Pre-2015 Project Proposals

Producing a discrete approximation to a continuous problem is a common task in computer modelling. When applying the chosen approximation technique, the target model of computation (e.g., an imperative program, such as in Fortran, Matlab, R, etc.) influences the solution structure. In the imperative case, the solution may eventually become a series of mutations on an array. We are working on programming languages which can describe efficient and safe numerical procedures more abstractly in higher-level, non-imperative form. The potential benefits of this are to produce code which is easier to understand, reason about, and can exploit different execution environments (such as parallel architectures).

We are looking for a PMC student with an interest in analysis. The project will involve characterising key techniques and approaches for deriving numerical approximations to dynamical systems models (e.g., Navier-Stokes). We would like to develop appropriate approximations in high-level forms which are still amenable to useful optimisation by the compiler. The student will have the opportunity to learn about the programming language research side of the project, and will assist the design and evaluation process.

Project 1: Numerical Simulation of Flapping Flight

The experimental and numerical investigation of animal flight is an active field of research. Moreover, there is an increasing interest in micro air vehicles designed as ornithopters. Nevertheless, the aerodynamic phenomena of moving wing structures at intermediate Reynolds numbers are poorly understood and require more in-depth studies.

The successful applicant will be contributing towards the development of software for the simulation of airflow over two- and three-dimensional wing structures. Good programming skills (C/C++) and familiarity with numerical techniques beneficial.

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Project 2: Parameter identification and optimisation of flexible wing structures

The optimal design of micro air vehicles, which are based on flapping flight locomotion, is an open question and still under active investigation. Part of such studies is the numerical analysis of the structural behaviour of flexible wings. But numerical simulations typically require well-defined input data which describe the computational model. Unfortunately, even in the simple case of elastic shell models, necessary data such as the material stiffness and thickness will only be known to lie within a wide range, if they are available at all.

Given the results of experimental studies, the aim of this project is to calibrate the computational model. Optimisation techniques will be used in order to achieve a best fit with the reference behaviour of the wing structures. A good understanding of numerical optimisation techniques and certain level of proficiency with MATLAB or similar software tools beneficial

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Other possibilities might exist to work on projects involving differential geometric variational formulations of nonlinear elasticity and/or applications of Gamma convergence to problems in computational mechanics. (Contact Cyrus Mostajeran (csm54@cam.ac.uk) for further details.)

When elastic, rubber, conical shells are point loaded as shown in Figure 1(in attached), these shells may form folds as shown in Figure 2. When the shells fold as shown in Figure 2, they display 3 interesting phenomena.

This PMC project would involve investigating one of these phemonema: Why do the folds form nonaxisymmetrically? Mahadevan et. al first observed the formation of these ridges in point indented elastic shells of positive Gaussian curvature.(Vaziri & Mahadevan, 2008). Nasto et. al. further examined the formation of these ridges in spherical shells and suggested that these ridges form due to a local instability. They do not give a general predictive model but suggest that geometric frustration as described by (Dias, Dudte, Mahadevan, & Santangelo, 2012) might underpin the formation of non-axisymmetrical folds.

The student will perform indentation experiments and compare stiffening effects with other well documented stiffening(Lazarus, Florijn, & Reis, 2012). The student can also further explore whether existing models such as the Love-Kirchoff and Mindlin-Reissner can be extended to apply to describe these nonlinear phenomena.

Dias, M. A., Dudte, L. H., Mahadevan, L., & Santangelo, C. D. (2012). Geometric Mechanics of Curved Crease Origami, (1), 1–5.
Lazarus, a., Florijn, H. C. B., & Reis, P. M. (2012). Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells. Physical Review Letters, 109(14), 144301. doi:10.1103/PhysRevLett.109.144301
Nasto, A., Ajdari, A., Lazarus, A., Vaziri, A., & Reis, P. M. (2013). Localization of deformation in thin shells under indentation. Soft Matter, 9(29), 6796. doi:10.1039/c3sm50279a
Reissner, E. (2014). A Note on Membrane and Bending Stresses in Spherical Shells. Journal of the Society for Industrial and Applied Mathematics, 4(4), 230–240.
Vaziri, A., & Mahadevan, L. (2008). Localized and extended deformations of elastic shells. Proceedings of the National Academy of Sciences of the United States of America, 105(23), 7913–8. doi:10.1073/pnas.0707364105

This project will look at developing an algorithm quickly deriving collision free paths for articulated vehicles. This is difficult because the configuration of the vehicle at a given point depends on the path history as well as the point along the path.

This will build on the work done by Tim Hennock last summer. The project ran for six weeks and produced some useful outputs and lots of ideas for future work. There is a lot of scope for the student to guide the project themselves, as there are many ways this could be approached.

See attached pdf for more info.

