Applied Mathematics MSc Projects 2015–2016

Imperial College London

Posted 1 December, 2015, Revised December 2, 2015

Projects with Prof Mauricio Barahona

General topics: Dynamics and graph theory. Network analysis. Stochastic processes on graphs.

Optimization. Dimensionality reduction, geometric projections for high-dimensional data. Com-

munity detection on graphs

Areas of application: Social networks, financial and economic data, coarse-graining and segmenta-

tion of images, bioinformatics

Some examples of possible projects:

1. Theory of graph-theoretical data analysis: a series of possible projects on the conceptual and

mathematical extensions of techniques for the representation of data as graphs, and the coarse-

graining of such representations. These include topics such as:

* time-varying networks and their partitions: detecting break points using multidimensional algo-

rithms

* development of notions of robustness for multiscale graph partitions: node and edge deletion,

statistical predictability and bootstraps

* optimal sparsification of graphs that preserve structural and spectral properties

* anomaly detection using graph-theoretical notions of manifold reconstruction

2. Finding roles and communities in directed networks based on flow profiles: Application to

Twitter networks (to find roles in information propagation. This project consists of the analysis of

networks constructed from a large database of ‘tweets’, collected over a period of more than a year.

We will construct sequences of networks (e.g., ‘retweet’, word adjacencies, followers) encompassing

small periods of time and analyse their features as they change in time. The methods we use rely

on concepts from graph theory, dynamical systems, and stochastic processes.

Partly in collaboration with: colleagues at Oxford, and at the Big Data Analysis Unit at Imperial.

3. Nonlinear dimensionality reduction for high-dimensional data: Application of recently developed

techniques in our group for the analysis of high dimensional data using graph theoretical techniques

linked to geometric constructions, as well as the use of diffusion dynamics on graphs for community

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detection. Application to:

(a) transcriptomics profiles of cellular responses to chemical compounds that can originate cancer,

(b) single-cell RNA profiles of cell types related to development and stem cells,

(c) analysis of behavioral time-series of motion of C elegans: Over 10,000 videos of freely mov-

ing C. elegans, a nematode worm, with sufficient resolution will be analysed to obtain reduced

representations of complex postural times series. What is the dimensionality of motional behavior?

Partly in collaboration with: Syngenta (a), the Sanger Institute at Cambridge (b), Dr Andre Brown

Imperial MRC Clinical Sciences Centre (c).

4. Coarse-graining and segmentation of images: applying community detection techniques to graphs

derived from images will be used to address problems in medical imaging and segmentation, and/or

stylistic changes in paintings throughout Art History.

Partly in collaboration with: Daniel Rueckert in Computing, and the Data Science Institute at

Imperial.

Stochastic dynamics of structured cell populations – Dr Philipp Thomas and Prof

Mauricio Barahona

Individual cells of a population can usually be distinguished by several dynamical features, such as

size, cell cycle stage or their protein content. Based on these features cells make decisions whether

to grow or divide which can be described as a branching process similar to those encountered in

ecology. The mathematical description of such structured populations differs from those of Markov

chains in that the state of the system is not characterized by the number of cells alone but by a

function describing the number of cells with a certain feature. As a consequence the probability of

observing the population in a certain state is a functional for which a functional master equation

can be written down.

This project will use simple models to investigate the solutions to these equations. A possible

starting point could be to derive moment equations. Unlike for Markov chain models, where the

moment dynamics satisfy ordinary differential equations, the moments of structured population

models are governed by partial differential equations. For simple systems these may be solved

analytically using the method of characteristics, in all other cases we will make use of perturbative

and/or numerical methods to extract the desired statistical information. Some basic knowledge of

stochastic processes is advantageous but not necessary.

Geometry of phases in quantum spin systems –Dr. Sania Jevtic and Dr. Ryan Barnett

Quantum spin systems exhibit a variety of interesting phenomena. When they are represented

by ground states of spin-lattice Hamiltonians, the presence of entanglement can lead to quantum

phase transitions occurring at zero temperature. In some cases it is possible to visualise these phase

transitions by plotting expectation values of terms that appear in the Hamiltonian [1].

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In this project, we will work towards constructing new Hamiltonians and depicting their ground-

state properties. Our Hamiltonians will be for a system of interacting spin-1/2 particles (quantum

systems living in a 2-dimensional Hilbert space) with interactions occurring between at most two

nearest-neighbour particles. Many interesting Hamiltonians are of this form (for instance those

appearing in the Ising and Heisenberg models). In this case, one can view these interaction terms

as arising from observables for a single spin-1 system (living in a 3-dimensional Hilbert space).

This spin-1 particle state can be nicely visualised when we draw different 3-dimensional projections

of the real (8-dimensional) vector that describes the state. These projections can form a variety

of convex shapes: a cone, a sphere, even an “obese tetrahedron” [2]. The central idea then is to

convert these 3-dimensional objects representing the single spin-1 particle into observables for a

pair of spin-1/2 systems (there is a well-defined way of achieving this). From this we obtain a new

Hamiltonian for the spin-1/2 particles.

Now we can use our knowledge of the spin-1 system to say something about the ground states of

this new Hamiltonian: they should be the states that correspond to the extremal points of the

convex shapes (the cones or spheres, etc). From this, one can investigate entanglement properties

and various other features of the spin-1/2 system.

If time permits, we will study the thermodynamic limit, that is, consider the Hamiltonians that

we construct as acting on an infinite number of spin-1/2 systems. This will allow us the use of

efficient numerical techniques to calculate the ground state energies of these Hamiltonians, and

thus investigate phase transitions.

References

[1] V. Zauner and D. Draxler and Y. Lee and L. Vanderstraeten and J. Haegeman, and F. Ver-

straete, “Symmetry Breaking and the Geometry of Reduced Density Matrices”, arXiv:1412.7642

[quant-ph]

[2] S. K. Goyal and B. N. Simon and R. Singh and S. Simon, “Geometry of the generalized Bloch

sphere for qutrit”, arXiv:1111.4427 [quant-ph]

Topological edge states in synthetic dimensions – Dr Ryan Barnett

Topological insulators are materials which have insulating interiors but conducting edge states [1].

As the existence of such edge states is due to a non-trivial topological invariant, they are robust

with respect to disorder.

There recently has been a large-scale effort to realise analogues of topological insulators with ul-

tracold atoms. For this case, atoms are to be thought of as the electrons in the material, while

standing light waves replace the ionic crystal lattice comprising the solid. About two months ago,

the first realisation of topological edge states in an atomic gas was reported [2]. The interpretation,

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however, had a curious twist: the internal spin degrees of freedom of the atoms were thought of as

an extra “synthetic” spatial dimension.

In this project, we will analyse this experiment using field-theoretical methods. We will then

investigate particular variations of the protocol with an aim to realise other systems of interest like

the so-called Weyl semimetal.

[1] X.-L. Qi and S.-C. Zhang, Physics Today (January 2010).

[2] B. K. Stuhl et al., Science 349, 1514 (2015)

Projects with John W. Barrett

(1) Numerical Approximation of a Differential Equation on a Moving Curve

This project will consider the finite element approximation of some simple differential equations

on a moving curve. At first in the case when the motion of the curve is given a priori. Then to

problems, where the motion of the curve and the solution of the differential equation are coupled.

A simple model case of the latter arises in diffusion induced grain boundary motion:

V = κ+ u2 and − uss + V u = 0 on Γ(t).

Here Γ(t) is the evolving curve (grain boundary) with normal velocity V and curvature κ, s denotes

arclength and u is the vapour concentration.

For an overview of this area, see the review article [1].

