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Time-periodic shear layers occur naturally in a wide range of applications from engineering to physiology. Transition to turbulence in such flows is of practical interest and there have been several papers dealing with the stability of flows composed of a steady component plus an oscillatory part with zero mean. In such flows a possible instability mechanism is associated with the mean component so that the stability of the flow can be examined using some sort of perturbation-type analysis. This strategy fails when the mean part of the flow is small compared with the oscillatory component which, of course, includes the case when the mean part is precisely zero.

This difficulty with analytical studies has meant that the stability of purely oscillatory flows has relied on various numerical methods. Until very recently such techniques have only ever predicted that the flow is stable, even though experiments suggest that they do become unstable at high enough speeds. In this talk I shall expand on this discrepancy with emphasis on the particular case of the so-called flat Stokes layer. This flow, which is generated in a deep layer of incompressible fluid lying above a flat plate which is oscillated in its own plane, represents one of the few exact solutions of the Navier-Stokes equations. We show theoretically that the flow does become unstable to waves which propagate relative to the basic motion although the theory predicts that this occurs much later than has been found in experiments. Reasons for this discrepancy are examined by reference to calculations for oscillatory flows in pipes and channels. Finally, we propose some new experiments that might reduce this disagreement between the theoretical predictions of instability and practical realisations of breakdown in oscillatory flows.

Collapsing bridges and exploding rockets have been associated with vibrations in resonance with natural frequencies. As well, the stability of the solar system and the existence of solutions of Schrödinger\'s equation and the wave equation are problematic in the presence of resonances. Such resonances can be avoided, or at least mitigated, by using ideas from Diophantine approximation, a branch of number theory. Applications of Diophantine approximation to these problems will be given and will include a connection with LISA (Laser Interferometer Space Antenna), a space-based gravity wave detector under construction.

The pricing of electricity involves several interesting features. Apart from daily, weekly and seasonal fluctuations, power prices often exhibit large spikes. To some extent this is because electricity cannot be stored. We propose a model for power prices in the Alberta market. This involves a diffusion process modified by a factor related to a Markov chain which describes the number of large generators on line. The model is calibrated and future contracts priced.

Honey dripping from an upturned spoon is an everyday example of a flow that extends and breaks up into drops. Such flows have been of interest for over 300 years, attracting the attention of Plateau and Rayleigh among others. Theoretical understanding has, however, lagged behind experimental investigation, with major progress being made only in the last two decades, driven by industrial applications including ink-jet printing, spinning of polymer and glass fibres, blow-moulding of containers, light bulbs and glass tubing, and rheological measurement by fibre extension. Albeit, the exact details of the final stages of breakup are yet to be fully resolved. An aspect that is relatively unexplored is the evolution of drop and filament from some initial configuration, and the influence of initial conditions on the final breakup. We will consider a drop of very viscous fluid hanging beneath a solid boundary, similar to honey dripping from an upturned spoon, using methods that allow examination of development and behaviour from early time, when a drop and filament begin to form, out to large times when the bulk of the fluid forms a drop at the bottom of a long thin filament which connects it with the upper boundary. The roles of gravity, inertia and surface tension will be examined.

The breakup of a mass of fluid into drops is a ubiquitous phenomenon in daily life, the natural environment and technology, with common examples including a dripping tap, ocean spray and ink-jet printing. It is a feature of many generic industrial processes such as spraying, emulsification, aeration, mixing and atomisation, and is an undesirable feature in coating and fibre spinning. Surface-tension driven pinch-off and the subsequent recoil are examples of finite-time singularities in which the interfacial curvature becomes infinite at the point of disconnection. As a result, the flow near the point of disconnection becomes self-similar and independent of initial and far-field conditions. Similarity solutions will be presented for the cases of inviscid and very viscous flow, along with comparison to experiments. In each case, a boundary-integral representation can be used both to examine the time-dependent behaviour and as the basis of a modified Newton scheme for direct solution of the similarity equations.

The standard diffusion equation is obtained by Einstein's method and its generalisation, Fokker-Plank-Kolmogorov-Feller theory. The time between jumps in Einstein derivation is constant.

We discuss random walks with residence time distribution, which occurs for flows of solutes and suspensions/colloids in porous media, CO2 sequestration in coal mines, several processes in chemical, petroleum and environmental engineering. The rigorous application of the Einstein's method results in new equation, containing the time and the mixed dispersion terms expressing the dispersion of the particle time steps.

Usually, adding the second time derivative results in additional initial data. For the equation derived, the condition of limited solution when time tends to infinity provides with uniqueness of the Caushy problem solution.

The solution of the pulse injection problem describing a common tracer injection experiment is studied in greater detail. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic (advection-dispersion) equation. This is in agreement with the experimental observations and predictions of the direct simulation.