Path Planning PMC Proposal

I am working on various mathematical models towards a robust re-definition of musical metre and would welcome a trained mathematician's view on the results. The length of the project will depend on how much room for improvement the mathematician identifies.

There is also the possibility of working together on new subsidiary projects including the analysis of metrical information in a corpus of musical works.

Many microscopic algae have evolved to swim and bias their motion in response to useful environmental stimuli. They can actively track light (phototaxis), which they need to grow. Passive mechanisms also orient cells: gravity causes bottom-heavy algae to swim up (gravitaxis). In flows algae swim at an angle dependent on the balance between gravitational and viscous torques on a cell (gyrotaxis). Thus, beautiful ‘plumes’ of cells emerge in down-welling flows as a result of the bias on individual swimmers.

The basic mechanism for gyrotaxis was proposed by Kessler in the 80s, but only recently has a stochastic theory of biased swimming been derived allowing to quantitatively predict suspension behaviour [1, 2]. The theory has recently been applied to show how the dispersion of gyrotactic swimmers in pipes and channels is very different from that of passive tracers (such as colloids or dissolved chemicals) [2]. It is currently being tested in our lab. Theoretically, unsatisfactory aspects of the current framework remain to be resolved by improved models (e.g. the width of a plume is currently predicted to decrease without limit to zero with increasing flow). Further, theoretical treatments so far neglect a biologically critical fact: algae grow! This neglect is acceptable for timescales smaller than the cell division time, but needs to be considered to predict the growth of swimming algae in the environment or in bioreactors.

This project will lay the foundations of improved models of gyrotactic populations of algae swimming in photobioreactors of different geometries. For example, swimming in fluid flows within a loop photobioreactor will be analysed. This reactor is a device to grow algae that consists of two vertical tubes, one bubbled providing the CO2 the algae need to grow and driving rising flow, the other with a steady down-flow. A part III project is running exploring the problem of coupling the swimmer biofluid dynamics (using the current models) and growth from a computational view point (direct numerical simulation, individual based models). The PMC summer project will aim to improve the swimmer biofluid dynamics and extend the description of growth. The approach will be analytical/numerical (cell and nutrient transport PDEs solved numerically) and, depending on the student’s skills, also computational (extending the part III simulations).

Experiments to investigate the growth of the gyrotactic alga Chlamydomonas reinhardtii are ongoing at DAMTP in collaboration with Kyriacos Leptos and Ray Goldstein and Alison Smith in Plant Sciences. It will be exciting to test any new model predictions resulting from the project experimentally. There will also be the chance to work with Dr Croze’s collaborators by Dr Yongyun Hwang (Imperial college) and Prof Tim Pedley (DAMTP). They have been developing some new models of the effect of flow in pipes on instabilities in suspensions of swimmers [3].

[1] R. N. Bearon, M. A. Bees & O. A. Croze Phys. Fluids 24, 121902 (2012) http://dx.doi.org/10.1063/1.4772189
[2] O. A. Croze, G. Sardina, M. Ahmed, M. A. Bees & L. Brandt J. R. Soc. Interface 10 20121041 (2013) http://dx.doi.org/10.1098/rsif.2012.1041
[3] Y. Hwang and T. J. Pedley J. of Fluid Mech., 738, 522-562 (2014) http://dx.doi.org/10.1017/jfm.2013.604

The Square Kilometre Array is envisaged to be the world’s largest and most sensitive radio telescope operating from 50 MHz to 10 GHz using multiple innovative collector technologies [1]. In the low frequency regime (50 – 650 MHz), the collectors will consist of millions of phased array antennas not so dissimilar to TV aerials. These are clumped into arrays of roughly 250 elements and are then combined using different amplitude and phase weighting to form a beam on the sky (beamforming). One of the most important aspects of the low frequency telescope design is the location of these antennas on the ground since this has a major effect on the shape of the beam, primarily its beamwidth and side-lobe profile.

Over the past decades many studies have been carried out to optimise the position of the antennas mainly with respect to reducing or maintaining a certain side-lobe level or profile. Recently a study was carried out to optimise the array response with respect to multiple parameters including but not limited to side-lobe level and beamwidth [2]. Here a project is proposed to continue the study of array optimisation using other methods. There will be various software available for use in this project including the code used in [2], Xarray [3] and OSKAR [4] which can be used to validate the results of the configuration design in an interferometric context. The student will be required to use MATLAB for this optimisation.

[1] www.skatelescope.org [2] T. Clavier, N. Razavi-Ghods, et. al., A Global-Local Synthesis Approach for Large Non-regular Arrays, IEEE Transactions on Antennas and Propagation (http://www.mrao.cam.ac.uk/~nima/new/ [3] sites.google.com/site/xarraytool [4] www.oerc.ox.ac.uk/~ska/oskar2

Mapping 3D microtubule dynamics in early flower morphogenesis.