(2) Finite Element Approximation ofthe p−Laplacian as “p→∞”

The limit of the p−Laplacian

−∇ · (|∇u|p−2∇u) = f in Ω, u = g on ∂Ω

as “p → ∞” leads to a critical state problem. These types of problems arise in the mathematical

modelling of sandpiles and superconductors, see e.g. [2], and are of variational type. We will

consider the finite element approximation of such problems. In addition, we wish to compare the

above to the ∞−Laplacian, which is not of variational type, and its numerical approximation, see

e.g. [3].

Prerequisites

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These projects will involve analysis and computation (e.g. Matlab).

Essential Course: e.g. M5MA47 Finite Elements: Numerical Analysis and Implementation. De-

sirable Courses: e.g. M5M8 Advanced Topics in PDEs, M5M9 Applied Functional Analysis, and

M5SC Scientific Computing.

References

[1] K. Deckelnick, G. Dziuk and C. M. Elliott Computation of geometric partial differential equations

and mean curvature flow, Acta Numerica, (2005), 139–232.

[2] J. W. Barrett and L. Prigozhin, Sandpiles and superconductors: nonconforming linear finite

element approximations for mixed formulations of quasi-variational inequalities, IMA J. Numer.

Anal., 35, (2015), 1–38.

[3] A. M. Oberman Finite difference methods for the infinity Laplace and p−Laplace equations, J.

Comp. Appl. Math., 254, (2013), 65–80.

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Projects with Dr Pavel Berloff

Applied Mathematics MSc Projects

Response of Simple Flows to Localized Transient Forcing

Oscillating forcing in the ocean can represent nonlinear effect of transient synoptic eddies on the large-scale circula-tion. Such a forcing excites propagating planetary waves that partially rectify into steady large-scale currents (Haidvogeland Rhines 1983). Effect of the mean background flows on this rectification process remains to be understood, and theoutcome can provide guidance for emerging stochastic parameterizations of the oceanic eddies. Recent progress in thisdirection was achieved in terms of the formal theory for the transient forcing “footprints” (Berloff 2015), but extension ofthis theory beyond simple zonal flows is the main subject of this Project.

Berloff, P., 2015: Dynamically consistent parameterization of mesoscale eddies. Part I: Simple model. Ocean Modelling,87, 1–19.Haidvogel, D., and P. Rhines, 1983: Waves and circulation driven by oscillatory winds in an idealized ocean basin.Geophys. and Astrophys. Fluid Dyn., 25, 1–63.

Data-Driven Statistical Stochastic Models

This Project is about statistical/stochastic modelling of multiscale processes, such as those in geophysical turbulence.Is it possible to construct a stochastically driven model that can emulate statistics of the observed multiscale process,including its long-time autocorrelations? A mathematical formalism — multilayer stochastic model framework — wasdeveloped recently and applied by Kondrashov and Berloff (2015). In this formalism, the vector of macroscopic variablesis predicted on the main level of the model and driven by some hidden (microscopic) variables. The hidden variables thataffect the macroscopic variables are predicted by the system of linear stochastic equations driven by the “next” hiddenvariables, and so on. The “last” hidden variables degenerate to the white noise and the sequence of models terminates.The Project will focus on mastering the underlying formalism and applying it to some relatively simple problems.

Kondrashov, D., and P. Berloff, 2015: Stochastic modeling of decadal variability in ocean gyres. Geophys. Res. Lett., 42,1543-1553.

Analysis of a model of laser confinement of an atomic cold gas – Prof Carrillo

We will study a model proposed in [1,2,3] for confinement of atomic gases by an array of laser

beams. These models lead to nonstandard drift-diffusion like equations with not too well-known

properties. We will develop a numerical scheme to solve those equations based on splitting dimen-

sional techniques together with the schemes in [4] and try to analyse the qualitative properties

of the solutions. If time allows, we will analytically try to prove some properties of simplified

equations.

1. Seminal paper: J. Dalibard, Opt. Comm. 68, 203 (1988).

2. Long-range one-dimensional gravitational-like interaction in a neutral atomic cold gas, M.

Chalony, J. Barr, B. Marcos, A. Olivetti, and D. Wilkowski. Phys. Rev. A 87, 013401.

3. Non-equilibrium Phase Transition with Gravitational-like Interaction in a cloud of Cold Atoms,

J. Barr, B. Marcos, and D. Wilkowski, Phys. Rev. Lett. 112, 133001.

4. A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. J.A.

Carrillo, A. Chertock, Y. Huang, Commun. Comput. Phys. 17 (2015), no. 1, 233258.

Evolutionary trees and the distances between them – Dr Caroline Colijn and Dr

Michelle Kendall

Exciting developments in gene sequencing technology have meant that gene sequences are being

used to understand many aspects of biology and evolution. One of the most common starting points

in looking at these data is to make phylogenetic trees: trees in which the leaves (tips) correspond to

the observed sequences, and internal nodes correspond to inferred common ancestors. For example,

this is how we understand the ”tree of life” and interpret the evolution of primates. In the last 10

years, scientists increasingly use the same kinds of tools to understand the evolution of infections.

But there are many possible trees for a given set of data, and there are various software tools

available to make trees. Not all of the data will be consistent with a tree, because sometimes

genetic information is passed ”horizontally” from one organisms to another. The space of possible

trees is large, and different trees can tell different stories about evolution.

In our group, we have developed a new metric to compute the distances between trees. There are

several projects that are related to these ideas, exploring one or more of the following questions:

(1) Making the best trees: How can we identify parts of the data that did evolve on a tree-like

way? If we remove those parts do we consistently obtain ”better” trees than if we do not?

(2) What do we expect trees to look like: There are different ways to simulate and model evolution-

ary trees. At the moment we do not understand how different two random trees are expected to

be under different models. With this kind of understanding we could begin to make claims about

some trees being ”outliers” and not fitting a given model, which would open exciting options for

tree-based inference of evolutionary patterns.

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(3) Polar bears and tree metrics There are many datasets available in which people infer sets of trees

and draw conclusions from them. We have found that comparing their trees reveals ”tree islands”:

clusters of trees, each cluster telling a slightly different story about the dataset. One project option

is to explore recent tools to incorporate fossils into tree inference; there is a step-by-step tutorial

for this available. We would explore the structure of the tree space of bears using this tutorial

combined with our tree metric.

These are just a few of the project options related to phylogenetic trees in our group to give you

a flavour of the kinds of topics in the group. There is quite a lot of flexibility to make a project

suited to your own interests and skills.

Modelling the dynamics of tuberculosis in human hosts – Dr Caroline Colijn and Dr

Vahid Shahrezaie

Every year there are over a million deaths worldwide from tuberculosis, a disease that is chronic,

hard to treat, and is evolving increasing levels of drug resistance. The dynamics of tuberculosis

infections inside human hosts are not very well understood, and in particular, how two strains

(eg a drug-resistant and drug-sensitive strain) interact is unknown. Tuberculosis bacteria are also

capable of entering into a ”persisting” state where they essentially do not grow or die, affecting the

impact of drug treatments. In this project we will use an ODE model to describe the dynamics of

tuberculosis bacteria and host immune system cells. The model has already been developed, and

we’ve found that it nicely captures the distinction between latent TB (not too many bacteria) and

active TB (lots of bacteria). Intriguingly, the competitive interactions between two strains in the

model appear to be different in these two modes. In the project, we will continue to develop the

model and use it to explore persistence and inter-strain competition in tuberculosis.

Monte Carlo Markov Chain (MCMC) methods for dispersal source location – Dr Colin

Cotter

Description: This project is about the inverse problem of locating the source of a tracer release, for

example the release of a pollutant in an urban environment, based on time series of measurements

at a small set of points. This project will investigate MCMC methods for obtaining the PDF of

the source, using fi nite element methods built using the Firedrake library in Python. Students

considering this project should have an enthusiasm for numerical methods, high level programming,

and uncertainty quantification.

