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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.

The Sunderbans is a region of deltaic isles formed in the mouth of the Ganges River on the border between India and Bangladesh. As the largest mangrove forest in the world it is a world heritage site, however it is also home to several remote communities who have long inhabited some regions. Many of the inhabited islands are low-lying and are particularly vulnerable to flooding, a major hazard of living in the region. Determining suitable levels of protection to be provided to these communities relies upon accurate assessment of the flood risk facing these communities. Only recently the Indian Government commissioned a study into flood risk in the Sunderbans with a view to determine where flood protection needed to be upgraded.

Flooding due to rainfall is limited due to the relatively small catchment sizes, so the primary causes of flooding in the Sunderbans are unnaturally high tides, tropical cyclones (which regularly sweep through the bay of Bengal) or some combination of the two. Due to the link between tidal anomaly and drops in local barometric pressure, the two causes of flooding may be highly correlated. I propose stochastic methods for analysing the flood risk and present the early work of a case study which shows the direction of investigation. The strategy involves linking several components; a stochastic approximation to a hydraulic flood routing model, FARIMA and GARCH models for storm surge and a stochastic model for cyclone occurrence and tracking. The methods suggested are general and should have applications in other cyclone affected regions.

On homogeneous spaces G/P, where G is a semi-simple Lie group and P is a parabolic subgroup (the ordinary sphere or projective spaces being examples), invariant operators, that is operators between certain homogeneous bundles (functions, vector fields or forms being amongst the typical examples) that are invariant under the action of the group G, have been studied extensively. Especially on so called hermitian symmetric spaces which arise through a 1-grading of the Lie algebra of G there exists a complete classification of first order invariant linear differential operators even on more general manifolds (that allow a so called almost hermitian structure).

This talk will introduce the notion of an invariant bilinear differential pairing between sections of the aforementioned homogeneous bundles. Moreover we will discuss a classification (excluding certain totally degenerate cases) of all first order invariant bilinear differential pairings on manifolds with an almost hermitian symmetric structure. The similarities and connections with the linear operator classification will be highlighted and discussed.

These well-known differential operators are, of course, important in applied mathematics. This is just the tip of an iceberg. I shall indicate some of what lies beneath the surface. There are links with topology, physics, symmetry groups, finite element schemes, and more besides. This talk will touch on these different topics by means of examples. Little prior knowledge will be assumed beyond the equality of mixed partial derivatives.

Digital gene expression (DGE) technologies measure gene expression by counting sequence tags. They are sensitive technologies for measuring gene expression on a genomic scale, without the need for prior knowledge of the genome sequence. As the cost of DNA sequencing decreases, the number of DGE datasets is expected to grow dramatically. Various tests of differential expression have been proposed for replicated DGE data using over-dispersed binomial or Poisson models for the counts, but none of the these are usable when the number of replicates is very small. We develop tests using the negative binomial distribution to model overdispersion relative to the Poisson, and use conditional weighted likelihood to moderate the level of overdispersion across genes. A heuristic empirical Bayes algorithm is developed which is applicable to very general likelihood estimation contexts. Not only is our strategy applicable even with the smallest number of replicates, but it also proves to be more powerful than previous strategies when more replicates are available. The methodology is applicable to other counting technologies, such as proteomic spectral counts.

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.

To model real data sets using second order stochastic processes imposes that the data sets verify the second order stationarity condition. This stationarity condition concerns the unconditional moments of the process. It is in that context that most of models developed from the sixties' have been studied; We refer to the ARMA processes (Brockwell and Davis, 1988), the ARCH, GARCH and EGARCH models (Engle, 1982, Bollerslev, 1986, Nelson, 1990), the SETAR process (Lim and Tong, 1980 and Tong, 1990), the bilinear model (Granger and Andersen, 1978, Guégan, 1994), the EXPAR model (Haggan and Ozaki, 1980), the long memory process (Granger and Joyeux, 1980, Hosking, 1981, Gray, Zang and Woodward, 1989, Beran, 1994, Giraitis and Leipus, 1995, Guégan, 2000), the switching process (Hamilton, 1988). For all these models, we get an invertible causal solution under specific conditions on the parameters, then the forecast points and the forecast intervals are available.

Thus, the stationarity assumption is the basis for a general asymptotic theory for identification, estimation and forecasting. It guarantees that the increase of the sample size leads to more and more information of the same kind which is basic for an asymptotic theory to make sense.