Lisa Willis, Ray Wightman, Elliot Meyerowitz, Henrik Jonsson

In plants as in animals, microtubules play fundamental roles both in cell division, when they accurately redistribute newly synthesized chromosomes between the two daughter cells, and in shuttling material about the cell. But unlike in animals, microtubules in plants are thought to play a further fundamental role in directing morphogenesis: in certain zones of tissue, microtubules can be seen to bundle and orient in particular directions and it is this orientation which is thought to determine how a cell grows to acquire its specific shape and so ultimately to determine the shape of a plant organ (see [1] for a brief review). The tensorial signal which causes microtubules to align in a particular direction, and how precisely microtubules respond to this signal, are major questions in plant research. The main hypothesis is that microtubules align in the direction of maximum mechanical tension, which is set up by the tissue's local geometry and anisotropy of its structural materials such as cellulose.

This project is to collect experimental data on the dynamic reorientations of microtubules in 3D during the earliest stages of flower morphogenesis and to correlate this data with the distribution of a second protein, PIN1, that also plays an important role in morphogenesis [2]. The student would be trained to: 1. grow plants and dissect young floral organs; 2. use our top of the range confocal microscope to collect 3D time-lapse images of microtubules and PIN1 during the first 4 days of flower development; 3. use appropriate image processing software for analysis. This will take a few weeks, while the student will work closely with a postdoc to collect the first data sets. Then the student may wish to study the effect of chemical perturbations that should alter the distribution of mechanical stresses across the tissue (e.g. colchicine which depolymerises microtubules, and arabinase which stiffens cell walls). Or, instead, there is the option to participate in the development of 3D computational models of plant tissue mechanics for comparison with experimental data.

[1] Plant Morphogenesis: a role for mechanical signals. Dumais, J. Current Biology 19(5): 207--208 [2] Alignment between PIN1 polarity and microtubule orientation in the shoot apical meristem reveals a tight coupling between morphogenesis and auxin transport. Heisler, M.G., Hamant, O., Krupinski, P., Uyttewaal, M., Ohno, C., Jonsson, H., Traas, J. & Meyerowitz, E.M. PLoS Biology 2010 19;8(10):e1000516.

Our lab works on the genetics and evolution of transmissible cancers, which are cancers that can be transmitted by touch. There are only two known naturally occurring transmissible cancers and these are the Tasmanian devil facial tumour disease (DFTD, affecting Tasmanian devils, and the canine transmissible venereal tumour (CTVT), affecting dogs. These two cancers are transmitted between individuals by the transfer of living cancer cells. The project will focus on the genetics of the two cancers, through computational analysis of sequence data. This will allow possible identification of differences between genomes and other unexpected rearrangements. Main focus of the project is bioinformatics.

The problem we are interested in is that of classifying higher order umbilics on surfaces. Simple (cubic) umbilics are well understood, but we have a conjecture that higher order umbilics can be treated as the limit of collections of simple ones as they converge towards a single point. The formal aim of the project is to prove or disprove this conjecture, but there is scope for exploring the small print, which we suspect might exist.

Our initial motivation was to extract a sparse quad mesh from a given dense triangulation by using umbilics as feature points and ridges as feature lines, as they join umbilics (or form closed loops). In order to understand the discrete case on meshes, we first need to understand the situation in the smooth case on differentiable surfaces. There is thus also further scope for extending the continuous case results to the discrete case.

Many social, economic and financial processes have come to have the character of potential epidemics. A simple example is the adoption of technological, operational or policy innovations among a population of firms. The epidemiological aspect of diffusion was brought to economics and marketing in the late Sixties. In the simplest diffusion model the adoption of an innovation by any customer makes adoption by other customers more likely, through infective word-of-mouth spread of information. The original model has a monopolistic setting and suppresses competition. More recently a literature has developed that generalizes diffusion in competitive settings, focussing on determining market share trajectories of competing firms.

In the market context the notion of tipping applies to the emergence of persistent dominance of one among many, in a context that is competitive and symmetric, a-priori. Examples include the evolution of market shares, geographic agglomeration, and the diffusion of innovations, standards and ideas. If the reinforcement process is strong, markets can get locked-in to inefficient end states due to early random events and history dependence. A classic example is the battle between QWERTY and DVORAK keyboard formats which ended with the market locked into the inefficient QWERTY configuration. Other often mentioned examples include the battle between operating systems, and digital formats. Some examples above have been explained in terms of non-linear positive feedback driving an adoption processes.

The issues around diffusion have become very important in Economics and Management. Building on prior work, the objective of this project is use Generalised Polya urn models as the basic building blocks to develop models of processes leading to various degrees of (eventual) dominance in contexts where innovations (firms) compete with each other. A number of applications across economic, financial and social processes follow.