Discontinuous Galerkin methods for weak Lagrangian solutions of the semi-geostrophic

equations – Dr Colin Cotter

Description: The semi-geostrophic (SG) equations were originally proposed by Hoskins and Brether-

ton (1980) as an approximation of the atmosphere equations in a regime relevant to the formation

of weather fronts. They have weak solutions with singularities corresponding to fronts. In the last

20 years, a beautiful analytic structure of these equations has been discovered in the context of the

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optimal transportation theory developed by Brenier and Villani. In this project we will consider

recently developed numerical methods for the Monge-Ampere equation by Awanou, coupling them

with the dynamical part of the SG equations, which still have a relevance in studying how numerical

weather prediction models behave in the presence of fronts. The model will be developed using

the Firedrake library in Python. Students considering this project should have an enthusiasm for

numerical methods, high level programming and geophysical fluid dynamics.

Invisibility and homogenization – Professor Richard Craster

It has recently been discovered that one can, under some circumstances render an object invisible

to either light or sound waves at specfic frequencies. This project will focus on homogenization

models of layered media in optics or acoustics and how these result in an effective material which

is ”invisible. This can be explored with a variety of techniques that could be adjusted to suit

the student: transformation optics uses analytical approaches based on coordinate transformations

and lenses such as the Maxwell fish-eye lens would be considered. Other approaches could involve

numerical techniques and/or asymptotics. The project could involve collaboration with a group

from Physics and there will opportunities to interact with them, attend group meetings, talks in

the topic and work within an active group of PhD students (some of whom did the MSc in previous

years) and postdocs. It is an exciting and very topical area of research with plenty of applications

and opportunities available.

Optomechanical coupling – Professor Richard Craster

Since the work of Maldovan and Thomas (Applied Physics Letters 88, 251907, 2006) identified that

optomechanical devices could be made there has been a explosion of activity with current successes

in the optical cooling of laser mirrors, nano mechanical resonators and much more it is clear that

many devices can be made that use this coupling and the physics community have embraced this

new technology. Indeed it has led to the development of a new subfield self titled as phoxonic crystals

as the theory requires the properties of both photonic and phononic crystals. The key idea being

that if one can create a localised resonance in both systems then, despite the massive mismatch

in frequency scales, one can couple the systems leading to large transfer of energy. At present the

vast majority of work in this new field of phoxonic crystals is semi-empirical, experimental and

designs are driven by intuition. The field is ripe for some mathematical modelling and this project

is aimed at understanding this coupling, designing better or new phoxonic crystals and improving

the available modelling. There is the possibility to interact with Physicists at Imperial and with a

research group of PhD students and postdoctoral researchers working in closely related topics.

A model for the formation of networks of blood capillaries in living tissues – Prof

Pierre Degond, Dr Benjamin Aymar, Dr Sara Merino

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In this project, we are interested in the self-organization of networks of blood capillaries in living

tissues. We model this process by a set of simple heuristic rules. Three different kinds of agents are

considered: (i) blood pressure is subject to a diffusion equation (the Darcy model) ; (ii) capillaries

are modelled as small line segments that locally modify the coefficients in Darcy’s equation, (iii)

Oxygen transport by the blood is modelled by a convection-diffusion equation ; gradients of the

Oxygen concentration may locally modify the capillary distribution. Numerical simulations show

that a vascularization network spontaneously emerges from these simple rules and that different

morphologies can be obtained varying the model parameters [1]. However, the model is still incom-

plete in several aspects. One of them is that entire regions of the tissue are ignored by the network.

The Msc project would pursue this work in two direction which are not mutually exclusive to each

other

One direction is to improve the biological relevance of the model by introducing a fourth agent

representing chemical signals emitted by the regions lacking Oxygen and whose functions are to

recruit new blood vessels in order to increase the Oxygen supply. The student will design the model

and will introduce it in the existing code. This will be performed under the guidance of the author

of the code, Benjamin Aymard who will provide all the necessary assitance.

A second direction is the theoretical analysis of the model. For this purpose, macroscopic models

for the density and mean capillary orientation will be derived following the methodology of [2] and

coupled to the equations for the blood flow and Oxygen concentrations. The resulting model will

be analyzed mathematically using tools of Partial Differential Equations and stability theory.

[1] B. Aymard, P. Degond et al, Emergence of vascularization networks, in preparation.

[2] P. Degond, F. Delebecque, D. Peurichard, Continuum model for linked fibers with alignment

interactions, http://arxiv.org/abs/1505.05027

Contact: Pierre Degond: pdegond@imperial.ac.uk, Sara Merino: s.merino-aceituno@imperial.ac.uk,

Benjamin Aymard : b.aymard@imperial.ac.uk

A new model for pedestrians including congestion – Prof Pierre Degond, Dr Ewelina

Zatorska

In this project, we are interested in the modelling of pedestrian flows by means of macroscopic

equations (i.e., continuous models). Such models consist of partial differential systems for the mean

density and/or the mean velocity of the pedestrians mainly in two space dimensions and occasionally

also in one-dimensional settings. Many such models have been proposed in the literature but, in

general, they fail to describe the regions where the concentration of pedestrians is high and where

safety is the most at risk.

The model we propose is inspired by the so-called Aw-Rascle (AR) model of road traffic [1], which

considers the mean car density and the mean desired velocity of drivers. The actual velocity of

traffic is less than the desired velocity by a quantity (the velocity offset) which accounts for the

effect of congestion [2]. In the AR model, this velocity offset is taken as a direct function of the local

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car density. In the proposed new model (work in collaboration with Andrea Tosin, from Politecnico

di Torino) the velocity offset is taken as the spatial gradient of a congestion cost function depending

on the density and possibly also on the actual speed of pedestrians. This leads to a nonlinear system

of parabolic type that seems at least formally well-posed.

The project has three different aspects that can be more or less developed according to the student’s

taste:

(i) The mathematical analysis of the model (existence and uniqueness of solutions in a one-

dimensional setting), following [3].

(ii) The realisation of numerical simulations in one-dimension to compare the behaviour of this

model with that of the AR model.

(iii) The asymptotic analysis of a singular perturbation problem related to this model by either

analytic or numerical methods. This analysis will help better characterizing the transition between

free and congested traffic.

References

[1] A. Aw and M. Rascle, Resurrection of ”Second Order” Models of Traffic Flow, SIAM J. Appl.

Math., 60 (2000) 916-938.

[2] P. F. Berthelin, P. Degond, M. Delitala, M. Rascle, A model for the formation and evolution of

traffic jams, Arch. Rat. Mech. Anal., 187 (2008), pp. 185-220.

[3] D. Bresch, C. Perrin, E. Zatorska. Singular limit of the Navier-Stokes system leading to a

free/congested zones two-phase model. C. R. Math. Acad. Sci. Paris 352 (2014), pp 685-690.

Contact: Pierre Degond: pdegond@imperial.ac.uk, Ewelina Zatorska: e.zatorska@imperial.ac.uk

Modelling the interplay between economics and climate change through mean-field

games – Prof Pierre Degond, Dr Mirabelle Muuls

Overview of the topic: Economics and climate change are closely entangled, both because economic

activity is leading to dangerous levels of greenhouse gases emissions and because climate change

has strong economic impacts. As an example of this interrelation, firms are likely to choose their

production sites taking into account the risk of extreme climate events. Another example would

be the fact that outsourcing their production to low-cost countries could affect the environmental

innovation and competitiveness of western firms. The project aims at modelling these effects. In

particular, given the large number of agents involved, the project will develop a mean-field games

(MFG) approach.

