for the moments of characteristic polynomials of random unitary matrices. This leads to a rather unexpected concequence of the Keating and Snaith conjecture: the moments of the zeta-function turn out to be connected with some increasing subsequence problems (such as the last passage percolation problem).

"Infinities of stable periodic orbits near robust cycling"

Abstract
We consider the dynamical behaviour of systems with robust heteroclinic cycles to saddles that may be periodic or chaotic. We differentiate attracting cycles into types that we call phase-resetting and free-running depending on whether the cycle always approaches a given saddle along one or many trajectories. At loss of stability of attracting cycling, we show in a phase-resetting example the existence of an infinite family of periodic orbits that accumulate on the cycling, whereas for the free-running example loss of stability of the cycling gives rise to a single quasiperiodic or chaotic attractor.
Joint work with: Peter Ashwin and Alastair Rucklidge

"On the equidistribution of non-closed horocycles"

Abstract
Let M be a hyperbolic surface of finite area (but non-compact). It was proved by Dani and Smillie, using ergodic theoretic methods, that any given non-closed horocycle on M is in fact asymptotically equidistributed with respect to the area measure on M. In my talk, I will present precise rate-of-convergence results concerning this asymptotic equidistribution, obtained using the spectral theory of the Laplace operator on M.

"The impact and spreading of drops on surfaces"

Abstract
The impact and spreading of drops on surfaces occurs in many industrial processes and natural phenomena, and has been extensively studied over many years. However, there are well known problems with the classical approach to the modelling of flows involving moving contact lines [Dussan and Davis (1974), J. Fluid Mech., 65 71]. Results of recent careful experiments [Blake et al. (1999), Physics of Fluids, 11, 1995] mean that for a spreading drop the evolution of the contact angle is dependent on, for example, the volume and initial speed of the drop, and not just on the flow field local to the contact line. The implication is that all moving contact line models appearing in the literature apart from one are flawed in their formulation. The single remaining model has its foundations in non-equilibrium thermodynamics [Shikhmurzaev (1993), Int. J. Multiphase Flow, 19, 589]. Our work is concerned with the application of this interfacial theory to the drop impact and spreading problem, with the aim of giving a more accurate and widely applicable description than all other models can provide. A combination of asymptotic and numerical techniques applied to modeling of the initial stages of drop impact will be presented.
Joint work with: SP Decent, AC King and YD Shikhmurzaev

"Adaptive finite element approximation of hyperboli"

Abstract
We consider the a posteriori error analysis of hp-version finite element approximations of hyperbolic partial differential equations. An hp-adaptive algorithm is then developed which relies on a Sobolev-index estimation strategy. The theoretical results are illustrated by numerical experiments.
Joint work with: Paul Houston

"Quantum chaos on graphs"

Abstract
Quantum graphs have recently been introduced as model systems to study quantum problems with a chaotic classical limit. I will review recent developments on quantum graphs and generalise these concepts to quantum propagation on directed graphs for which a unistochastic transition matrix exists.

"Noncommutative knot theory"

Abstract
We explain how classical knot invariants, like the Alexander polynomial and twisted signatures, can be generalized into a noncommutative setting. The idea is to use arbitrary solvable coverings of the knot complement instead of just the infinite cyclic cover. This leads to invariants of knots up to isotopy (higher Alexander polynomials) as well as knot concordance (higher signatures). It turns out that the n-th level signature vanishes if the knot bounds a grope of height n in the 4-ball. Pushing such a grope into 3-space leads to a new geometric notion of "grope cobordism" which turns out to be closely related to the Vassiliev invariants of knots.
This program has two conceptual analogies to Alain Connes' noncommutative geometry: Von Neumann algebras turn out to be a central tool for detecting the higher signatures, and the combinatorics of gropes in 3-space is best explained in terms of certain Feynman diagrams, just like in the Connes-Kreimer approach to renormalization. Hence the choice of our title. The major difference to Connes' theory is that we do not invent new objects, but just study classical knots with noncommutative tools.

