**Project descriptions**** **

**Project 1**: Network Science | Mason Porter, Mikko Kivela, Sang Hoon Lee

Networks are the language of connectivity, as they describe how entities (called "nodes") are connected to each other by ties (called "edges"). Many dynamical processes---such as epidemics, traffic, and rumors, propagate on networks; and the structure of networks has a profound but poorly understood effect on such dynamics. This project will entail the study of network structure and/or dynamical processes on networks.

There are a diverse set of possible topics available (see http://www.maths.ox.ac.uk/groups/ociam/research/networks and Mason Porter's website, http://people.maths.ox.ac.uk/~porterm/, for numerous examples of past and current research on networks), and we are happy to discuss possible projects with interested students.

**Project 2**: Vertex-based modelling of epithelial regeneration in the wing disc | Alexander Fletcher, Ruth Baker

A paradigm multiscale phenomenon is the development, maintenance and repair of epithelial tissues, which line many of the cavities and surfaces of structures in the body. Epithelia are composed of a sheet of cells of similar height that are connected via cell-cell adhesion. The capacity of epithelial cells to modulate intercellular junctional contacts is essential for tissue form and function. Junction remodelling underlies many of the dramatic tissue reorganizations occurring during development. While many genes that play a role in epithelial junction remodelling have been identified, the physical mechanisms involved in rearrangement of epithelial cell packing are not well understood.

Computational modelling can help investigate the contributions of cell mechanics, adhesion, and cortical contractility to the development of specific packing geometries. The network of adherens junctions may be described by a two-dimensional vertex model in which cells are modelled as polygons with vertices at which cell edges meet. The movement of each vertex is determined through an energy function describing contributions due to cell elasticity, actin-myosin bundles, and adhesion molecules. Very recently, Bayesian statistics has been applied to this modelling framework, with the development of an inverse problem framework that allows for estimation of key model parameters. Parameter estimates have been shown to be consistent with other force readouts, for example from experiments which use a laser to destroy cortical actin cables.

This project will develop a vertex-based model to describe the dynamics of regeneration and pattern repair in the proliferating wing disc epithelium of the fruit fly Drosophila melanogaster. This work is motivated by live-imaging data obtained experimentally by our collaborator Dr Jeremiah Zartman (University of Notre Dame). These data show vertex movements in the Drosophila wing disc are observed during regeneration following wound healing through laser ablation. By constructing a vertex-based model of cell dynamics within the epithelium and comparing model simulations to experimental results, we may study the force balances and determine key parameter values.

**Project 3**: Interaction between an elastic shell and a thin liquid film on a curved substrate | Peter Howell, Philippe Trinh

This study is principally motivated by the need to understand the deformation and relaxation of a soft contact lens caused by interaction with the tear film. During a blink, the upper eyelid pushes the lens closer to the cornea, deforming the lens and pushing the tear film underneath the lens outwards. Between blinks, the lens relaxes to its original shape, and this should effectively pull fluid from the periphery back inwards. However, if the inter-blink period is too short, then the lens will not return fully to its initial state, and the central tear film can thin after multiple blinks. This thinning may eventually lead to rupture of the tear film and direct contact between the lens and the cornea, and is one cause of discomfort in wearing contact lenses.

Several previous models for contact lenses have been proposed, and mathematical models have also been derived for the tear film that include realistic eye curvature. However, the interaction between a deformable elastic lens and the thin viscous tear film is still not fully understood when the curvature of the eye is taken into account.

This is a novel problem in "elasto-hydrodynamic lubrication", in which the dynamics is driven by elastic stress due to a mismatch in curvature between the lens and the substrate, mediated by the thin viscous film between them. Such problems are of significant scientific interest and are relevant to a wide range of industrial processes and in small scale biological and microelectromechanical systems.

Methodology: The lens will be modelled as a naturally curved elastic shell that deforms under a lubrication pressure which is coupled to the flow in the liquid tear film via Reynolds' equation. The significant complications associated with lens deformation and substrate curvature will be ameliorated by initially restricting to two space dimensions and neglecting surface tension (i.e. assuming that the lens is small compared with the elasto-capillary length). This will result in a sixth-order degenerate parabolic PDE for the thickness of the tear film beneath the lens with appropriate boundary conditions. The model will be solved numerically, and the results will be validated by studying limiting cases analytically (for example: small perturbations of steady states; limits where the lens stiffness tends to zero or to infinity). The dependence of the dynamics upon the lens and substrate properties will be parameterised. If time permits, external forces acting on the lens will be incorporated in the model to simulate lens insertion or removal.

The completion of the project will require a range of mathematical techniques, including mathematical modelling, perturbation theory, and numerical methods for solving partial differential equations using software such as COMSOL and Matlab. In addition, the project student will acquire invaluable transferable skills, for example, the abilities to read and apply challenging research literature, to work in small teams and to communicate scientific material orally and in writing.

**Project 4**: Continuous production of solid metal foams | Peter Stewart

Porous metallic solids, or solid metal foams, are extremely useful in many engineering applications, as they can be manufactured to be strong yet exceedingly lightweight. However, industrial processing methods for manufacturing such foams are problematic and unreliable, and it is not currently possible to control the porosity distribution of the final product *a priori*.