Motivation of the problem: The study of the interrelation between climate change and firm strate-

gies is booming. However, there are still many poorly investigated questions such as the two

examples given in section 2.1 above. The application of MFG to economics is also booming but

most of such attempts are focussing on the energy market or mining industries. MFG provide more

tractable models than discrete agent approaches when the number of agents is large. However,

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there are still difficulties in incorporating realistic utility functions. An approximation of MFG, the

so-called Best Replied Strategy (BRS) will be investigated.

Statement of the problem: We will start from [1] which provides a well suited MFG model for firm

growth prone to rigorous mathematical analysis and we will develop it in two directions. First,

we will modify the utility functions to incorporate the interplay with climate change within the

context of the two examples stated in section 2.1, and second we will simplify the MFG approach

of [1] by using the BRS as in [2]. The simplification brought by the BRS should keep the model

tractable in spite of the modifications brought to the utility functions.

Where the project can lead: The project could lead to a PhD thesis where the BRS model including

climate change would be used to derive macroscopic equations of firm behaviour. These macroscopic

equations will lend themselves to numerical computations allowing for a comparison between the

model outcomes and firm data (the supervisors have access to two large databases). Mathematical

questions pertaining to the validity of the MFG approach (in the limit of a number of agents tending

to infinity) and that of the macroscopic model (in the limit of large spatio-temporal scales) will be

investigated. Modern data model comparison tools will be developed to calibrate the model on the

data and possibly provide predictions along various types of scenarios. The outcome will be models

of firm behaviour that could shed new light on how climate change influences and is influenced

by firm behaviour. These outcomes will be of high policy relevance and also useful to companies

willing to better understand the challenges that arise from climate change and how they should

respond to it.

Literature:

[1] M. Huang & S. L. Nguyen, Mean field games for stochastic growth with relative consumption,

submitted.

[2] P. Degond, M. Herty, J-G Liu, Mean field games and model predictive control, submitted.

arXiv:1412.7517

Supervisors’ Name and Institution:

Pierre Degond, Department of Mathematics, Imperial College London, pdegond@imperial.ac.uk

Mirabelle Muuls, Grantham Institute - Climate Change and the Environment, Imperial College

London, m.muuls@imperial.ac.uk

Integrable Nonlinear PDEs – Professor J. Elgin

Certain nonlinear PDEs have the property that they can be solved analytically using a transform

technique, in much the same way that we use the Fourier transform to solve linear PDEs like

the heat equation. These nonlinear PDEs include the Burgers Equation, the Korteweg de Vries

Equation (KdV), the Nonlinear Schroedinger Equation (NLS); they all have features in common

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though have application to very different physical systems, and all have been studied extensively

in recent years. The project will look at one aspect of one or more of these equations.

Studies on the Toda lattice – Professor J. Elgin

The Toda lattice is a discrete integrable nonlinear lattice with properties similar to those found in

certain nonlinear PDEs such as the Korteweg-de Vries equation. The project will explore certain

types of solution for this lattice, working towards a family of solutions known as finite gap or quasi-

periodic solutions.

Asynchronous Networks and Event Driven Dynamics – Prof Michael Field

An asynchronous network is a network of interacting dynamical systems where the network struc-

ture may be state and time dependent and we allow, for example, nodes to stop, synchronize with

other nodes and later restart. This type of network is characteristic of those encountered in biology

and modern technology and does not fit into the classical framework of systems of analytic ordinary

differential equations. The topics we have in mind are (1) Braess’s paradox for transport and prod-

uct networks: how adding connectivity can reduce performance; (2) Modelling transport networks

using asynchronous networks and the identification of network characteristics that lead to poor

performance (the District and Circle lines come to mind); (3) Inventory oscillations in production

networks and functional asynchronous networks. In all cases the focus will be on obtaining good

examples and intuitive explanations for the phenomena being considered.

PT-symmetric Quantum Mechanics (Dr Christopher Ford)

Background

In both classical and quantum mechanics the potential energy, V , is assumed to be real. In classical

mechanics a complex potential energy would lead to apparently absurd equations of motion and

in quantum mechanics a complex V would lead to complex energy levels and non-unitary time

evolution.

In the 1990s it was realised that the Schrodinger equation with the imaginary potential energy

V (x) = ix3

actually yields real and positive energy levels. Bender and Boettcher considered a more general set

of complex potentials of the form

V (x) = −(ix)N ,

where N is a positive number (not necessarily integer). This potential has the property that it is

PT-symmetric meaning that the potential energy is invariant under a combined reversal of space

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and time coordinates. Using this PT-symmetry Bender and Boettcher argued that the quantum

theory based on this complex potential has real energy levels if N ≥ 2.

Remarkably, it is possible to see a hint of this transition in the classical trajectories derived from

the potential. As the potential is complex the solutions of the equation of motion trace curves in

the complex plane. If N ≥ 2 there are closed orbits in the complex plane when the energy, E, is

real. If N < 2 there are no closed orbits for real E.

Objectives

To study the Schrodinger equation for integer N and to understand the relationship between the

existence of closed classical orbits and real quantum energy levels.

References

[1] C. Bender, S. Boettcher and P. Meisinger, Journal of Mathematical

Physics, Volume 40, number 5 (1999). This paper and other useful information is available at the

homepage of Carl Bender (University of Washington in St. Louis).

Yang-Mills in a Box (Dr Christopher Ford)

Background

Yang Mills theory is a non-linear generalisation of electrodynamics. In the Standard Model of

particle physics interactions between spin 12 particles are mediated by Yang-Mills fields. However,

sixty years after the theory was developed a full understanding of quantum Yang-Mills theory is

lacking. The main difficulty is to extract the infrared or long-distance properties. The problem

is present regardless of the matter content. Indeed, it is still there in the case of pure Yang-Mills

theory which comprises Yang-Mills fields without any additional matter fields.

In the early 1980s Gerard ’t Hooft [1] suggested that one could get a grip on these infrared problems

by quantising the theory on a Euclidean four-torus rather than R4. A four-torus can be viewed

as a four-dimensional box with opposite ‘faces’ identified. ’t Hooft also considered classical Yang-

Mills theory on a four torus. He found some simple solutions with remarkable properties including

constant field strength, self-duality and fractional topological charge.

A four-torus has four periods (or ‘edge-lengths’ when viewing the torus as a box) and classical

solutions are known to exist for arbitrary periods. However, ’t Hooft’s solutions are only valid for

a restricted choice of periods. Morover, the solutions that ’t Hooft wrote down remain the only

known analytical solutions of Yang-Mills theory on a torus.

Objectives

To understand the Lagrangian formulation of pure Yang-Mills theory and the definition and con-

struction of self-dual solutions. To understand how this works for a Euclidean four-torus and study

14

’t Hooft’s solutions. For the gauge group SU(2), investigate the properties of charge 12 solutions

with arbitrary periods.

Reference

[1] G. ’t Hooft, “Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus,”

Commun. Math. Phys. 81, 267 (1981).

15

[1] P.A.M. Dirac, Proc. R. Soc. Lond. A 117 (1928) 610.

[2] P.A.M Dirac, Proc. R. Soc. Lond. A 126 (1930) 360.

[3] R. Jackiw and C. Rebbi, Phys. Rev. D 13 (1976) 3398.

[4] R. Jackiw, ‘Fractional charge from topology in polyacetylene and graphene’ AIP Conf.

Proc. 939 (2007) 341.

Large amplitude asymptotics of the elliptic sinh-Gordon equation - Dr J.

Gibbons

The PDE

uxx + uyy = 2 sinh(2u)

has a large family of exact solutions, expressible as rational functions of exponentials.