The goal of this project is to develop, to test and to analyse different non-linear reconstruction
strategies for a specific MRI application in chemical engineering, and the
effect of different parameter choices (e.g. different initial values) on the
solution.

Disasters uproot the social infrastructure of a community. Relief efforts exacerbate the problem by providing emergency shelters where one’s identity is not acknowledged, where one is expected to be a stranger. What is lacking is the capacity to preserve existing social networks and draw upon the skill sets, histories and dispositions of individuals to constitute engines for positive, self-generated community involvement. Research is needed into how relief organizations can empower displaced individuals to become active participants in their own disaster recovery.

My research identifies an opportunity to rewire the social infrastructure of displaced communities, not to reflect a pre-disaster state, but in a manner that reimagines what social interactions are best suited for an interim context of living and recovery. This involves bridging two traditionally independent fields of study—evacuation studies (movement of people) and human settlement studies (living conditions)—and demonstrating that they are causally related, that by choreographing people's movement to emergency shelters one can maintain social infrastructure in the wake of physical devastation. The research, as having real-world value, is compelled by a technological catalyst: the use of personal mobile devices to direct human movement and thereby alter the social dynamics and perception of "place” in post-disaster landscapes.

The research will explore the local and systemic benefits/drawbacks, ethical considerations and privacy issues inherent to a new program that perpetuates friendships, kinships and socio-cultural networks at evacuation sites. Along with surveys designed to expose viable social constructs, my research will also demonstrate how other methodologies, such as microscopic traffic simulations, can be integrated into and benefit from a more socially conscious evacuation plan.

I am confronted with a range of problems at the intersection of mathematics and computer science. Shortest-distance diagrams play an important role in evacuations (e.g., the use of weighted, multiplicative Voroni cells to create shortest-distance polygon charts; this is geometry-based mathematics). The same can be said about combinatorial optimization (implementation of the "multiple knapsack problem” to assign demographic subsets of a population to best-fit shelters) and network analysis (understanding the impact of multiple inputs/strategies to simultaneously influence both the "network clearance time” and the possible emergence of demographic specific shelters). I am also using cluster analysis as a strategy to identify shelter location.

Brief Description of Project

This project is about a fascinating feature of fluid dynamics. When a fast moving jet of water strikes a wall, it spreads out radially until a point where it forms a hydraulic jump. Beyond the jump the flow is slow moving. If the jet impinges on a vertical wall, the liquid will form a falling film which can change shape, form braids, split of form dry patches or narrow into several rivulets.

The aim of this project is to develop a model for the flow behaviour of wide falling films, which is desired in cleaning and coating operations. We started this in 2012, where an MPhil student showed that the most popular model in the literature is wrong. Since then, we have come up with a reformulated version of that model and the aim of that project is to compare the new model with experimental results. Additionally, there is ample opportunity to broaden the scope of this modelling work as experimental observations made in the GK Batchelor laboratory in DAMTP revealed a broad spectrum of different flow phenomena. There is also scope within this project to continue the collaboration with the fluids group in DAMTP.

Genome wide association studies have successfully identified many variants associated with common diseases, but the stringent levels of significance required to limit false positives means there are presumed to be many more variants to be discovered. One means of identifying these from existing data is to exploit pleiotropy, and use information that a variant has been associated in one disease when testing for association in another, related, disease or trait. Brad Efron's work on Bayesian empirical false discovery rates (1) has been extended to this context (2), but one of the assumptions made in the method is that the evidence for association with the two diseases at truly null SNPs is independent, which means each disease GWAS must be conducted with distinct sets of controls. A shared control design, which pools the controls into one larger group, is more powerful and commonly employed by groups of researchers studying related diseases, for example, the ImmunoChip consortium (3). There is therefore a need to extend this methodology to the shared control design.

The method of "decoupling” GWAS signals from two scans with distinct controls (4) will be employed to extend the empirical FDR method to the shared control design, and applied to association scans of multiple autoimmune traits based on the ImmunoChip. The placement will offer the chance to learn about the statistics underlying Bayesian empirical FDR methods, and to develop the R code needed to combine them with the decoupling approach. The project requires some experience with R and working in a unix/linux environment, and would suit students with an interest in statistical genetics. We expect this work will lead to the identification of novel autoimmune associated variants, a software package that would enable other researchers to deploy the method in their disease areas, and a publication. The first and second are likely to be achieved within the timeframe of the rotation, with the student's contribution appropriately recognised in a following paper. Support will be offered both with regards to statistical methods and code development.

During the project, the student will be considered a member of Professor John Todd’s Multidisciplinary Diabetes and Inflammation Laboratory, which has been at the forefront of genetic discoveries in type 1 diabetes in recent years. The project will be supervised by Dr Chris Wallace and Oliver Burren, who is responsible for the development of ImmunoBase, a website for integrated display and annotation of results from the ImmunoChip studies.