In this talk, we will survey a number of different free boundary problems involving slow viscous (Stokes) flows in which surface tension is active on the free boundary. Both steady and unsteady flows will be considered. Motivating applications range from industrial processes such as viscous sintering (where end-products are formed as a result of the surface-tension-driven densification of a compact of smaller particles that are heated in order that they coalesce) to biological phenomena such as understanding how organisms swim (i.e. propel themselves) at low Reynolds numbers. Common to our approach to all these problems will be an analytical/theoretical treatment of model problems via complex variable methods -- techniques well-known at infinite Reynolds numbers but used much less often in the Stokes regime. These model problems can give helpful insights into the behaviour of the true physical systems.

A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated to this equation is obtained, also by reduction.

Suspension/colloid transport in porous media is a basic phenomenon in environmental, petroleum and chemical engineering. Suspension of particles moves through porous media and particles are captured by straining or attraction. We revise the classical equations for particle mass balance and particle capture kinetics and show its non-realistic behaviour in cases of large dispersion and of flow-free filtration. In order to resolve the paradoxes, the pore-scale model is derived. The model can be transformed to Boltzmann equation with particle distribution over pores. Introduction of sink-source terms into Boltzmann equation results in much more simple calculations if compared with the traditional Chapman-Enskog averaging procedure. Technique of projecting operators in Hilbert space of Fourier images is used. The projection subspace is constructed in a way to avoid dependency of averaged equations on sink-source terms. The averaging results in explicit expressions for particle flux and capture rate. The particle flux expression describes the effect of advective particle velocity decrease if compared with the carrier water velocity due to preferential capture of "slow" particles in small pores. The capture rate kinetics describes capture from either advective or diffusive fluxes. The equations derived exhibit positive advection velocity for any dispersion and particle capture in immobile fluid that resolves the above-mentioned paradox. Finally, we discuss validation of the model for propagation of contaminants in aquifers, for filtration, for potable water production by artesian wells, for formation damage in oilfields.

Noncommutative geometry (in the sense of Alain Connes) involves replacing a conventional space by a "space" in which the algebra of functions is noncommutative. The simplest truly non-trivial noncommutative manifold is the noncommutative 2-torus, whose algebra of functions is also called the irrational rotation algebra. I will discuss a number of recent results on geometric analysis on the noncommutative torus, including the study of nonlinear noncommutative elliptic PDEs (such as the noncommutative harmonic map equation) and noncommutative complex analysis (with noncommutative elliptic functions).

The study of wall-bounded turbulent flows has received great attention over the past few years as a result of high Reynolds number experiments conducted in new high Reynolds number facilities such as the Princeton "superpipe", the NDF facility in Chicago and the HRNBLWT at the University of Melbourne. These experiments have brought into question the fundamental scaling laws of the turbulence and mean flow quantities as well as revealed high Reynolds number phenomena, which make extrapolation of low Reynolds number results highly questionable. In this talk these issues will be reviewed and new results from the HRNBLWT and atmospheric surface layer on the salt-flats of Utah will be presented documenting unique high Reynolds number phenomena. The implications for skin-friction drag reduction technologies and improved near-wall models for large-eddy simulation will be discussed.

The classical Hele-Shaw flow problem is related to Laplacian growth and null-quadrature domains. A generalisation is constructed for power-law fluids, governed by the p-Laplace equation, and a number of results are established that are analogous to the classical case. Both fluid clearance and bubble extinction is considered, and by considering two extremes of extinction behaviour, a rather complete asymptotic description of possible behaviours is found.

This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor-corrector Euler methods.

In 1970, two independent studies (by Wunsch and Phillips) of the behaviour of a linear density-stratified viscous fluid in a closed container demonstrated a slow flow can be generated simply due to the container having a sloping boundary surface This remarkable motion is generated as a result of the curvature of the lines of constant density near any sloping surface, which in turn enables a zero normal-flux condition on the density to be satisfied along that boundary. When the Rayleigh number is large (or equivalently Wunsch's parameter $R$ is small) this motion is concentrated in the near vicinity of the sloping surface, in a thin `buoyancy layer' that has many similarities to an Ekman layer in a rotating fluid.

A number of studies have since considered the consequences of this type of `diffusively-driven' flow in a semi-infinite domain, including in the deep ocean and with turbulent effects included. More recently, Page & Johnson (2008) described a steady linear theory for the broader-scale mass recirculation in a closed container and demonstrated that, unlike in previous studies, it is possible for the buoyancy layer to entrain fluid from that recirculation. That work has since been extended (Page & Johnson, 2009) to the nonlinear regime of the problem and some of the similarities to and differences from the linear case will be described in this talk. Simple and elegant analytical solutions in the limit as $R \to 0$ still exist in some situations, and they will be compared with numerical simulations in a tilted square container at small values of $R$. Further work on both the unsteady flow properties and the flow for other geometrical configurations will also be described.