"Understanding historical and predicting future climate change"

Abstract
> >Observations of 20th century temperature show a early century warming, >followed by constant temperatures then till the mid-1970s. Waring then >resumed in the mid-70s. 1998 was the warmest year on record and >paleo-data suggests that the late 20th century was unusually warm >relative to the last millennium. > >Coupled Ocean Atmosphere General Circulation models attempt to >simulate climate from basic physical principals but still contain many >uncertainties. Thus the equilibrium warming for doubled concentrations >of pre-industrial CO2 is believed to lie in the range 1.5 to 4.5 >Kelvin. Using the observed temperature record and a variety of >simulations we estimate the contribution, and its uncertainty, to 20th >century temperature change for anthropogenic and natural forcings. >>From these uncertainties and best estimates we then estimate the >likely range of future temperature change under a variety of different >forcing scenarios. > To be supplied

"Modelling rate independent evolution problems"

Abstract
Plastic behaviour of crystals is typically modelled as a rate independent process, i.e. the solution depends only on the load increments, but not on the rate at which the load is increased. A mathematical framework for the treatment of the Cauchy problem in the case of finite deformations will be presented. In the case of a single slip plane elasto-plastic microstructures occur. It will be shown, how an effective theory, which takes only averages over the spatial oscillations into account, can be constructed.

"Edge waves and crack diffraction in thin plates"

Abstract
Thin plate theory gives rise to many interesting (and challenging) mathematical problems, many of which are of considerable importance in the field of non-destructive evaluation. One such problem is that of flexural wave diffraction by a crack in a thin plate. We will begin with a brief summary of the existing work in this area, and of the problems which currently remain unsolved. The level of difficulty of a problem of this type is dependent upon the complexity of the material (isotropic/orthotropic), the presence (or lack of) symmetries in the system, and the model used (Kirchoff/Mindlin).
In the cases which have already been solved, guided waves were found to propagate along the free edges of the crack. A discussion of such phenomena, commonly known as 'edge waves' will form the main body of this talk. Although edge waves usually arise as one component of the solution of a broader problem, the possibility of their existence in a given system can be investigated directly. The procedure for such an investigation will be briefly described, and a summary of results concerning the existence of edge waves propagating along the free edges of a crack in an orthotropic Kirchoff plate will be given.
We will conclude with an outline of our current work and plans for future research in this area.
Joint work with: I David Abrahams

"Title to be supplied"

Abstract
To be supplied

"The topology of surface bundles"

Abstract
Given a family of (complex) vector spaces one can understand its topology by calculating its characteristic classes, the Chern classes. Some twenty years ago D. Mumford started the systematic study of characteristic classes for families of surfaces. We give an overview of recent developments.

"Backward errors and condition numbers for singly and doubly structured eigenvalue problems"

Abstract
Condition numbers characterize the sensitivity of solutions to problems and backward errors can be used to assess the stability and quality of numerical algorithms. Recently, efforts have been concentrated on deriving new structure-preserving algorithms for the solution of structured eigenvalue problems, for both the dense case and for the large and sparse case. It is therefore of interest to develop backward errors and condition numbers that fully respect the inherent structure of these problems.
We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the structured backward error is within a small factor of the unstructured backward error. In this talk, we collect, unify and extend existing work on this subject.

"Structural stability in one-dimensional dynamics"

Abstract
The C^k structural stability conjecture for unimodal maps is proved. We discuss some aspects of this result.

"A characterization of Morita equivalence pairs"

Abstract
We characterize the pairs of operator spaces which occur as pairs of Morita equivalence bimodules between non-selfadjoint operator algebras in terms of the mutual relation between the spaces. We obtain a characterization of the operator spaces which are completely isometrically isomorphic to imprimitivity bimodules between some strongly Morita equivalent (in the sense of Rieffel) C*-algebras. As corollaries, we give representation results for such operator spaces.

Abstract
The Pade method has been widely used in applied mathematics to accelerate the convergence of power series. Often, the method works very well. Sometimes, the results are disappointing. By using an analogy with continued fractions, we seek to understand the convergence properties of the method. In particular, we introduce and study the dynamical properties of a map that is the Pade counterpart of the well- known Gauss map.
Joint work with: Philip G Drazin

"Stability of Infinite-Dimensional Sampled-Data Systems"

Abstract
Suppose that a static state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space is a Hilbert space. This is true for finite dimensional spaces. In the infinite dimensional case, if the feedback is not compact, then it is easy to find counter-examples. Therefore we restrict attention to compact feedback. We show that the result is true if the input operator is bounded, or if the system is analytic. We sue semigroup techniques and complex analysis for operator-valued functions.