Large-amplitude, long-wave solutions to the equation, rescaled as

ε2(uxx + uyy) = 2 sinh(2u)

Where 0 < ε ≫ 1, may be expressed in terms of solutions of the eikonal equation

|∇S|2 = 1

with appropriate boundary conditions. These solutions have singularities, however. The

aim of the project is to investigate the solutions near these singularities.

Exact Solutions of the Benney Equations - Dr J. Gibbons

The Benney moment equations are a model describing long waves on a shallow perfect

fluid with a free surface y = h(x, t). They are a completely integrable system. The

moments of the horizontal fluid velocity u(x, y, t),

An =

! t

0h(x, t)un dy

satisfy

(An)t + (An+1)x + n(An−1)(A0)x = 0.

The system has infinitely many conservation laws and symmetries, and also has some im-

portant families of exact solutions. The project will be to describe some of these solutions.

Quıest que le FEEC: expressive, high-performance mathematical computa-

tion Dr David Ham

FEEC, the finite element exterior calculus, is a beautiful abstraction for the finite

element method which provides a powerful mechanism for deriving accurate and stable

numerical schemes. In an ideal world, a mathematician would state a system to be simu-

lated in the elegant high-level terms of the FEEC, and the corresponding high performance,

parallel, low-level implementation would be generated and run automatically. The goal of

this project is to make this happen.

The project will develop a computer language which embodies the abstractions of

FEEC and supports symbolic manipulation of the language; such as differentiation, fac-

torisation and cancellation of terms in the differential forms that form the basis of FEEC.

10

Dirichlet boundary conditions for Dolfin-adjoint – Dr David Ham

Systems of PDEs are solved numerically to simulate many natural and man-made phenomena.

However we frequently wish to go beyond simulation to study the sensitivity of a system to its

inputs, optimise the design a system, or to analyse the stability of the simulated system. For all of

these purposes, it is very useful to be able to calculate the derivative of some functional of the PDE

solution with respect to given input parameters. The process can be achieved with considerable

computational efficiency by differentiating the PDE and integrating by parts to form the adjoint

PDE.

Forming numerical solutions to the adjoint PDE is a grand challenge problem in scientific comput-

ing, since it involves a complex interplay between symbolic mathematics and sophisticated data

interactions. However recent work at Imperial has produced Dolfin-adjoint, a system for automat-

ically computing the adjoint solutions to finite element PDEs. This is a huge step forward in this

field, and Dolfin-adjoint was this year awarded the Wilkinson Prize, the world’s most prestigious

numerical software award. However, Dolfin-adjoint cannot yet compute the adjoint to Dirichlet

boundary conditions. Your task, should you take on this project, is to derive the appropriate com-

putations and extend Dolfin-adjoint to achieve them. Success would mean a significant improvement

in capability for the users of Dolfin-adjoint, and the possibility of a journal paper documenting this

advance.

Qu’est que le FEEC: expressive, high-performance mathematical computation – Dr

David Ham

FEEC, the finite element exterior calculus, is a beautiful abstraction for the finite element method

which provides a powerful mechanism for deriving accurate and stable numerical schemes. In an

ideal world, a mathematician would state a system to be simu- lated in the elegant high-level terms

of the FEEC, and the corresponding high performance, parallel, low-level implementation would

be generated and run automatically. The goal of this project is to make this happen.

The project will develop a computer language which embodies the abstractions of FEEC and sup-

ports symbolic manipulation of the language; such as differentiation, fac- torisation and cancellation

of terms in the differential forms that form the basis of FEEC.

Once the high-level symbolics and language have been developed, the project could move on to

transforming this representation into high-performance elemental kernels suitable for numerical

simulation.

Execution of these simulations will be carried out within the simulation framework exposed by

Firedrake (www.firedrakeproject.org). This imposes the need to build the FEEC language within

Python and it will be an interesting challenge to maintain as much mathematical expressivity as

possible while staying compatible with Python.

17

MSc project topics with Professor Darryl D Holm in Geometric Mechanics

The shape of water, metamorphosis and infinite-dimensional geometric mechanics Whenever we say the words ”fluid flows” or ”shape changes” we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler’s fluid equations tell us that water flows a particular way. Namely, it flows to get out of its own way as efficiently as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but also the optimal metamorphosis of any shape into another follows one of these optimal paths. This project will study the commonalities between fluid dynamics and shape changes and will use the methods that are most suited to fundamental understanding - the methods of geometric mechanics. In particular, the main approach will use momentum maps and geometric control for steering along the optimal paths from one shape to another. The approach will also use emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff. The EPDiff equation governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures. Extreme events in potential vorticity gradients that are stirred, but not mixed The evolution at http://www.met.rdg.ac.uk/Data/CurrentWeather/ shows satellite data indicating the development of large gradients of both potential vorticity (PV) and potential temperature (PT) in the Earth’s stratosphere. Recent work of Gibbon and Holm posted at http://arxiv.org/abs/0911.1476 proposes an equation governing the dynamics of this process. The numerical integration of this equation is required in order to investigate this PV and PT stirring dynamics in the stratosphere and to compare it with these observations.

Entropy and evolving networks

Supervisor: Professor Henrik Jeldtoft Jensen

Co-evolving agents, such as biological organism or economical companies,constitute a time dependent network of interactions or interdependencies.It has long been proposed that the evolution of ecological networks can becharacterised by information theoretic (or entropic) measures(1,2,3). Theproject will use analytic and simulation techniques to study the time depen-dence of such entropic measures in simple evolutionary models in which thetopology of the ecological network changes with time (4,5).

[1] Ulanowicz, R. E. (2011). The Central Role of Inform- ation Theoryin Ecologe. In Dehmer, M., Emmert- Streib, F., and Mehler, A., editors,Towards an In- formation Theory of Complex Networks: Statistical Methodsand Applications. Birkha user, New York..[2] Ulanowicz, R. E. and Hannon, B. (1987). Life and the production ofentropy. Proceedings of the Royal Soci- ety London B, 232:181192.[3] D. Jones, H.J. Jensen and P. Sibani, Mutual information in the TangledNature model. Ecological Modelling, 221, 400-404 (2010).[4] S. Laird and H.J. Jensen, Correlation, selection and evolution of speciesnetworks. Ecological Modelling 209, 149-156 (2007).[5] S. Laird and H. J. Jensen, A non-growth network model with exponentialand 1/k scale-free degree distribution. Europhys. Lett 76, 710-716 (2006).

Complex systems and generalised entropy

Supervisor: Professor Henrik Jeldtoft Jensen

It has been argued that the statistical mechanics of complex systems requiresa generalisation of Boltzmann’s entropy(1). The q-statistics of Tsallis focuson the need of the entropy to be extensive, although it isn’t entirely clearto what extent the extensivity is a fundamental requirement(2). The ques-tion remains: under which circumstances the generalised Tsallis statisticsis necessary? The project will study this question by first investigate thealleged need for an entropy to be extensive. Secondly the project will con-sider the Restricted Random Walker Model(3) and analyse the functionalform of the return time distribution derived in the continuum limit, see Eq.

(18) in [4]. Numerical evidence indicates that the return distribution is aq-exponential for one parameter value but not for others. Understandingthis finding better at the analytic level will add to our understanding of thestatus of q-statistics.

[1] Constantino Tsallis, Introduction to Nonextensive Statistical Mechanics.Springer 2009.[2] E.T. Jaynes, in Maximum Entropy and Bayesian Methods, page 22 Edi-tors C.R. Smith and J. Erickson and P.O. Neudorfer.Kluwer Academic Pub-lishers. 1992.[3] U. Tirnakli, H.J. Jensen and C. Tsallis, Restricted random walker modelas a new testing ground for the application of q-statistics. EPL, 96, 40008(2011).[4] J. Zand, U. Tirnakli and H.J. Jensen, On the relevance of q-distributionfunctions: The return time distribution of restricted random walker. J.Phys. A: Math. Theor. 48 425004 (2015)

Disentangling entangled quantum states – Dr Sania Jevtic

Typical quantum systems are entangled (correlated) and this makes their joint state hard to express

as matrices because the number of matrix elements grows exponentially with the number of systems.