Turbulence is characterised by three-dimensional unsteady fluid motion over a wide range of spatial and temporal scales. It is important in many problems of technological and scientific interest, such as drag reduction, energy production and climate prediction. In this talk, I will explain why turbulent flows are difficult to predict and describe a modern mathematical model of turbulence based on a random collection of fluid vortices.

The flows generated in many modern micro-devices possess very little convective inertia, however, they can be highly unsteady and exert substantial hydrodynamic forces on the device components. Typically these components exhibit some degree of compliance, which traditionally has been treated using simple one-dimensional elastic beam models. However, recent findings have suggested that three-dimensional effects can be important and, accordingly, we consider the elastohydrodynamic response of a rapidly oscillating three-dimensional elastic plate that is immersed in a viscous fluid. In addition, a preliminary model will be presented which incorporates the presence of a nearby elastic wall.

The unsteady flow of a viscous fluid through a curved pipe is a widely occuring and well studied problem. The stability of such flows, however, has largely been overlooked; this is in marked contrast to flow through a straight-pipe, examination of which forms a cornerstone of hydrodynamic stability theory. Importantly, however, flow through a curved pipe exhibits an array of flow structures that are simply not present in the zero curvature limit, and it is natural to expect these to substantially impact upon the flow's stability. By considering two very different kinds of flows through a curved pipe, we illustrate that this can indeed be the case.

The current financial crisis has been in part precipitated by the growth of complex credit derivatives and their mispricing. This talk will discuss some of the background to the `credit crunch', as well as the models and methods used currently. We will then develop an alternative view of large basket credit derivatives, as functions of a stochastic partial differential equation, which addresses some of the shortcomings.

This presentation concerns the fluid-structure interaction (FSI) that occurs between a fluid flow and an arbitrarily deforming flexible boundary considered to be a flexible panel or a compliant coating that comprises the wetted surface of a marine vehicle. We develop and deploy an approach that is a hybrid of computational and theoretical techniques. The system studied is two-dimensional and linearised disturbances are assumed. Of particular novelty in the present work is the ability of our methods to extract a full set of fluid-structure eigenmodes for systems that have strong spatial inhomogeneity in the structure of the flexible wall.

We first present the approach and some results of the system in which an ideal, zero-pressure gradient, flow interacts with a flexible plate held at both its ends. We use a combination of boundary-element and finite-difference methods to express the FSI system as a single matrix equation in the interfacial variable. This is then couched in state-space form and standard methods used to extract the system eigenvalues. It is then shown how the incorporation of spatial inhomogeneity in the stiffness of the plate can be either stabilising or destabilising. We also show that adding a further restraint within the streamwise extent of a homogeneous panel can trigger an additional type of hydroelastic instability at low flow speeds. The mechanism for the fluid-to-structure energy transfer that underpins this instability can be explained in terms of the pressure-signal phase relative to that of the wall motion and the effect on this relationship of the added wall restraint.

We then show how the ideal-flow approach can be conceptually extended to include boundary-layer effects. The flow field is now modelled by the continuity equation and the linearised perturbation momentum equation written in velocity-velocity form. The near-wall flow field is spatially discretised into rectangular elements on an Eulerian grid and a variant of the discrete-vortex method is applied. The entire fluid-structure system can again be assembled as a linear system for a single set of unknowns - the flow-field vorticity and the wall displacements - that admits the extraction of eigenvalues. We then show how stability diagrams for the fully-coupled finite flow-structure system can be assembled, in doing so identifying classes of wall-based or fluid-based and spatio-temporal wave behaviour.

Jeffery–Hamel flows describe the steady two-dimensional flow of an incompressible viscous fluid between plane walls separated by an angle $\alpha$. They are often used to approximate the flow in domains of finite radial extent. However, whilst the base Jeffery–Hamel solution is characterised by a subcritical pitchfork bifurcation, studies in expanding channels of finite length typically find symmetry breaking via a supercritical bifurcation.

We use the finite element method to calculate solutions for flow in a two-dimensional wedge of finite length bounded by arcs of constant radii, $R_1$ and $R_2$. We present a comprehensive picture of the bifurcation structure and nonlinear states for a net radial outflow of fluid. We find a series of nested neutral curves in the Reynolds number-$\alpha$ plane corresponding to pitchfork bifurcations that break the midplane symmetry of the flow. We show that these finite domain bifurcations remain distinct from the similarity solution bifurcation even in the limit $R_2/R_1 \rightarrow \infty$.

We also discuss a class of stable steady solutions apparently related to a steady, spatially periodic, wave first observed by Tutty (1996). These solutions remain disconnected in our domain in the sense that they do not arise via a local bifurcation of the Stokes flow solution as the Reynolds number is increased.