"Wave packet pseudomodes of matrices and differential operators"

Abstract
Nonhermitian differential operators and "twisted Toeplitz matrices" with varying coefficients have unexpected spectral and pseudo-spectral properties. For complex pseudo-eigenvalues z far from any true eigenvalues, they may have pseudo-eigenmodes in the form of exponentially decaying wave packets that are indistinguishable physically from true eigenmodes. This phenomenon can be analysed by making use of the "symbol" that describes the dual dependence of such operators on both space and wave number. This talk will present several examples and applications, outlining links to WKB analysis, Anderson and Hatano-Nelson localisation, pseudodifferential operators, and Lewy/Hormander nonexistence of solutions to certain PDE. It represents joint work with M. Embree and T. Wright, and we have benefited crucially from the ideas of E. B. Davies and M. Zworski.

"Hypersonic flow past bodies in dispersive MHD"

Abstract
We consider the problem of a steady two-dimensional hypersonic flow past a thin wedge in a dispersive dissipationless medium. Our main example is motion of a cold plasma in a transversal magnetic field. The characteristic feature of such a flow is formation of a finite-amplitude undular bore (collisionless shock). We analyse the problem with the aid of the Whitham method of slow modulations and derive dispersive shock conditions replacing traditional shock adiabat of the dissipative hydrodynamics.
Joint work with: G El and V Khodorovskii

"Efficient and reliable iterative methods for linear systems"

Abstract
Krylov subspace methods offer extremely good possibilities for the solution of large sparse linear systems of equations. An overview of the currently most competitive algorithms will be presented: CG, GMRES, GMRESR, Bi-CGSTAB, and Bi-CGSTAB(ell).
For general systems, some of the popular methods often show an irregular type of convergence behavior, which may lead to a delay in the convergence but also to a dercreased accuracy of the method. We will discuss a simple strategy for the reliable updating in iterative methods so that small updated residuals really reflect accurate approximations. We will also discuss a technique that helps to overcome a local stagnation phase in the Bi-CGSTAB methods.
Krylov subspace methods are usually applied with preconditioning. We will briefly discuss some modern developments in preconditioning. In particular parallel preconditioners will be highlighted: a domain decomposition approach and a reordering technique with sparsified Schur complements.

"Singularities of projections"

Abstract
We consider the topology of the singular sets of the projection of a smoothly embedded closed manifold. In particular, the question of which 'alterations' of these sets are realisable by a C^0 small isotopy of the original embedding.

"Conjugate Natural Convection in Porous Media "

Abstract
The analysis of convective heat transfer from horizontal and vertical bodies embedded in porous media has received considerable attention in recent years. Various technological applications, particularly in geothermal systems and heat transfer analysis of extended surfaces, motivated these investigations.
The general features of conjugate convection in a porous medium are addressed here. The analysis is concerned with natural convection problems, specifically natural convection above and below a horizontal flat plate and from a plate or a cylindrical vertical fin. In order to model the incompressible flow in the porous medium, the two-dimensional steady state equations of the Darcy flow model are employed. Numerical solutions of the full governing equations and solutions based on the boundary-layer approximation are considered for the fluid-porous medium region, whilst in the finite plate or fin the two-dimensional heat conduction equation is employed, so that both the longitudinal and the transverse conduction effects are included. The equations in the fluid-porous medium region are coupled to the heat conduction equation in the finite plate or fin, all equations are approximated by finite differences, and the coupled equations are solved by an iterative procedure, in which the equations for the two media are solved successively. The effect of the non-dimensional parameters appearing in the problems are discussed.

"New multi-critical matrix models"

Abstract
We study a class of Hermitian matrix models with an action containing nonpolynomial terms of the Freud-weight type. By tuning the parameters in the action to criticality we obtain that the eigenvalue density vanishes as an arbitrary real power at the origin, thus defining a new class of multicritical matrix models. We study also the corresponding class of universality and the critical behaviour at the edge of the spectrum.
Joint work with: G Akemann

"Deformed root systems and quantum integrability"

Abstract
Calogero-Moser system which describes the interacting particles on the line has integrable generalisations related to any root system. It turned out that in the quantum case there are also non-symmetric integrable generalisations related to certain deformations of the root systems. The nature of these deformations is still mysterious but they have already appeared in different problems (WDVV equation, symmetric superspaces). I will discuss all this in the talk.