In this project the student will consider an entirely new method of solid foam production, where bubbles of gas are introduced continuously into a molten metal flowing through a heat exchanger; foaming and solidification can then occur almost simultaneously, allowing the foam structure to be controlled pointwise. The aim of the project is to construct a simple mathematical model for a gas bubble moving in liquid filled channel ahead of a solidification front, to predict optimal conditions whereby the gas bubble is drawn toward the phase boundary, hence forming a porous solid.

This project will require some background in fluid mechanics and a combination of analytical and numerical techniques for solving partial differential equations.

**Project 5**: Stochastic and continuum models of tumours | Helen Byrne, Philip Maini, Fabian Spill

Cancer is a complex disease, involving many different processes, including uncontrolled cell growth and, frequently, invasion to other parts of the body. While its complexity makes cancer challenging to describe using mathematics, several different approaches have been proposed. These include deterministic models, formulated as systems of differential equations and stochastic models that account for random events, such as mutations, random cell death or random movement of cells.

The aim of this project is to develop stochastic models of tumour growth and derive from them deterministic equations which describe the evolution of their mean values and higher moments. In this way we aim to identify situations in which the deterministic, mean field equations provide an accurate description of the original, stochastic models. Consider, for example, a situation in which tumour cells have a higher probability of proliferating than dying. On average, we might expect that the number of tumour cells will increase with time. However, when the number of cells is small, there is a nonzero probability of extinction due to random cell death and we conclude that the deterministic description is likely to be accurate only for large cell populations.

The ideal candidate for this project should have experience of analysing partial and/or stochastic differential equations and enthusiasm for helping to develop and analyse new models of tumour growth. Knowledge of mathematical biology and/or master equations is desirable but not essential. For informal enquiries please contact Helen Byrne (byrneh@maths.ox.ac.uk), Philip Maini (maini@maths.ox.ac.uk) or Fabian Spill (spill@maths.ox.ac.uk).

**Project 6**: Extending the life of electric motors | Ian Hewitt, Cameron Hall, Colin Please

The electric motors that are used to power cars degrade over time and have to be replaced. To improve the performance and viability of electric cars, the manufacturers would like to extend the lifetime of the motors. Degradation is due in part to the repeated expansion and contraction of the wire coils in the motor when it heats and cools. It is desired to calculate the stresses and strains that occur as a result of differential expansion of the copper wires and their polymer casing. This would provide insights into where abrasion and eventual failure occur, and suggest design strategies to minimize the damage. The results of this work may be used to support activity in the Department of Engineering Science.

This project will use linear elasticity and homogenization theory to study the maximum stress that results when a confined array of wires expands due to heating. The project will require some background in partial differential equations and continuum mechanics, and an interest in learning about applications of mathematics to material science. It will involve a combination of analytical work and scientific computing.

**Project 7**: Bivariate trending O-U | Terry Lyons

Through the Mathematical Institutes links to the Oxford-Man Institute of Quantitative Finance and their co-location with the hedge fund Man Group, we have been able to develop a project opportunity that has both academic and practitioner relevance. The project will be to review the Lo and McKinlay Bivariate Trending O-U paper and process in light of current research influences; using maximum likelihood, fit the model to futures data and confirm good fit to underlying correlation term structure; and if time allows, derive the optimal portfolio policy under a given utility for this process.

Tasks include: Discussion of the O-U trending process (construction, parameters, black-scholes formula for this process, limitations for empirical applications, inconsistency with the empirical observations - returns from assets can be positively autocorrelated); Analysis of the first order autocorrelation of returns from the stock as a function of holding time(different behaviour if we change the parameters of detrended log-price process and time-varying expected return factor); Estimating coefficients of this model using real data; Utilise this model to option pricing and compare with BSThink of the hedging strategies.

Supervision will be provided by Prof Terry Lyons, Director, Oxford-Man Institute (also Mathematical Institute) and Dr Remy Cottet, Senior Research Analyst, Man Research Laboratory (co-located with OMI), with some additional input by our Deputy Director, Marek Musiela, who has an interest in this field. The relationship between the Man and OMI has enabled us to identify a project that has potential interest to both academics and practitioners and as such Dr Cottet is happy to provide supervision and guidance during the project term.

**Project 8**: Cloud modelling | Andrew Fowler, Ian Hewitt

In numerical models of weather prediction and future climate prediction, one of the least well understood ingredients is the behaviour and prediction of clouds. Clouds have a double effect on climate and weather. A thick cloud layer leads to increased albedo (i.e., planetary reflectivity), and this causes increased reflection of received solar short wave radiation, and causes cooling. On the other hand, clouds also enhance the greenhouse effect, whereby long wave radiation emitted by the Earth is trapped and re-radiated downwards: this causes warming, as is easily observed in the evening (clear skies give colder temperatures at night).

In predicting future climate due to carbon change in the atmosphere, the parameterisation of the effects of cloud cover is of central importance. But the problem is of supreme difficulty, because cloud cover varies on a spatial scale which is typically much less than the grid scale of numerical weather prediction models.

This project will consider very simple models to describe varying cloud cover in a vertical column of atmosphere. The model will include the effects of updraughts and downdraughts, moisture content of the air, and the consequent formation of clouds in conditions of supersaturation. The model will consist of reaction-diffusion type partial differential equations, and a principal aim of the project will be to gain an understanding of cloud physics, construct a mathematical model, non-dimensionalise and simplify it, and provide a numerical solution. A background in differential equations is necessary, and some understanding of fluid mechanics is useful but not essential.