Now non-stationarity modelling has also a long tradition in econometrics. This one is based on the conditional moments of the data generating process. It appears mainly in the heteroscedastic and volatility models, like the GARCH and related models, and stochastic volatility processes (Ghysels, Harvey and Renault 1997). This non-stationarity appears also in a different way with structural changes models like the switching models (Hamilton, 1988), the stopbreak model (Diebold and Inoue, 2001, Breidt and Hsu, 2002, Granger and Hyung, 2004) and the SETAR models, for instance. It can also be observed from linear models with time varying coefficients (Nicholls and Quinn, 1982, Tsay, 1987).

Thus, using stationary unconditional moments suggest a global stationarity for the model, but using non-stationary unconditional moments or non-stationary conditional moments or assuming existence of states suggest that this global stationarity fails and that we only observe a local stationary behavior.

The growing evidence of instability in the stochastic behavior of stocks, of exchange rates, of some economic data sets like growth rates for instance, characterized by existence of volatility or existence of jumps in the variance or on the levels of the prices imposes to discuss the assumption of global stationarity and its consequence in modelling, particularly in forecasting. Thus we can address several questions with respect to these remarks.

1. What kinds of non-stationarity affect the major financial and economic data sets? How to detect them?

2. Local and global stationarities: How are they defined?

3. What is the impact of evidence of non-stationarity on the statistics computed from the global non stationary data sets?

4. How can we analyze data sets in the non-stationary global framework? Does the asymptotic theory work in non-stationary framework?

5. What kind of models create local stationarity instead of global stationarity? How can we use them to develop a modelling and a forecasting strategy?

These questions began to be discussed in some papers in the economic literature. For some of these questions, the answers are known, for others, very few works exist. In this talk I will discuss all these problems and will propose 2 new stategies and modelling to solve them. Several interesting topics in empirical finance awaiting future research will also be discussed.

Enrico Betti (1823-1892) is recognized in the mathematics community for his pioneering contributions to topology. An equally important contribution is his formulation of the reciprocity theorem applicable to elastic bodies that satisfy the classical equations of linear elasticity. Although James Clerk Maxwell (1831-1879) proposed a law of reciprocal displacements and rotations in 1864, the contribution of Betti is acknowledged for its underlying formal mathematical basis and generality. The purpose of this lecture is to illustrate how Betti's reciprocal theorem can be used to full advantage to develop compact analytical results for certain contact and inclusion problems in the classical theory of elasticity. Inclusion problems are encountered in number of areas in applied mechanics ranging from composite materials to geomechanics. In composite materials, the inclusion represents an inhomogeneity that is introduced to increase either the strength or the deformability characteristics of resulting material. In geomechanics, the inclusion represents a constructed material region, such as a ground anchor, that is introduced to provide load transfer from structural systems. Similarly, contact problems have applications to the modelling of the behaviour of indentors used in materials testing to the study of foundations used to distribute loads transmitted from structures. In the study of conventional problems the inclusions and the contact regions are directly loaded and this makes their analysis quite straightforward. When the interaction is induced by loads that are placed exterior to the indentor or inclusion, the direct analysis of the problem becomes inordinately complicated both in terns of formulation of the integral equations and their numerical solution. It is shown by a set of selected examples that the application of Betti's reciprocal theorem leads to the development of exact closed form solutions to what would otherwise be approximate solutions achievable only through the numerical solution of a set of coupled integral equations.

Atherosclerosis, characterised by plaques, is the most common arterial disease. Plaques tend to develop in regions of low mean wall shear stress, and regions where the wall shear stress changes direction during the course of the cardiac cycle. To investigate the effect of the arterial geometry and driving pressure gradient on the wall shear stress distribution we consider an idealised model of a curved artery with uniform curvature. We assume that the flow is fully-developed and seek solutions of the governing equations, finding the effect of the parameters on the flow and wall shear stress distribution. Most previous work assumes the curvature ratio is asymptotically small; however, many arteries have significant curvature (e.g. the aortic arch has curvature ratio approx 0.25), and in this work we consider in particular the effect of finite curvature.

We present an extensive analysis of curved-pipe flow driven by a steady and unsteady pressure gradients. Increasing the curvature causes the shear stress on the inside of the bend to rise, indicating that the risk of plaque development would be overestimated by considering only the weak curvature limit.

The theory of partial differential equations can say much about generic bifurcations from spatially homogeneous steady states, but relatively little about generic bifurcations from unimodal steady states. In many applications, spatially homogeneous steady states correspond to low-energy physical states that are destabilized as energy is fed into the system, and in these cases standard PDE theory can yield some impressive and elegant results. However, for many macroscopic biological systems such results are less useful because low-energy states do not hold the same priviledged position as they do in physical and chemical systems. For example, speciation -- the evolutionary process by which new species are formed -- can be seen as the destabilization of a unimodal density distribution over phenotype space. Given the diversity of species and environments, generic results are clearly needed, but cannot be gained from PDE theory. Indeed, such questions cannot even be adequately formulated in terms of PDEs. In this talk I will introduce 'Pod Systems' which can provide an answer to the question; 'What happens, generically, when a unimodal steady state loses stability?' In the pod system formalization, the answer involves elements of equivariant bifurcation theory and suggests that new species can arise as the result of broken symmetries.