I will discuss two problems in porous media flow.

On a tropical island, fresh water may sit in the soil beneath the ground, floating on the ocean's salt water. This water is a valuable resource for the inhabitants, but requires sufficient rainfall to recharge the lens. In this paper, Green's functions are used to derive an integral equation to satisfy all of the conditions except those on the interfaces, which are then solved for numerically. Conditions under which the lens can be maintained will be described. This is work I did with an Honours student, Sue Chen, who is now at U. Melbourne.

In the second problem, I will discuss an "exact" solution to a problem in withdrawal from an unconfined aquifer. The problem formulation gives rise to a singular integral equation that can be solved using a nice orthogonality result I first met in airfoil theory. This is work with Hong Zhang from Griffith University.

Meshless methods such as vortex methods (VMs) and smoothed particle hydrodynamics (SPH) schemes offer many advantages in fluid flow computations. Particle-based computations naturally adapt to complex flow geometries and so provide a high degree of computational efficiency. Also, particle based methods avoid CFL conditions because flow quantities are integrated along characteristics. There are many approaches to improving numerical methods, but one of the most effective routes is quantifying the error through the direct estimate of residual quantities. Understanding the residual for particle schemes requires a different approach than for meshless schemes but the rewards are significant. In this seminar, I will outline a general approach to understanding convergence that has been effective in creating high spatial accuracy vortex methods, and then I will discuss some recent investigations in the accuracy of diffusion operators used in SPH computations. Finally, I will provide some sample Navier-Stokes computations of high Reynolds number flows using BlobFlow, an open source implementation of the high precision vortex method.

Many complex datasets suffer from missing data, and interpolating these missing elements is a key task in data analysis. Moreover, it is often the case that we see only a linear combination of the desired measurements, not the measurements themselves. For instance, in network management, it is easy to count the traffic on a link, but harder to measure the end-to-end flows. Additionally, typical interpolation algorithms treat either the spatial, or the temporal components of data separately, but in many real datasets have strong spatio-temporal structure that we would like to exploit in reconstructing the missing data. In this talk I will describe a novel reconstruction algorithm that exploits concepts from the growing area of compressive sensing to solve all of these problems and more. The approach works so well on Internet traffic matrices that we can obtain a reasonable reconstruction with as much as 98% of the original data missing.

I will discuss a model for drug diffusion that involves a Stefan problem with a "kinetic undercooling". I like Stefan problems, so I like this model. I like drugs too, but only legal ones of course. Anyway, it turns out that in some parameter regimes, this sophisticated moving boundary problem hardly works better than a simple linear undergraduate model (there's a lesson here for mathematical modelling). On the other hand, for certain polymer capsules, the results are interesting and suggest new means for controlled drug delivery. If time permits, I may discuss certain asymptotic limits that are of interest from a Stefan problem perspective. Finally, I won't bring any drugs with me to the seminar, but I'm willing to provide hugs if necessary.

Simulations in which the system of interest is essentially mimicked are termed explicit numerical simulations (ENS). Direct numerical simulation (DNS) of turbulence is a well known and long-standing example of ENS. Such simulations provide a basis for elucidating fundamentals in a way that is impossible experimentally and formulating and parameterizing engineering models with reduced experimentation. In this presentation, I will first outline the concept of ENS. I will then report a number of ENS-based studies of various multiphase fluid systems and flows in porous media. In the first of these studies, which is concerned with flow of suspensions in porous media accompanied by deposition, ENS is used to demonstrate the significant inadequacies of the classical trajectory models typically used for the study of such problems. In the second study, which is concerned with elucidating the change in binary droplet collision behaviour with Capillary number (Ca) and Reynolds number (Re), a range of collision scenarios are revealed as a function of Ca and Re and it appears that the boundaries between these scenarios in the Ca-Re space are not distinct but, rather, smeared. In the final study, it is shown that ENS an be used to predict ab initio the hydrodynamic properties of single phase flow through porous media from the Darcy to the turbulent regimes.

Animals typically search for food and mates, while avoiding predators. This is particularly critical for keystone organisms such as intertidal gastropods and copepods (i.e. millimeter-scale crustaceans) as they typically rely on non-visual senses for detecting, identifying and locating mates in their two- and three-dimensional environments. Here, using stochastic methods derived from the field of nonlinear physics, we provide new insights into the nature (i.e. innate vs. acquired) of the motion behavior of gastropods and copepods, and demonstrate how changes in their behavioral properties can be used to identify the trade-offs between foraging for food or sex. The gastropod Littorina littorea hence moves according to fractional Brownian motions while foraging for food (in accordance with the fractal nature of food distributions), and switch to Brownian motion while foraging for sex. In contrast, the swimming behavior of the copepod Temora longicornis belongs to the class of multifractal random walks (MRW; i.e. a form of anomalous diffusion), characterized by a nonlinear moment scaling function for distance versus time. This clearly differs from the traditional Brownian and fractional Brownian walks expected or previously detected in animal behaviors. The divergence between MRW and Levy flight and walk is also discussed, and it is shown how copepod anomalous diffusion is enhanced by the presence and concentration of conspecific water-borne signals, and is dramatically increasing male-female encounter rates.

