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Faculty of Mathematics

 

Previous Academic PMC Proposals

This page is an archive of proposed PMC projects from previous years.

Click on each project for more details.


Deriving new programming constructs from numerical analysis techniques

Contact Name Dominic Orchard
Contact CRSid dao29
Lab/Department Computer Laboratory
Address 15 JJ Thomson Avenue
Cambridge
CB3 0FD
Period of the Project 8-10 weeks
Brief Description of Project

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.

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Skills Required Strong numerical analysis skills and good knowledge of commonly used discretisation techniques. Programming skills, enough to implement solution techniques (any language will
suffice).
Skills Desired Interest in programming and programming languages.
Any Other Information  

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Using biological and palaeontological observations to calculate evolutionary history

Contact Name Martin Smith
Contact CRSid ms609
Lab/Department Earth Sciences
Address Downing Street
Cambridge
CB2 3EQ
Period of the Project Eight weeks
Brief Description of Project Historical biology is primarily concerned with the inference of evolutionary history and relationships. Whereas molecular data (such as genetic sequences) can be readily interrogated under a range of models, the reconstruction of phylogenetic trees from morphological data – the mainstay of palaeontologists and conventional taxonomists – has outstanding problems. This project seeks to (i) adapt existing models of morphological evolution to incorporate peculiarities of morphological data sets; (ii) to quantify the quality of input data; and (iii) to assess the level of support for conflicting phylogenetic hypotheses.
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Skills Required The skills required will depend on which component of the project the student chooses to pursue.
Skills Desired It would be advantageous to have familiarity with Markov Chain Monte Carlo methods, Bayesian Inference, or information theory; some level of programming experience would also be useful.
Any Other Information  

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Numerical simulation of flapping flight (+ other projects)

Contact Name Fehmi Cirak
Contact CRSid fc286@cam.ac.uk
Lab/Department Engineering (CSMLab)
Address Department of Engineering
Trumpington Street
Cambridge
Cambridgeshire
CB2 1PZ
Period of the Project July - September
Brief Description of Project

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.)

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Skills Required The required skills depend on the specifics of the selected project. Project 1 requires very good
programming skills and familiarity with C++. Project 2 requires good understanding of numerical analysis and optimisation techniques and a certain level of proficiency with MATLAB. Other projects may require significant knowledge of differential geometry and analysis.
Skills Desired Very good numerical and analytic skills. Passion and genuine interest.
Any Other Information  

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Arathi Ramachandran

Contact Name Arathi Ramachandran
Contact CRSid ar590
Lab/Department AIM Lab/ Materials Science and Metallurgy
Address Department of Materials Science & Metallurgy
University of Cambridge
27 Charles Babbage Road
Cambridgeshire
CB3 0FS
Period of the Project 3 months
Brief Description of Project

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

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Skills Required no special skills
Skills Desired mechanical testing; solid mechanics useful
Any Other Information  

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Path Planning for Articulated Vehicles

Contact Name Amy Rimmer
Contact CRSid ajr80
Lab/Department Engineering
Address BEO-10
Department of Engineering
CB21PZ
Period of the Project Summer 2014
Brief Description of Project

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.

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Path Planning PMC Proposal

Skills Required Project will involve MATLAB but anyone with some programming experience should be fine. No specific Maths skills required.
Skills Desired Familiarity with search algorithms
Any Other Information Any questions, please email Amy Rimmer (ajr80)

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Mathematical Models for a Theory of Musical Metre

Contact Name Mark Gotham
Contact CRSid mrhg2
Lab/Department Centre for Music and Science / Music Faculty
Address 11 West Road
Cambridge
CB3 9DP
Period of the Project Probably short
Brief Description of Project

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.

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Skills Required Broad knowledge of mathematics and ability to identify useful avenues to explore.
Skills Desired Musically literate.
Any Other Information  

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Functions with Cyclic Permutational Invariance

Contact Name Tim Hele
Contact CRSid tjhh2
Lab/Department Theoretical Chemistry
Address University Chemical Laboratory
Lensfield Road
Cambridge
Cambridgeshire
CB2 1EW
Period of the Project 8 weeks (negotiable)
Brief Description of Project Investigating functions which are invariant under cyclic permutation of their arguments, and their associated permutationally variant functions. This is in order to accurately approximate the quantum dynamics of physical systems using classical dynamics, where the observable is invariant under cyclic permutation of the co-ordinates of path-integral beads.
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Skills Required Enthusiasm
Skills Desired Specialism in algebra, algebraic geometry or algebraic number theory. Some knowledge of time-dependent quantum mechanics, Feynman path-integrals, or classical dynamics would be useful but not essential.
Any Other Information Some funding may be available for applications received before March, but the successful applicant will be expected to apply for relevant funding from their college.