Being entangled means that we cannot write the states of such many-particle quantum systems in a

“separable” way, that is, as a mixture over products of single-particle quantum states which grows

only linearly in system size. However, if we only have access to a limited set of measurements that

we can implement on the system, then “generalised separable” descriptions, even for entangled

states, can be obtained. This gives us an efficient description of the entangled state with respect

to this restricted set of measurements. The task for this project is to explore these “generalised

separable” decompositions.

A quantum state is a unit-trace, positive-semidefinite operator ρ acting on a Hilbert space. In this

project, we will restrict to the bipartite case where ρ is the joint state of two quantum systems A

and B, each of which has a Hilbert space dimension d (with d finite). The state ρ can be decomposed

into linear combinations of tensor products of local operators Ai and Bi each acting solely on

the Hilbert spaces of A and B:

ρ =∑

i

wiAi ⊗Bi. (1)

There are many such decompositions for a given ρ. As a special case, if ρ is not entangled, then the

wi form a probability distribution (∑

iwi = 1 and wi ≥ 0) and Ai and Bi are quantum states. On

the other hand, if ρ is entangled, then we would like to find its “most useful” generalised separable

decomposition characterised by three features: (i) the wi form a probability distribution; (ii) the

sets Ai and Bi contain positive trace operators, and (iii) these sets are as close as possible to

the sets of quantum states.

Condition (ii) is important if we want the generalised separable decomposition to be valid for

quantum measurements. Satisfying condition (iii) means that we have an efficient description for

ρ for the largest restricted set of measurements. The notion of “as close as” is made precise by

optimising over the sizes of the operator sets SA = Ai and SB = Bi. Specifically, we want

to minimise the product ||SA||.||SB|| where ||...|| is some operator norm. These ideas have been

initiated in [1, 2]. This project primarily requires familiarity with linear algebra.

References

[1] H. Anwar, S. Jevtic, O. Rudolph, and S. Virmani, NJP 17, 093047 (2015)

[2] H. Anwar, S. Jevtic, O. Rudolph, and S. Virmani, arXiv:1511.03196

Simulation Based Inference for Molecular Machines – Dr David Rueda (MRC), Dr

Tom Ouldridge, Dr Nick Jones

21

While nanomachines offer the prospect of technological revolution the characterization of their

dynamics remains a challenge. We will attempt to understand the underlying potential surfaces

associated with high resolution single molecule data gathered in the laboratory of David Rueda.

We will develop a simulation based inference pipeline that allows us, given a belief about an under-

lying energy surface, to simulate real data. This approach draws on tools from stochastic processes,

statistical physics and simulation based inference (we will likely use Approximate Bayesian Com-

puting). Deploying this suite of tools will be a unique piece of progress in single molecule data

analysis and so, beyond allowing us to understand the behaviour of a particular nanomachine, the

will thus be of interest to a large scientific community: solid progress is likely to lead to publication

in widely consumed journals. Concrete examples will be drawn from a variety of Biological systems

that play key roles in cellular function and disease, such as polymerases, helicases or deaminanases.

4D Cell Dynamics – Dr Silvia Santos (MRC), Dr Nick Jones

The cell-cycle is the sequence of steps associated with cell division. It is a cannonical process in

q-bio and its malfunction is linked to cancer. The group of Silvia Santos has collected durations of

each of the 4 successive stages of the cell cycle for large sets of individual cells. A population of cells

thus becomes a set of points in this 4D duration space. If cells were strictly identical then each cell

would be mapped to a single point in this space: but in fact there is pronounced variability. What

is the character of this variation? Of what dynamics is it the consequence? Dr Santos has further

collected these 4D point clouds for cancerous cells and stem cells. The student will investigate the

structural differences between the point-clouds associated with each of these distinct cell types and

attempt to construct simple models to account for the structure within the data and between cell

lines. The group of Nick Jones has previously argued that the spread of these point clouds can

be attributed to mitochondrial variability, the student will be given opportunities to explore this.

Dr Santos can also collect data for the cell-cycle-stage durations of a single cell through multiple

divisions: we can thus access dynamics in this 4D space. The data collected in the Santos lab is

unique and yet it addresses a core problem in biology: solid progress is likely to lead to publication

in widely consumed journals.

Hybrid bases for solving Laplace’s equation in 3D – Dr Eric E Keaveny and Prof

Darren Crowdy

Laplace’s equation is one of the most widely encountered equations, arising in the mathematical

theories of electromagnetism, fluid mechanics, and chemical transport just to name a few cases.

In 2D, we can rely on powerful mapping techniques and potential theory to find closed form solu-

tions to Laplace’s equation. These techniques also inform many computational methods to solve

Laplace’s equation when complicated boundaries are involved. One such technique is a hybrid basis

collocation scheme that combines solution bases for two different domain mappings. Our initial

results indicate that the hybrid basis technique can be extended to 3D to compute solutions to

22

Laplace’s equation in multiply connected domains with spherical inclusions. The project will entail

implementing this novel scheme and studying how to choose the combination of basis functions.

The computational aspects of the project will be performed using MATLAB.

Gyrotactic instabilities in swimming suspensions – Dr Eric E Keaveny

The collective dynamics of swimming microorganisms can play an important role in ensuring their

survivability. This holds true for algae cells that can be cultured in bioreactors to create biofuels.

These single celled organisms exhibit interesting dynamics due to the effect of gravity that causes

them to swim in a particular direction. At the population level, this effect, known as gyrotaxis,

can lead to interesting instabilities of cell concentration profiles.

This project entails exploring gyrotactic instabilities using a computational model that treats the

movement and interaction of each cell making up the population. By varying the parameters

associated with gravity and cell swimming, we aim to quantify the onset of this instability and

compare the results with continuum theories for cell concentration evolution. Prior knowledge

of the programming language C will be useful but not mandatory. This project will be done in

collaboration with Dr Yongyun Hwang in Aeronautics.

Crawling on gel surfaces – Dr Eric E Keaveny

Experiments using the worm and model organism C. elegans offer an unprecedented opportunity

to connect the underlying genetic make up of an organism with its behaviour. For C. elegans,

this entails observing the movements of its various mutant strains as they crawl on a gel surface.

While mechanics and mechanical modelling plays a key role in making this connection quantitative,

accurate modelling of the worm-gel interaction has only just begun.

This project entails exploring the effectiveness of recently postulated nonlinear drag models to

capture the interaction between the worm and the gel surface. This will be done by interfacing

the model with data provided by Dr Andre Brown at Imperial MRC Clinical Sciences Centre. The

aim of the project will be to compare experimental trajectories with those given by the model

and to find gait patterns where the model may under perform. This project will entail using and

developing MATLAB programs. This project will be done in collaboration with Dr Andre Brown.

The dynamics of Brownian suspensions – Dr Eric E Keaveny

Brownian motion is the random motion of microscopic objects immersed in liquids. Due to hydro-

dynamic interactions, the Brownian motions of the particles, though random, are coupled together.

This coupling plays an important role in suspension dynamics and the evolution of the particle

concentration profile. It can also produce changes in the mechanical properties of the suspension

and the forces that are needed to make suspensions flow.

Using recently developed techniques to rapidly compute coupled Brownian motion, this project will

entail examining suspension dynamics when they are subject to external potentials. These results

will then be compared with those given by continuum PDE models for the particle concentration.