The no-slip boundary condition was considered to have been experimentally established for the flow of simple liquids over solid surfaces in the early 20th century. Nonetheless the refinement of a number of measurement techniques has recently led to the observation of nano- and microscale violations of the no-slip boundary condition by simple fluids flowing over non-wetting surfaces. However it is important to distinguish between intrinsic slip, which arises solely from the chemical interaction between the liquid and a homogeneous, atomically flat surface and effective slip, typically measured in macroscopic experiments, which emerges from the interaction of microscopic chemical heterogeneity, roughness and contaminants. Here we consider the role of both intrinsic and effective slip boundary conditions in nanoscale and microscale fluid flows using a theoretical approach, complemented by molecular dynamics simulations, and experimental evidence where available. Firstly, we consider nanoscale flows in small capillaries, including carbon nanotubes, where we have developed and solved a generalised Lucas-Washburn equation that incorporates slip to describe the uptake of droplets. We then consider the general problem of relating effective slip to microscopic intrinsic slip and roughness, and discuss several cases where we have been able to solve this problem analytically. Finally, we look at applications of these results to carbon nanotube growth, self-cleaning surfaces, catalysis, and putting insulation in your roof.

Horizontal convecton refers to flows driven by uneven heating on a horizontal forcing boundary. Flows exhibiting these characteristics are prevalent in nature, and include the North-South Hadley circulation within the atmosphere between warmer and more temperate latitudes, as well as ocean currents driven by non-uniform heating via solar radiation.

Here a model for these generic convection flows is established featuring a rectangular enclosure, insulated on the side and top walls, and driven by a linear temperature gradient applied along the bottom wall. Rayleigh number dependence of heat transfer through the forcing boundary is computed and compared with theory. Attention is given to transitions in the flow, including the development of unsteady flow and three-dimensional flow: the effect of these transitions on the Nusselt-Rayleigh number scaling exponents is described.

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the populations size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Here we will discuss the use of diffusion approximations to perform estimation in several scenarios, with successively reduced assumptions: (i) under the assumption of stationarity (the process had been evolving for a very long time with constant parameter values); (ii) transient dynamics (the assumption of stationarity is invalid, and thus only constant parameter values may be assumed); and, (iii) time-inhomogeneous chains (the parameters may vary with time) and accounting for observation error (a sample of the true state is observed).

Standard analytical solutions to elliptic boundary value problems on asymmetric domains are rarely, if ever, obtainable. Several years ago I proposed a solution technique to cope with such complicated domains. It involves the embedding of the original domain into one with simple boundaries where the classical eigenfunction solution approach can be used. The solution in the larger domain, when restricted to the original domain is then the solution of the original boundary value problem. In this talk I will present supporting theory for this idea, some numerical results for the particular case of the Laplace equation and the Stokes flow equations in two-dimensions and discuss advantages and limitations of the proposal.

Markov population processes are popular models for studying a wide range of phenomena including the spread of disease, the evolution of chemical reactions and the movements of organisms in population networks (metapopulations). Our ability to use these models effectively can be limited by our knowledge about parameters, such as disease transmission and recovery rates in an epidemic. Recently, there has been interest in devising optimal experimental designs for stochastic models, so that practitioners can collect data in a manner that maximises the precision of maximum likelihood estimates of the parameters for these models. I will discuss some recent work on optimal design for a variety of population models, beginning with some simple one-parameter models where the optimal design can be obtained analytically and moving on to more complicated multi-parameter models in epidemiology that involve latent states and non-exponentially distributed infectious periods. For these more complex models, the optimal design must be arrived at using computational methods and we rely on a Gaussian diffusion approximation to obtain analytical expressions for Fisher's information matrix, which is at the heart of most optimality criteria in experimental design. I will outline a simple cross-entropy algorithm that can be used for obtaining optimal designs for these models. We will also explore the improvements in experimental efficiency when using the optimal design over some simpler designs, such as the design where observations are spaced equidistantly in time.