"Nonlocal eigenvalue problems"

Abstract
Cell-growth population models give rise to a new class of problems: nonlocal singular eigenvalue problems. The spectral properties of such problems are strikingly different from the local case pointing the need for new theory about spectra, orthogonality etc. A class of problems analogous to the classical second order singular case is analysed as a means of providing concrete results. In this case the spectrum is discrete with the first eigenvalue being related directly to the time constant of the temporal growth of the cell-growth cohort. This work has application in the growth of muscle and tumour cells.
Reference B. van Brunt, G.C. Wake, and H.K. Kim, "On a singular Sturm-Liouville problem involving an advanced functional differential equation", Euro Jnl of Applied Mathematics, (2001), Vol 12, pp.625-644.

"Tumour induced angiogenesis as a reinforced random walk"

Abstract
An avascular tumour has no active transport mechanism for nutrients and waste products, and grows to a limiting size. In contrast, a vascular tumour has an effective transport mechanism for nutrients and waste products and is able to sustain its growth. The development of a network of capillary blood vessels (angiogenesis) provides the bridge between these states. Here, we examine the migration of endothelial cells from an existing capillary towards a solid tumour, in response to chemical gradients, thereby initiating the growth of new capillaries. The migration of individual cells is modelled using the theory of reinforced random walks. We also begin to consider possible mechanisms for modelling cell proliferation (growth), a vital part of angiogenesis.

"The abdominal aortic aneurysm - a new mathematical model"

Abstract
An aneurysm is a localized dilation of the arterial wall. Growth of the aneurysm is associated with a weakening of the wall and the possibility of rupture. Stucturally, the two key components of the arterial wall are elastin and collagen. A model is proposed which captures two essential features of the development of abdominal aortic aneurysms, namely the degradation of the elastin and the remodelling of the collagen fibres. To gain an understanding of the remodelling, a uniaxial model was initially considered with the elastin and collagen arranged in parallel. Under the assumption that the collagen remodels in order to maintain an equilibrium level of strain, two key parameters, which relate to the density and the onset of recuitment of the collagen fibres, are found to be important. The uniaxial model yielded physiologically consistent results and its features were implemented into a 3D axisymmetric membrane model of the artery which incorporates the realistic microstructure of the arterial wall. Using a physiologically determined set of parameters to model the abdominal aorta and realistic remodelling rates for its constituents, the predicted dilations of the aneurysm are consistent with those observed in vivo.
The model has been generalised to consider non-axisymmetric deformations so that aneurysm development with spinal contact can be considered. It can also be usefully employed to consider generalised behaviour of aneurysms under different environmental conditions. For example, it predicts the increase in the rate of dilation in hypertensive conditions and the retraction of the aneurysm that follows a stent bypass operation.
Joint work with: NA Hill

"The structure of longitudinal vortices within the atmosphere"

Abstract
Curved, heated mixing layers support instabilities in the form of longitudinal counter-rotating vortices, similar to the Gortler vortices known to exist within curved boundary layers. These vortices have important consequences within many physical situations, for example they can be observed in the base of clouds generated by mountain lee-waves. The linear form of the governing equations are parabolic in nature and may be solved using a Crank-Nicholson marching scheme in the downstream coordinate. Here we present neutral curves generated using an artificial initial condition; we also examine the receptivity problem of a disturbance within the free stream impinging upon the mixing layer and hence triggering the vortex instability.
Joint work with: SR Otto and SO MacKerrell

"The Triple Pursuit"

Abstract
3 Particles A, B, C, move each at the same spped, A towards B, B towards C, C towards A. Given their initial, non-collinear positions, where do they meet? An assertion in 'The Penguin Dictionary of Curious and Interesting Geometry' is disproved.

"Particle transport over the alveolar surface"

Abstract
The fate of inhaled particles deposited in the liquid lining of lung airways is important in the development of certain diseases and in inhalation therapies. Experiments have shown how surface tension can cause a particle to be pulled downwards into the fluid lining, indenting underlying epithelial cells. With this motivation we consider the dynamics of a particle partially immersed in a laterally bounded liquid pool, as may arise in the corner of a polyhedral alveolus. We consider a two-dimensional geometry, with the particle modelled as a cylinder, and investigate the normal and lateral motion of the particle within the liquid pool. The speed of the motion is controlled by a balance between the net capillary force acting on the particle and the viscous drag force due to the flow in the narrow gap between the particle and the wedge wall, where lubrication theory can be applied. We find that a particle may either move to the corner of the wedge, to the nearest contact line, to a stable equilibrium position in between or become completely submerged. The behaviour of the particle is difficult to predict intuitively and depends on the fluid volume, particle size, wedge angle and initial position. This model provides a basis for the inclusion of effects such as deformation of underlying epithelial cells and stretching of the alveoli during breathing.
Joint work with: Oliver Jensen and Sarah Waters

"Optimal computation of radionuclide chain transportation"

Abstract
In this talk I will describe how a general class of multi-order paprmeter models may used to to provide diffuse interface models of a range of alloys in which there are several phases, eg, a eutectic alloy which involves two solid phases and a liquid phase. Numerical computations using these models exhibit a range of realistic complex solidification behaviour observed in real systems.