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Quantitative analysis of microtubule organisation

Contact Name Daniel St Johnston
Contact CRSid ds139
Lab/Department The Gurdon Institute
Address The Gurdon Institute
Tennis Court Rd
Cambridge
Cambs
CB1 7AD
Period of the Project 8-12 weeks
Brief Description of Project The formation of the main body axis in the fruit fly, Drosophila melanogaster, depends on the formation of a polarised microtubule cytoskeleton in the developing oocyte, which directs the motor-dependent transport of axis determinants to opposite ends of the cell. The goal of the project will be to use image analysis tools to quantify the distribution of microtubule minus ends around the cell cortex and to incorporate these data into a model of microtubule organisation and RNA transport in the oocyte.
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Skills Required Basic analytical and statistical skills
Skills Desired Experience with modelling would be useful.
Any Other Information  

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Biological fluid dynamics of swimming algae in photobioreactors

Contact Name Dr Otti Croze
Contact CRSid oac24
Lab/Department POM/Physics
Address BSS, Cavendish Laboratory, University of Cambridge
JJ Thomson Ave, Cambridge
Cambridgeshire
CB3 0HE
Period of the Project 2 months
Brief Description of Project

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

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Skills Required Analytical modelling, PDEs, solid knowledge of fluid dynamics, Mathematica and/or Matlab
Skills Desired Computational modelling skills, Matlab, knowledge of third generation programming language
(C++, fortran, python, ...), knowledge of mathematical biology and biofluid dynamics
Any Other Information Deadline: 1 May 2014

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Towards A Cell Cycle Phase-Resolved Genomic Atlas of the Functions of the Temporal Program of DNA Replication

Contact Name Philip Zegerman
Contact CRSid paz20
Lab/Department Gurdon Institute
Address Gurdon Institute
Tennis court rd
Cambridge
CB2 1QN
Period of the Project Summer
Brief Description of Project The stable inheritance of the genome in every cell division is a fundamental requirement for all organisms. In the Zegerman lab we are interested in how such stability is achieved. Utilising unique experimental systems in the budding yeast Saccharomyces cerevisiae we can systematically perturb aspects of genome duplication and analyse the subsequent effects on a range of cellular outcomes on a global scale. The integration of these large data sets requires the simultaneous decomposition of multiple matrices or higher order tensors. In collaboration with the mathematician Prof. Orly Alter (University of Utah) we will mathematically define and analytically study several possible frameworks for such simultaneous decomposition of multiple higher-order tensors. These novel decompositions will be used to model all of these multiple genomic datasets simultaneously. The models will test the ability of the novel mathematical frameworks to consistently uncover both the experimental variations and the known underlying biological reality within the data. The models will also be used to computationally predict previously unknown mechanisms for how cells divide.
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Skills Required Understanding of tensor decomposition. Handling of large datasets in the form of matrices.
Skills Desired Some interest in biology would be good.
Any Other Information  

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Pathway simulations of polyketide synthase multienzyme complexes

Contact Name Dr Bill Broadhurst
Contact CRSid rwb1002
Lab/Department Biochemistry
Address 80 Tennis Court Road
Cambridge
CB2 1GA
Period of the Project 8 weeks
Brief Description of Project Type I polyketide synthases are multi-enzyme complexes responsible for assembling a large number of pharmaceutically important natural products. They work as a modular assembly line, with each module fusing a new extender unit to a growing substrate chain and then modifying it. At the moment, we are concentrating on the role of the acyl carrier protein (ACP) domain, which behaves like the conveyor belt in this enzyme factory, shuttling the substrate between several enzyme active sites before passing it on to the next module. I would like to explore using sets of differential equations and/or discrete stochastic modelling to study the function and kinetics of polyketide synthase complexes at the single molecule level. Depending on how far we get during the project, the methods used could range from simple stochastic simulations to full-scale molecular dynamics studies of the motion of ACP domains restricted by flexible polypeptide tethers.
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Skills Required Computational methods or solving multiple sets of differential equations
Skills Desired Stochastic modelling
Any Other Information  

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Optimisation of phased arrays

Contact Name Nima Razavi-Ghods
Contact CRSid nr286
Lab/Department Physics
Address Cavendish AP Group
Thomson Avenue
Cambridge
Cambridgeshire
CB3 0HE
Period of the Project 8 weeks
Brief Description of Project

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

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Skills Required Analytical and mathematical reasoning
Skills Desired Analytical and mathematical reasoning
Any Other Information  

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Mapping 3D microtubule dynamics in early flower morphogenesis

Contact Name Lisa Willis
Contact CRSid lw464@cam.ac.uk
Lab/Department Sainsbury Laboratory
Address Sainsbury Laboratory
Cambridge University
Bateman St
Period of the Project July 2014 - Sep 2014
Brief Description of Project

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.