Through these simulations we aim to highlight the role of the hydrodynamic coupling and the

23

ability of PDE models to capture it.

Stochastic modelling of hydrodynamic and crowding effects in biochemical reactions

– Dr Eric Keaveny and Dr Thomas Ouldridge

Bimolecular biochemical reactions are fundamental for a wide range of phenomena, from cellular

function to the assembly of artificial nanostructures and devices. These reactions are often modelled

at a mean-field level, in which detailed spatial information is neglected and the system is assumed to

be well-mixed. Recent work has shown that deviations from mean-field behaviour (spatio-temporal

correlations) could have dramatic effects, with (in particular) the possibility of rapid unbinding and

rebinding of reactants leading to novel behaviour [1,2,3].

Biochemical reactions occur in solution. In this environment, the motion of a particle disturbs

the surrounding fluid, which in turn influences the motion of other nearby particles. Moreover,

reactions occur in the presence of other molecules, often called ”crowders”, which can fundamentally

influence the diffusive mechanism by which reactants come into contact. Previous studies [1,3] have

neglected either one or both of these key features – this project will explore the combined influence

of hydrodynamic and crowding effects on biochemical reactions.

The project will involve using Dr Keaveny’s recently-developed ”Fluctuating force-coupling method”

[4] to study models of biochemical reactions. The results will be compared to approaches that ne-

glect hydrodynamics, crowding, or both to identify the key differences. Finally, the student will

explore the consequences of these differences for typical biochemical systems, such as enzymatic

cascades.

[1] Spatio-temporal correlations can drastically change the response of a MAPK pathway. K.

Takahashi et al., Proc. Natl. Acad. Sci. USA, 107:2473-2478, 2010.

[2] A general mechanism for competitor-induced dissociation of molecular complexes. T. Para-

manathan et al., Nat. Commun.: 5:5207, 2014.

[3] Intracellular Facilitated Diffusion: Searchers, Crowders, and Blockers, Brackley et al., Phys.

Rev. Lett. 111, 108101, 2013.

[4] Simulating Brownian suspensions with fluctuating hydrodynamics, B. Delmotte and E. E. Keav-

eny, http://arxiv.org/abs/1507.02185

Dynamics near termination points of homoclinic orbits – Prof Jeroen Lamb and Dr

Rene Medrado

The proposal is to study the following problem in bifurcation theory: in a two-parameter dynamical

system, the end-point of a curve in the parameter space where homoclinic (bi-asymptotic) solutions

to an equilibrium exist, has been observed to coincide with the centre of a spiralling curve of

24

parameter values for which we find periodic solutions with strong stability. The proposal is to

study this phenomenon with numerical and analytical techniques.

Period doubling bifurcation with bounded noise - Prof Jeroen Lamb

The interplay between stochastic input (noise) and nonlinear dynamics is a topic of increasing

interest in many branches of science. The proposal is to study case studies in dimension one

and two of discrete time random dynamical systems with bounded noise, where the underlying

deterministic dynamics displays a so-called period-doubling bifurcation. The objective is to analyse

these examples in the context of the existing literature and identify additional dynamical features,

in directions where theory is yet to be developed. The analysis may involve a combination of

analytical and numerical approaches.

Hele-Shaw moving free boundary flows around solid obstacles – Dr Jonathan Marshall

Hele-Shaw flows constitute an important class of free boundary problems, relevant to a wide range

of interesting physical processes, such as flows in porous media. They are the subject of what has

become a vast body of research, and many exact solutions are known for them. However, exact

results for Hele-Shaw moving free boundary flows around solid obstacles are relatively scarce. This

project will involve investigating such problems, building on recent research which has led to new

results in this area. An interest in complex analysis would be helpful. Also, computation in Matlab

or an equivalent program will be required.

Mean first passage time for narrow escape problems – Dr Jonathan Marshall

The narrow escape problem in two-dimensions describes the motion of a Brownian particle confined

in a bounded planar domain whose boundary is almost entirely reflecting except for small open

windows through which the particle can escape. Such problems can be used to model numerous

physical processes. In many such contexts, it is of interest to determine to expected time taken for a

particle to escape from the domain, known as the mean first passage time (MFPT). Mathematically,

this problem is described by Poissons equation with mixed Dirichlet-Neumann boundary conditions,

taking the asymptotic limit as the open windows shrink in size. The aim of this project is to

investigate this problem in new geometries, applying new techniques to derive approximations to

the MFPT. An interest in complex analysis would be helpful. Also, computation in Matlab or an

equivalent program will be required.

25

MSc in Applied Mathematics Dr Robert NurnbergMSc Projects 2015-16

1. Image segmention and image restoration with active contours

Two fundamental tasks in image processing are image segmentation and image smoothing.A natural strategy is to combine the two processes in a single step, following the idea of theseminal work by [MS]. They introduced the following optimization problem: Find a minimizer(u, S) of the functional

EMS(u, S) = σHd−1(S) + λ

∫

Ω(u− u0)2 dLd +

∫

Ω\S|∇u|2 dLd . (1)

Here, given an image u0 : Rd ⊃ Ω → R, the task is to find its set of discontinuities S, and apiecewise smooth approximation u : Ω→ R of u0.

As the Mumford–Shah problem is difficult to tackle in its original form, several simplifiedmodels have been proposed. Chief among them are the models by [CV] and [TYW] which, intheir simplest forms, assume that S is a closed curve Γ that partitions Ω into two regions: Ω1

and Ω2. Moreover, u is assumed to be constant or smooth in each of the two regions. Hence(1) reduces to

E(u,Γ) = σHd−1(Γ) + λ2∑

i=1

∫

Ω(ui − u0)2 dLd +

2∑

i=1

∫

Ωi

|∇ui|2 dLd . (2)

Possible numerical approaches to minimize (2) can be found in e.g. [DMN] and [Ben], wherethe latter work uses a piecewise constant approximation of u and active contours based on[BGN]. The aim of this project is to build on the work in [Ben], but with u = u1XΩ1 +u2XΩ2

being a piecewise smooth approximation.

Prerequisites: Good knowledge of finite differences and finite elements. Programming skillsin C, MATLAB or Python.

References

[Ben] Heike Benninghoff. Parametric Methods for Image Processing Using Actice Contours withTopology Changes. PhD thesis, University Regensburg, Regensburg, 2015.

[BGN] John W. Barrett, Harald Garcke, and Robert Nurnberg. On the variational approximationof combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput.,29(3):1006–1041, 2007.

[CV] T. F. Chan and L. A. Vese. Active contours without edges. IEEE Trans. Image Process.,10(2):266–277, 2001.

[DMN] Gunay Dogan, Pedro Morin, and Ricardo H. Nochetto. A variational shape optimizationapproach for image segmentation with a Mumford-Shah functional. SIAM J. Sci. Comput.,30(6):3028–3049, 2008.

[MS] David Mumford and Jayant Shah. Optimal approximations by piecewise smooth functionsand associated variational problems. Comm. Pure Appl. Math., 42(5):577–685, 1989.

MSc in Applied Mathematics Dr Robert Nurnberg

[TYW] A. Tsai, Jr. Yezzi, A., and A. S. Willsky. Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification. IEEETrans. Image Process., 10(8):1169–1186, 2001.

2. Fast algorithms to find the distance between two ellipsoids

This project will be concerned with the geometric problem of finding the distance betweentwo ellipsoids. The problem can be formulated as

min ‖x− y‖ s.t. x ∈ E1, y ∈ E2 , (1)

where Es = x : q1(x) ≤ 0 and E2 = y : q2(y) ≤ 0 are two given ellipsoids determined bythe two quadratic functions

q1(x) = 12 x

T A1 x+ bT1 x+ α1 and q2(y) = 12 y

T A2 y + bT2 y + α2,

with positive symmetric matrices A1 and A2, vectors b1 and b2 and scalars α1 and α2.