In this talk I will review existing analytical results for priority queueing systems with Poisson incoming flows, general service times and a single server which needs some (random) time to switch between requests of different priority. Specifically, I will discuss analytical results for the busy period and workload of such systems with a special structure of switchover times. The results related to the busy period can be seen as generalisations of the famous Kendall-Tak\'{a}cs functional equation for $M|G|1$: being formulated in terms of Laplace-Stieltjes transform, they represent systems of functional recurrent equations. I will present a methodology and algorithms of their numerical solution; the efficiency of these algorithms is achieved by acceleration of the numerical procedure of solving the classical Kendall-Tak\'{a}cs equation. At the end I will identify open problems with regard to such systems; these open problems are mainly related to the modelling of switchover times.

One of the beautiful aspects of Mathematics is that seemingly disparate areas can often have deep connections. This talk is about the fundamental connection between probability density estimation, diffusion processes, and partial differential equations. Specifically, we show how to obtain efficient probability density estimators by solving partial differential equations related to diffusion processes. This new perspective leads, in combination with Fast Fourier techniques, to very fast and accurate algorithms for density estimation. Moreover, the diffusion formulation unifies most of the existing adaptive smoothing algorithms and provides a natural solution to the boundary bias of classical kernel density estimators. This talk covers topics in Statistics, Probability, Applied Mathematics, and Numerical Mathematics, with a surprise appearance of the theta function. This is joint work with Zdravko Botev and Joe Grotowski.

We consider two different position jump processes: (i) a random walk on a lattice (ii) the Euler scheme for the Smoluchowski differential equation. Both of these reduce to the diffusion equation as the time step and size of the jump tend to zero. We consider the problem of adding chemical reactions to these processes, both at a surface and in the bulk. We show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which stochastic model for diffusion is used.

The distribution of individual orbits of unipotent flows in homogeneous spaces are well understood thanks to the work work of Marina Ratner. It is conjectured that this property is preserved on restricting the times from the integers to primes, this being important in the study of prime numbers as well as in such dynamics. We review progress in understanding this conjecture, starting with Dirichlet (a finite system), Vinogradov (rotation of a circle or torus), Green and Tao (translation on a nilmanifold) and Ubis and Sarnak (horocycle flows in the semisimple case).

For steady flows, the boundaries of Lagrangian Coherent Structures are segments of manifolds connected to fixed points. In the general unsteady situation, these boundaries are time-varying manifolds of hyperbolic trajectories. Locating these boundaries, and attempting to meaningfully quantify fluid flux across them, is difficult since they are moving with time. This talk uses a newly developed tangential movement theory to locate these boundaries in nearly-steady compressible flows.

Various physical processes take place on multiply-connected domains (domains with some number of 'holes'), such as the stirring of a fluid with paddles or the extrusion of material from a die. These systems may be described by partial differential equations (PDEs). However, standard numerical methods for solving PDEs are not well-suited to such examples: finite difference methods are difficult to implement on multiply-connected domains, especially when the boundaries are irregular or moving, while finite element methods are computationally expensive. In this talk I will describe a fast and accurate numerical method for solving certain PDEs on two-dimensional multiply-connected domains, considering Laplace's equation as an example. This method takes advantage of complex variable techniques which allow the solution to be found with spectral accuracy provided the boundary data is smooth. Other advantages over traditional numerical methods will also be discussed.

The problem of variables separation in the linear stability equations, which govern the disturbance behavior in viscous incompressible fluid flows, is discussed. Stability of some unsteady nonparallel three-dimensional flows (exact solutions of the Navier-Stokes equations) is studied via separation of variables using a semi-analytical, semi-numerical approach. In this approach, a solution with separated variables is defined in a new coordinate system which is sought together with the solution form. As the result, the linear stability problems are reduced to eigenvalue problems for ordinary differential equations which can be solved numerically. In some specific cases, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for methods used in the hydrodynamic stability theory. Exact solutions of the stability problems for some stagnation-type flows are presented.

Optical fibres are used extensively in modern telecommunications as they allow the transmission of information at high speeds. Microstructured optical fibres are a relatively new fibre design in which a waveguide for light is created by a series of air channels running along the length of the material. The flexibility of this design allows optical fibres to be created with adaptable (and previously unrealised) optical properties. However, the fluid flows that arise during fabrication can greatly distort the geometry, which can reduce the effectiveness of a fibre or render it useless. I will present an overview of the manufacturing process and highlight the difficulties. I will then focus on surface-tension driven deformation of the macroscopic version of the fibre extruded from a reservoir of molten glass, occurring during fabrication, which will be treated as a two-dimensional Stokes flow problem. I will outline two different complex-variable numerical techniques for solving this problem along with comparisons of the results, both to other models and to experimental data.

In this talk, I will describe a mechanism for determining instability of standing wave solutions to a class of inhomogeneous nonlinear Schrodinger (NLS) equations. The inhomogeneity in this case means that the equations will spatially alternate between NLS and the so-called Gross-Pitaevskii equation. Such equations are useful in 1-D models of Bose-Einstein Condensates (BECs). The mechanism is inherently topological and therefore robust, leading to its application to a number of different soliton solutions, such as gap solitons, surface gap solitons, and dark soliton among others.