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Skills Required Basic knowledge of biology and programming in e.g. matlab/python/c++ would be an advantage, but there are no strict prerequisites for this project.
Skills Desired An interest in plant development and a desire to try working in a lab!
Any Other Information  

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Genetic diversity and evolution in transmissible cancers

Contact Name Andrea Strakova
Contact CRSid as2112
Lab/Department Department of Veterinary Medicine
Address Madingley Road
West Cambridge Site
Cambridgeshire
CB3 0ES
Period of the Project Summer 2014 (exact dates to be agreed)
Brief Description of Project

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.

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Skills Required Basic programming skills
Skills Desired Enthusiasm for biological concepts underlying the work, which will mainly involve computational analysis of large amounts of data.
Interest in bioinformatics.
Any Other Information Supervisor - Dr Elizabeth Murchison

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Higher-order umbilics on surfaces and meshes

Contact Name Neil Dodgson, Jiri Kosinka
Contact CRSid nad10, jk520
Lab/Department Computer Laboratory
Address 15 JJ Thomson Avenue
Cambridge
CB3 0FD
Period of the Project About 8 weeks; flexible
Brief Description of Project

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.

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Skills Required Differential geometry
Skills Desired Ability to explain articulately the results found
Any Other Information  

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Random processes with reinforcement in competitive systems

Contact Name Paul Kattuman
Contact CRSid pak13
Lab/Department Judge Business School
Address Judge Business School University of Cambridge
Trumpington Street
Cambridge
CB2 1AG
Period of the Project mid-June to mid-August (flexible)
Brief Description of Project

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.

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Skills Required Interest in modelling economic/financial/social processes (applications of probability theory)
Skills Desired Interest in generalised urn schemes
Programming (Matlab or R)
Any Other Information  

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Differential Geometry in Multidisciplinary Design Optimization

Contact Name Craig Bakker
Contact CRSid ckrb2
Lab/Department Engineering Design Centre
Address Department of Engineering
Trumpington Street
Cambridge
Cambridgeshire
CB2 1PZ
Period of the Project  
Brief Description of Project We are currently using a differential geometry framework to investigate the coupled optimization problems found in Multidisciplinary Design Optimization (MDO) and to develop solution methods for those optimization problems. There are multiple projects that could be undertaken by PMC students depending on the number and interests of those students. One particular project is related to a PDE-based algorithm which we are developing for solving multi-objective optimization problems: we are looking for help in putting together a convergence proof for the algorithm. Another project involves integrating time-varying optimization (optimal control) problems with the current time-invariant results and formulation. Other potential projects include investigating uncertainty propagation in MDO, implementing ODE-based optimization methods in MDO, and relating structural descriptions of optimization problems (e.g. through graph representations) to local information (e.g. derivatives and manifold curvature). I encourage any interested students to contact me (ckrb2) if they want more information about any of these projects, have ideas regarding other potential intersections of optimization and differential geometry, or would like to know more about the research in general.
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Skills Required Differential Geometry
Real Analysis and Advanced Calculus
Nonlinear Optimization
Skills Desired The desired skills will vary with the particular project
Subject Areas: ODEs, PDEs, statistics and probability, graph theory, numerical
analysis
Some programming experience may be helpful - preferably Matlab, but possibly
Mathematica or Maple, too
Any Other Information There are multiple projects available, and we would be happy to take on multiple students if the interest is there.

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Resource pooling with discretionary service times – mathematical analysis

Contact Name Dr Houyuan Jiang
Contact CRSid hj231
Lab/Department Judge Business School
Address Judge Business School
University of Cambridge
Trumpington Street
Cambridge
CB2 1AG
Period of the Project July-September (flexible)
Brief Description of Project The goal of this project is to compare the performance measures of non-pooling and pooling systems with discretionary service times. Many research questions remain to be answered. Firstly, we can systematically and mathematically compare non-pooling and pooling systems based on some classical queueing models such as M/M/s, M/D/s, G/G/s, and Jackson networks. Secondly, we investigate if different performance measures such as the average waiting time and average number of customers waiting in the queue have the same directional changes. Thirdly, we can extend the above queueing analysis to queueing design and queueing control problems. It is envisaged that many more research questions will arise during the project period. See the uploaded file for more detail.
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Skills Required Queueing Theory, Mathematical Proofs
Skills Desired Stochastic Modelling
Any Other Information Please contact Houyuan Jiang for a meeting or a phone conversation.