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.

The utility of mathematical models relies on their ability to predict the future from a known set of initial states. But there are non-linear systems, like the weather, where future behaviours are unpredictable unless their initial state is known to infinite precision. This is the butterfly effect. I will show how to analyse functions to overcome this problem for the classical Painleve equations, differential equations that provide archetypical non-linear models of modern physics.

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.

Trajectory data from observations of a random walk process are often used to characterize macroscopic transport coefficients and to infer motility mechanisms in cell biology. New continuum equations describing the average moments of the position of an individual agent in a population of interacting agents are derived and validated. Unlike standard noninteracting random walks, the new moment equations explicitly represent the interactions between agents as they are coupled to the macroscopic agent density. Key issues associated with the validity of the new continuum equations and the interpretation of experimental data will be explored.

Within a continuous-time framework, this paper proposes a stochastic heterogeneous agent model (HAM) of financial markets with time delays to unify various moving average rules used in discrete-time HAMs. Intuitive conditions for the stability of the fundamental price of the deterministic model in terms of agents' behavior parameters and time delay are obtained. By focusing on the stabilizing role of the time delay, it is found that an increase in time delay not only can destabilize the market price, resulting in oscillatory market price characterized by a Hopf bifurcation, but also can stabilize an otherwise unstable market price. Numerical simulations show that the stochastic model is able to characterize long deviations of the market price from its fundamental price and excess volatility and generate most of the stylized facts observed in financial markets.

In this talk, I introduce recent work on macroscopic reduction for stochastic PDEs by an averaging method. Furthermore by using a special coupling boundary conditions, a macroscopic discrete approximation model can be derived.

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.

Partial differential equations are used to model populations (such as cells, animals or molecules) consisting of individuals that undergo two important processes: dispersal and settling. I will describe some general characteristics of these systems, as well as some of our recent projects.

The ordinary integral on L^1 functions on R^d unfortunately does not extend to a translation invariant linear form on the whole algebra of pseudodifferential symbols on R^d, forcing to work with ordinary linear extensions which fail to be translation invariant. Defect formulae which express the difference between various linear extensions, show that they differ by local terms involving the noncommutative residue. In particular, we shall show how integrals regularised by a "dimensional regularisation" procedure familiar to physicists differ from Hadamard finite part (or "cut-off" regularised) integrals by a residue. When extended to pseudodifferential operators on closed manifolds, these defect formulae express the zeta regularised traces of a differential operator in terms of a residue of its logarithm. In particular, we shall express the index of a Dirac type operator on a closed manifold in terms of a logarithm of a generalized Laplacian, thus giving an a priori local description of the index and shall discuss further applications.

Given a polynomial in two or more variables, one may study the zero locus from the point of view of different mathematical subjects (number theory, algebraic geometry, ...). I will explain how tropical geometry allows to encode all topological aspects by elementary combinatorial objects called "tropical varieties." Mohammed Abouzaid received a B.S. in 2002 from the University of Richmond, and a Ph.D. in 2007 from the University of Chicago under the supervision of Paul Seidel. He is interested in symplectic topology and its interactions with algebraic geometry and differential topology, in particular the homological mirror symmetry conjecture. Since 2007 he has been a postdoctoral fellow at MIT, and a Clay Mathematics Institute Research Fellow.

Fourier-Mukai transforms are a geometric analogue of integral transforms playing an important role in algebraic geometry. Their name derives from the construction of Mukai involving the Poincare line bundle associated to an abelian variety. In this talk I will discuss recent work looking at an analogue of this original Fourier-Mukai transform in the context of differential geometry, which gives an isomorphism between the invariant differential cohomology of a real torus and its dual.

Berger's classification of irreducibly represented Lie groups that can occur as holonomy groups of semi-Riemannian manifolds is a remarkable result of modern differential geometry. What is remarkable about it is that it is so short and that only so few types of geometry can occur. In Riemannian signature this is even more remarkable, taking into account that any representation of a compact Lie group admits a positive definite invariant scalar product. Hence, for any not too small n there is an abundance of irreducible subgroups of SO(n). We show that in other signatures the situation is quite different with, for example, SO(1,n) having no proper irreducible subgroups. We will show how this and the corresponding result about irreducible subgroups of SO(2,n) follows from the Karpelevich-Mostov theorem. (This is joint work with Antonio J. Di Scala, Politecnico di Torino.)

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.

We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.