We look at an equation arising from the differential-geometric problem of specifying the scalar curvature of a manifold.

Motivated by certain tea room discussions, I am going to (attempt to) model the flow of a viscous fluid under gravity over conical topography. The method used is 'Smoothed Particle Hydrodynamics' (SPH), which is an easy-to-use but perhaps limited-accuracy computational method. The model could be extended to include solidification and thermodynamic effects that can also be implemented within the framework of SPH, and this has the obvious practical application to the modelling of the coverage of ice cream with ice magic, I mean, lava flows. If I fail to achieve this within the next 4 weeks, I will have to go through a talk on SPH that I gave during honours instead.

Spatial-point data sets, generated from a wide range of physical systems and mathematical models, can be analyzed by counting the number of objects in equally sized bins. We find that the bin counts are related to the Polya distribution. New indexes are developed which quantify whether or not a spatial data set is at its most evenly distributed state. Using three case studies (Lagrangian fluid particles in chaotic laminar flows, cellular automata agents in discrete models, and biological cells within colonies), we calculate the indexes and predict the spatial-state of the system.

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. There is no guidance available in the mathematical biology literature with regard to which approach is more appropriate. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. We provide a link between individual-based and continuum models using a multiscale approach in which we analyse the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is a nonlinear diffusion equation related to the porous media equation. We show that there are several reasonable approaches for dealing with agent size effects, and that these different approaches are related mathematically through the concept of mean action time. We extend our results to consider proliferation and travelling waves where greater care must be taken to ensure that the continuum model replicates the discrete process. This is joint work with Dr Ruth Baker (Oxford) and Dr Scott McCue (QUT).

Although cancers seem to consistently evade current medical treatments, the body's immune defences seem quite effective at controlling incipient tumours. Understanding how our immune systems provide such protection against early-stage tumours and how this protection could be lost will provide insight into designing next-generation immune therapies against cancer. To engage this problem, we formulate a mathematical model of the immune response against small, incipient tumours. The model considers the initial stimulation of the immune response in lymph nodes and the resulting immune attack on the tumour and is formulated as a hybrid agent-based and delay differential equation model.

In financial derivatives pricing theory, it is very common to change the probability measure from historical measure "real world" into a Risk-Neutral measure as a development of the non arbitrage condition. Girsanov theorem is the most known example of this technique and is used when prices randomness is modelled by Brownian motions. Other genuine candidates for modelling market randomness that have proved efficiency in recent literature are jump process, so how can a change of measure be performed for such processes? This talk will address this question by introducing the non arbitrage condition, discussing Girsanov theorem for diffusion and jump processes and presenting a concrete example.

Rivers, floods and tsunamis are often very turbulent. Conventional models of such environmental fluids are typically based on depth-averaged inviscid irrotational flow equations. We explore changing such a base to the turbulent Smagorinski large eddy closure. The aim is to more appropriately model the fluid dynamics of such complex environmental fluids by using such a turbulent closure. Large changes in fluid depth are allowed. Computer algebra constructs the slow manifold of the flow in terms of the fluid depth h and the mean turbulent lateral velocities u and v. The major challenge is to deal with the nonlinear stress tensor in the Smagorinski closure. The model integrates the effects of inertia, self-advection, bed drag, gravitational forcing and turbulent dissipation with minimal assumptions. Although the resultant model is close to established models, the real outcome is creating a sound basis for the modelling so others, in their modelling of more complex situations, can systematically include more complex physical processes.

The drawing of viscous threads is important in a wide range of industrial applications and is a primary manufacturing process in the optical fiber and textile industries. Most of the materials used in these processes have viscosities that vary extremely strongly with temperature. We investigate the role played by viscous heating in the drawing of viscous threads. Usually, the effects of viscous heating and inertia are neglected because the parameters that characterize them are typically very small. However, by performing a detailed theoretical analysis we surprisingly show that even very small amounts of viscous heating can lead to a runaway phenomena. On the other hand, inertia prevents runaway, and the interplay between viscous heating and inertia results in very complicated dynamics for the system. Even more surprisingly, in the absence of viscous heating, we find that a new type of instability can occur when a thread is heated by a radiative heat source. By analyzing an asymptotic limit of the Navier-Stokes equation we provide a theory that describes the nature of this instability and explains the seemingly counterintuitive behavior.

The classical diffusion (thermal conductivity) equation was derived from the Master random walk equation and is parabolic. The main assumption was a probabilistic distribution of the jump length while the jump time is constant. Distribution of the jump time along with the jump length adds the second time derivative into the averaged equations, but the equation becomes ... elliptic! Where from to take an extra initial condition? We discuss how to pose the well-posed flow problem, exact 1d solution and numerous engineering applications. This is joint work with A. Shapiro and H. Yuan.