"Cube complexes and the conjugacy problem for graph braid groups"

Abstract
I will talk about "graph braid groups" which made their appearance recently in the thesis of Abrams. These are the fundamental groups of configuration spaces of finite sets of points in a graph. In his thesis Abrams showed that these groups are the fundamental groups of non-positively curved cubed complexes. I will try to explain in my talk how each of these cubed complexes may be mapped locally isometrically (and hence \pi_1-injectively) into a non-positively curved cubed complex whose fundamental group is a right angled Artin group, and how this leads to a solution of the conjugacy problem of the graph braid group in polynomial time. These considerations suggest, in fact, a very explicit polynomial time solution to the conjugacy problem for any group which is the fundamental group of a non-positively curved cubed complex.
Joint work with: John Crisp

"The overtopping of swash;"

Abstract
In coastal protection, whether against flooding or to prevent wave disturbance in harbours, it is often valuable to be able to estimate the flow of water over the top of a structure, or natural feature. When the mean level is below the crest of such a feature any overtopping is due to the run-up from incident waves. The solution of Peregrine and Williams (2001) will be discussed, with extensions being presented.
Joint work with: DH Peregrine

"Magnetic instability in a sheared azimuthal flow"

Abstract
We study the magneto-rotational instability of an incompressible flow in the Taylor--Couette geometry. The fluid rotates with angular velocity \Omega(r)=a+b/r^2 where r is the radius and a and b are constants. We find that an applied magnetic field destabilises the flow, in agreement with the results of Ruediger & Zhang (2001), such that it can violate the Rayleigh stability criterion. The parameter space is extended to large magnetic Prandtl numbers, motivated by the possibility of large values in the central region of galaxies (Kulsrud & Anderson 1992). In this regime we find that increasing the magnetic Prandtl number greatly enhances the instability; the stability boundary drops well below the Rayleigh line and tends toward the line of solid body rotation. Study at very small magnetic Prandtl numbers is motivated by current MHD dynamo experiments performed using liquid sodium and gallium. Our finding in this regime confirms Ruediger & Zhang's conjecture that the linear magneto-rotational instability and the nonlinear hydrodynamical instability (Richard & Zahn 1998) take place at Reynolds numbers of the same order of magnitude.
Joint work with: Carlo F Barenghi

"Inviscid flows through curved ducts "

Abstract
The interaction of compressibility and geometry of inviscid flows through curved ducts has received little attention, in contrast to centrifugally-induced viscous effects which are often of dominant concern to industry and the biomedical sciences. However, some industrial situations justify the study of inviscid flows through curved ducts and of those, some require the inclusion of compressibility in the model. Solutions for the steady 3D motion within and beyond a weak bend in a slender duct of rectangular cross-section are determined computationally and analytically for a perfect, inviscid, compressible fluid.

"Non-isothermal rivulet flow"

Abstract
The gravity-driven draining of a rivulet of viscous fluid down an inclined substrate occurs in a number of practical situations ranging from many industrial devices and coating processes to a variety of geophysical flows. In many of these situations heating or cooling effects are significant and so it is of considerable interest to investigate rivulet flows in which non-isothermal effects play a role.
In the this talk we use the lubrication approximation to investigate one such problem, namely the unsteady gravity-driven draining of a thin rivulet of Newtonian fluid with temperature-dependent viscosity down a substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere. Various models for the dependence of viscosity on temperature are considered.
In the first part of the talk we derive the general nonlinear evolution equation for a thin film of fluid with an arbitrary dependence of viscosity on temperature. We then use this equation to show that at leading order in the limit of small Biot number the rivulet is isothermal, as expected, but that at leading order in the limit of large Biot number (in which the rivulet is not isothermal) the governing equation can, rather unexpectedly, always be reduced to that in the isothermal case with a suitable rescaling. Solutions are presented in three situations in which the corresponding isothermal problem has previously been solved analytically. A detailed account of this work has recently been accepted for publication in J. Eng. Maths.
In the second part of the talk we investigate steady locally unidirectional draining down a slowly varying substrate in which the Biot number is small, but in which the variation of viscosity with temperature is sufficiently strong that thermoviscosity effects appear at leading order in the limit of small Biot number. A detailed account of this work has recently been submitted for publication.
In all the cases investigated we find that the effect of cooling the atmosphere is always to widen and deepen the rivulet, while the effect of heating the atmosphere is always to narrow and shallow it.
Joint work with: Brian Duffy

"Are graphs a model for Quantum Chaos?"