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Formalised Mathematics

Contact Name Lawrence Paulson
Contact CRSid lp15
Lab/Department Computer Laboratory
Address 15 JJ Thomson Avenue
Cambridge
CB3 0FD
Period of the Project 8 weeks starting mid-June
Brief Description of Project The formalisation of mathematical theorems (chosen jointly by the candidate and the supervisor) using the proof assistant Isabelle/HOL
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Skills Required Some knowledge of the principles of formal logic.
Skills Desired Experience with an interactive theorem prover, such as Isabelle, Coq or HOL4.
Any Other Information  

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Nonlinear Magnetic Resonance Imaging in Chemical Engineering

Contact Name Martin Benning
Contact CRSid mb941
Lab/Department Magnetic Resonance Research Centre
Address c/o Cavendish Stores
JJ Thomson Avenue
Cambridge
Cambridgeshire
CB3 0HE
Period of the Project  
Brief Description of Project Many modern MRI applications can be modeled as non-linear optimization problems, as for instance phase-encoded velocity imaging, or simultaneous MRI and auto-calibration of magnetic field inhomogeneities. However, many challenges arise when solving non-linear instead of linear optimization problems. In most cases, reconstruction algorithms cannot be guaranteed to find a unique, optimal solution. Therefore, analysing the impact of different initial values on the solution is crucial. Further, non-linear problems can be linearised in numerous different ways, leading to different algorithmic strategies with various different properties. Comparing these approaches is necessary to find a suitable strategy in order to tackle a specific application.

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.

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Skills Required Numerical Analysis,
Programming skills in Matlab
Skills Desired Inverse Problems,
(Convex) Optimization,
Image Processing - Variational and PDE Methods,
(Compressed Sensing, Fluid Dynamics)
Any Other Information It would be ideal if the applicant(s) can find time and funding for joining a 10 week project (it is however not necessary that the project takes place within 10 consecutive
weeks). The subject matter is relatively complex and therefore requires some
time to learn the ropes, before getting started with the actual project.

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Social Dynamics at Evacuation Sites

Contact Name Nathaniel Douglass
Contact CRSid nkd26
Lab/Department Geography
Address Clare Hall
Herschel Road
Cambridge
Cambridgeshire
CB3 9AL
Period of the Project  
Brief Description of Project

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.

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Skills Required

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.

Skills Desired While far from necessary, experience with Python could be useful. I have found several Python
scripts helpful in beginning to solve the more difficult problems.
Any Other Information Some of the strategies are quite unique (particularly those involving geometry); if you are
interested in publishing, there might be an opportunity to do so. Also, my
adviser is an expert in agent-based modeling, and I will be working on several
agent-based simulations over the summer—if you are interested in acquiring this
skill set, you are more than welcome to participate.

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Surface Diffusion and Open Quantum Systems

Contact Name Peter Townsend
Contact CRSid psmt2
Lab/Department Cavendish Lab, SMF Group
Address J J Thomson Avenue
Cambridge
Cambs
CB3 0HE
Period of the Project 8 weeks over summer
Brief Description of Project The aim of this project is to model the dynamics of an atom adsorbed (adatom) on a metal surface, in situations where classical mechanics is expected to fail. The proposed method is to treat the adatom as a small system in contact with a large reservoir (the metal crystal). The problem then falls into the domain of the theory of open quantum systems. The challenge is to characterize the reservoir in the right way. Classically, a common approach is to add stochastic and dissipative terms into the equation of motion for the particle, as in the generalised Langevin equation. We would like to work out the analogous quantum picture, using a realistic model of the electrons and nuclei in the metal rather than an abstract reservoir. In particular, we would like to compare the result of the theoretical investigation with data taken on the Cambridge surface spin-echo spectrometer, which is unique in being able to measure equilibrium surface dynamics on picosecond timescales.
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Skills Required Quantum Physics
Skills Desired Quantum Information Theory
Quantum Field Theory
Symbolic/numerical programming
Any Other Information  

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Flow patterns created by impinging jets

Contact Name Dr Bart Hallmark
Contact CRSid bh206
Lab/Department Chemical Engineering and Biotechnology
Address New Museums Site
Pembroke St
Cambridge
Cambridgeshire
CB2 3RA
Period of the Project TBC

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.

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Skills Required Interest in fluid mechanics
Skills Desired Potential requirement for numerical analysis
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Identification of Pleitrophic Autoimmune Disease Associated SNPs Using a Bayesian Empirical FDR Approach

Contact Name Mary Fortune
Contact CRSid mdf34
Lab/Department Cambridge Institute for Medical Research
Address University of Cambridge
Cambridge Biomedical Campus
Wellcome Trust/MRC Building
CB2 0XY
Period of the Project  
Brief Description of Project

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.

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Skills Required Experience in R.
Experience with working in a unix/linux environment.
Skills Desired Interest in statistical genetics.
Any Other Information  

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