We aim to solve the advection-diffusion-reaction equation on the surface of a sphere. In order to do this we will be required to utilise spherical harmonics, a set of solutions to Laplace's equation in spherical coordinates. Upon solving the equations, we aim to find a set of parameters that cause a localised concentration to be maintained in the flow, referred to as a hotspot. In this talk I will discuss the techniques that are required to numerically solve this problem and the issues that occur/how to deal with these issues when searching for hotspot solutions.

The analysis of turbulence in transient flows has applications across a broad range of fields. We use the flow of fluid in a toroidal container as a paradigm for studying the complex dynamics due to this turbulence. To explore the dynamics of our system, we exploit the numerical capabilities of semtex; a quadrilateral spectral element DNS code. Rainbows result.

It is a well-known fact that the methods of complex analysis provide great advantage in studying physical problems involving a harmonic field satisfying Laplace's equation. One example is in ideal fluid mechanics (infinite Reynolds number) where the absence of viscosity, and the assumption of zero vorticity, mean that it is possible to introduce a so-called complex potential -- an analytic function from which all physical quantities of interest can be inferred. In the opposite limit of zero Reynolds number flows which are slow and viscous and the governing fields are not harmonic it is much less common to employ the methods of complex analysis even though they continue to be relevant in certain circumstances. This talk will give an overview of a variety of problems involving slow viscous Stokes flows where complex analysis can be usefully employed to gain theoretical insights. A number of example problems will be considered including the locomotion of low-Reynolds-number micro-organisms and micro-robots, the friction properties of superhydrophobic surfaces in microfluidics and problems of viscous sintering and the manufacture of microstructured optic fibres (MOFs).

The study of flow in open helically-wound channels has application to many natural and industrial flows. We will consider laminar flow down helically-wound channels of rectangular cross section and with small torsion, in which the fluid depth is small. Assuming a steady-state flow that is independent of position along the axis of the channel, the flow solution may be determined in the two-dimensional cross section of the channel. A thin-film approximation yields explicit expressions for the fluid velocity in terms of the free-surface shape. The latter satisfies an interesting non-linear ordinary differential equation that, for a channel of rectangular cross section, has an analytical solution. The predictions of the thin-film model are shown to be in good agreement with much more computationally intensive solutions of the small-helix-torsion Navier-Stokes equations. This work has particular relevance to spiral particle separators used in the minerals processing industry. Early work on modelling of particle-laden thin-film flow in spiral channels will also be discussed.

Reaction-diffusion processes occur in many materials with microstructure such as biological cells, steel or concrete. The main difficulty in modelling and simulating accurately such processes is to account for the fine microstructure of the material. One method of upscaling multi-scale problems, which has proven reliable for obtaining feasible macroscopic models, is the method of periodic homogenisation. The talk will give an introduction to multi-scale modelling of chemical mechanisms in domains with microstructure as well as to the method of periodic homogenisation. Moreover, a few aspects of solving the resulting systems of equations numerically will also be discussed.

Conformal Killing spinors are the solutions of the conformally covariant twistor equation on spinors. Special cases are parallel and Killing spinors, the latter appear as eigenspinors of the Dirac operator on compact Riemannian manifolds of positive scalar curvature for the smallest possible positive eigenvalue. In the talk I will discuss geometric properties of manifolds admitting (conformal) Killing spinors. In particular, I will explain a local classification of the special geometric structures admitting conformal Killing spinors without zeros in the Riemannian as well as in the Lorentzian setting.

Many mathematical and engineering problems have a multiscale nature. There are a vast of theories supporting multiscale modelling on infinite domain, such as homogenization theory and centre manifold theory. To date, there are little consideration of the correct boundary conditions to be used at the edge of macroscale model. In this seminar, I will present how to derive macroscale boundary conditions for the diffusion system.

Buoyancy-affected turbulence represents a significant challenge to our understanding, yet it dominates many important flows that occur in the ocean and atmosphere. The presentation will highlight some recent progress in the characterisation, modelling and prediction of buoyancy-affected turbulence using direct and large-eddy simulations, along with implications for the characterisation of mixing in the ocean and the low-cloud feedback in the atmosphere. Specifically, direct numerical simulation data of stratified turbulence will be employed to highlight the importance of boundaries in the characterisation of turbulent mixing in the ocean. Then, a subgrid-scale model that captures the anisotropic character of stratified mixing will be developed for large-eddy simulation of buoyancy-affected turbulence. Finally, the subgrid-scale model is utilised to perform a systematic large-eddy simulation investigation of the archetypal low-cloud regimes, from which the link between the lower-tropospheric stability criterion and the cloud fraction interpreted.