Abstract
Quantum graphs are models of molecules and networks of quantum wires. Graphs have also recently been suggested as models for quantum chaotic systems. Generic graphs display spectral behaviour typical of such systems: their energy levels appear to be correlated like the eigenvalues of large random matrices. However, certain graphs (for example the so-called star graphs) do not exhibit this behaviour.
We give an introductory review of the connections between quantum graphs and quantum chaos. We present recent work on the value distribution of the spectral determinant for star graphs and show that the although the resulting distribution is not the Gaussian distribution characteristic of random matrices, we get the Cauchy distribution, which falls into the same class of stable distributions. We also give an expression for the value distribution of eigenfunctions of the Schroedinger operator on star graphs. We conjecture that these results also hold for a certain class of billiards known as Seba billiards.
Joint work with: JP Keating

"Walking with insects"

Abstract
Recent mathematical models of insects "Central Pattern Generators", which coordinate leg movement during locomotion, have concentrated on a rather simplistic 'six legs implies six oscillators' approach. These models however fail to simulate many observed locomotory patterns among hexapods. If instead we look at how insects have evolved to the current six-legged version we are lead to a more complex model of typically 22 coupled oscillators which, after some plausible symmetry constraints are placed on the structure, can be analysed through equivariant hopf-bifurcations (Z_11 X Z_2 invariant coupling for example).

"Visualization of Kleinian groups"

Abstract
We will show a collection of new images of limit sets of Kleinian groups representing surfaces of finite type mostly chosen from Maskit-Kra constructions of Teichmueller spaces of 5-times and 6-times punctured spheres, twice and thrice-punctured tori, and once-punctured genus two surfaces.
We will also show a new collection of colored images of automorphic functions on Kleinian groups, computed mostly at NCSA. The automorphic functions are computed as ratios of Poincare series evaluated at a grid of finely space points in the complex plane. These pictures illustrate the action of the group on the plane.

"Stability of Poisson equilibria"

Abstract
Poisson equilibria are equilibria of differential equations conserving a Poisson bracket. Conservation of energy and flow-invariance of the symplectic leaves often helps to establish Lyapunov stability of such equilibria. The situation is particularly easy if the space of symplectic leaves is Hausdorff which however is often not satisfied in applications. We will present new results which do not require such an assumption.

"Realization of Mu-synthesis: Spectral 2 by 2 case"

Abstract
In this talk,the realizations of spectral interpolation problem of 2 by 2 case are presented This is relevantnt to the structured uncertainty in H-infinity control.

"Theory of Weak Turbulence and its numeric and experimental confirmation"

Abstract
Weak Turbulence is realized in the systems of nonlinear dispersive waves with random phases.The most natural examples of such systems are ocean and atmoshere.In spite of the fact that the basic ideas of the Weak Turbulent Theory were formulated more than thirty year ago,time for its application to reality is coming only now.At the moment we have strong evidence in support of a radical view-point: the weak turbulent theory explains an essential part, if not majority, of experimental facts on nonlinear waves,accumulated in Physical Oceanography and in Physics of Atmoshpere during last decades. The main achevment of the Weak Tubulence is explanation of observed spectra of wind-driven gravity waves in deep sea. It explains very well also spectra of capilliary ripples.It explains satisfactory the spectra of internal waves in the stratified ocean and the spectra of inertia- gravity waves in atmosphere. In my talk I formulate basic results of the weak turbulent theory and give a review of comparision of this theory with the results of field, laboratory and numerical experiments.

"Cluster algebras: foundations and finite type classification"

Abstract
Cluster algebras are commutative algebras of a special kind introduced by Sergey Fomin and myself in an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups. The main structural result of the theory so far is a classification of cluster algebras of finite type which turns out to be yet another instance of the Cartan-Killing classification. We shall discuss main structural properties of cluster algebras and the main ingredients of the proof of the classification theorem.