The problem (1) appears to be simple, but it is highly nonlinear and so cannot be solved bysimple linear or quadratic programming methods. Solving (1) has applications in e.g. compu-tational biology, robotics, computer graphics, computer games and virtual reality, [SJKW].

Possible approaches to tackle (1) are a direct parameterization of the boundaries ∂E1 and∂E2, and then applying a Netwon-type algorithm similar to what is described in [Nur]. Analternative is to develop an algorithm that iteratively solves the simpler sub-problem of findingthe distance between a point and an ellipsoid, see also [Dai]. Both approaches should becompared to the algorithms in [LH, Kim].

Prerequisites: Good programming skills in MATLAB or Python.

References

[Dai] Yu-Hong Dai. Fast algorithms for projection on an ellipsoid. SIAM J. Optim., 16(4):986–1006, 2006.

[Kim] Ik-Sung Kim. An algorithm for finding the distance between two ellipses. Commun.Korean Math. Soc., 21(3):559–567, 2006.

[LH] Anhua Lin and Shih-Ping Han. On the distance between two ellipsoids. SIAM J. Optim.,13(1):298–308, 2002.

[Nur] R. Nurnberg. Distance from a point to an ellipsoid, 2006. www.ma.ic.ac.uk/~rn/

distance2ellipse.pdf.

[SJKW] Kyung-Ah Sohn, B. Juttler, Myung-Soo Kim, and Wenping Wang. Computing distancesbetween surfaces using line geometry. In 10th Pacific Conference on Computer Graphicsand Applications, Proceedings., pages 236–245, 2002.

Analysis and design of biomolecular networks – Dr Diego Oyarzun

Living cells control their internal machinery with complex feedback regulation systems. We have

a number of projects on the analysis and design of biomolecular networks. The projects generally

combine analysis and simulation with ideas from Systems & Control Theory applied to nonlinear

biochemical systems. The results from these projects will help us to understand how living systems

work, as well as to discover new-to-nature biomolecular circuits with useful functions for Synthetic

Biology.

Complex dynamics in cellular metabolism. Cells use feedback regulation to switch on or

off parts of their metabolism in response to environmental signals. The goal of this project is

to characterise the dynamics of metabolic systems with with different feedback regulatory archi-

tectures. Following our recent work based on piecewise affine dynamics on polytopes, we aim to

find links between combinations of positive/negative feedback and their emerging dynamics. We

will use these results to discover new regulatory systems with novel functions for Biotechnology.

The analysis will combine Filippov’s theory for discontinuous dynamical systems, graph-theoretic

analyses and singular perturbation theory.

References:

[1] Oyarzun, Chaves, Hoff-Hoffmeyer-Zlotnik. “Multistability and oscillations in genetic control of

metabolism”, J of Theoretical Biology, 295, 2012.

[2] Oyarzun, Chaves. “Design of a bistable switch to control cellular uptake”, J Royal Society

Interface, 2016 (goo.gl/UHbaPZ)

Stochastic fluctuations in metabolism. Intracellular fluctuations are an important source of

noise in biological systems. In this project we seek to analyse stochastic models for metabolic

systems. The goal is to identify the key parameters that control the attenuation or amplifica-

tion of stochastic fluctuations. The project will combine stochastic simulation and closed-form

approximation methods for the Chemical Master Equation.

Reference:

[3] Oyarzun, Lugagne, Stan. “Noise propagation in synthetic gene circuits for metabolic control”,

ACS Synthetic Biology, 2015.

Signal processing in cancer cells. We seek to characterise how membrane receptors transmit

signals from the environment into the cell. We will focus on the EGFR, a hormone receptor with a

critical role in the onset, diagnostics, and treatment of cancer. We recently derived a closed-form

for the `1 norm of receptor activation. In this project you will apply and extend this result to more

detailed models for cellular sensing, aiming to detect weak spots in the network that could be used

as future targets for cancer drugs. The project will combine nonlinear ODE analysis and parameter

fitting from time-series data.

28

Reference:

[4] Oyarzun et al. “The EGFR demonstrates linear signal transmission”, Integrative Biology, 8(6),

2014.

Population dynamics of cellular growth models. We seek to understand how individual

metabolic responses shape the dynamics of a microbial population. We will rely on a cellular

growth model developed with our collaborators in the U. of Edinburgh. We will extend the model

to account for stochastic cell division and random partitioning of the cellular content. The project

will be in close collaboration with our colleagues from Edinburgh.

References:

[5] Weiße et al. “Mechanistic links between cellular trade-offs, gene expression, and growth”, PNAS,

2015.

The hydrodynamics of immiscible multi-fluid viscoelastic shear flows – Prof D Papa-

georgiou and Dr P Ray

Multi-fluid flows of immiscible viscous fluids arise in a wide range of applications ranging from

physiological (fluid films lining the inside of lung alveoli) to technological phenomena (coating

flows encountered in advanced manufacturing, among many others). Considerable attention has

been focussed on Newtonian flows governed by the incompressible Navier-Stokes equations, but in

many instances the modeling of the fluids as Newtonian is inappropriate and viscoelastic effects

need to be incorporated.

This project is concerned with the linear and nonlinear stability of paradigm shear flows, and in

particular the stability of core-annular flows consisting of an arrangement of a viscous viscoelastic

fluid adjacent to a second immiscible fluid in a vertical pipe. The flow dynamics are then driven

by the balance between gravity, surface tension, and both viscous and elastic forces. The objective

is to study the stability of this class of flows using a combination of analytical and computational

techniques. In particular, the case of a thin viscoelastic near-wall layer will be considered with a

lower-viscosity Newtonian fluid in the central core of the pipe - this is a technologically-relevant

physical set-up to be addressed initially, at least. Asymptotic analysis of the governing equations

will be used to develop evolution equations valid when the near-wall layer is thin. These equations

are novel and expected to be very rich mathematically; in analogy with the Newtonian case -

see [1]-[3] for a related problem - nonlocal equations would emerge that can fully take into account

inertial effects in the central less-viscous layer. These equations are of the active-dissipative infinite-

dimensional dynamical systems type and can support a host of complex phenomena including

spatiotemporal chaos. Such phenomena are of interest in enhancing heat and mass transfer in

applications.

Learning outcomes There will be several learning outcomes emerging from this project including:

1. You are unlikely to have encountered viscoelastic flows which are extensions of the Navier-

29

Stokes models that you have seen in fluid dynamics. These are a little more complicated and

several models exist. In this project you will concentrate on flows described by the so-called

Oldroyd-B model which is well-documented and well-studied in many contexts. You will learn

about the origin of fluid viscoelasticity and its mathematical modeling.

2. You will also learn about mathematical models involving several immiscible viscous fluids and

in particular the nonlinear boundary conditions to be applied across moving interfaces. The

problems are of the moving boundary type with the position of interfaces determined as part

of the solution.

3. You will learn to use asymptotic multi-scaled methods applied to nonlinear partial differential

equations in order to derive reduced-dimension evolution PDEs.

4. You will learn to analyze and solve such equations numerically using different methods in-

cluding spectral discretizations in space and high-order time discretizations appropriate for

stiff systems.

Relevant Courses. The following courses (or equivalent) from the Applied Mathematics MSc

would be useful but not all of them are essential: : Fluid Dynamics I, Fluid Dynamics II, Hydro-

dynamic Stability, Computational Partial Differential Equations, Asymptotic Analysis.

[1] Bassom, Andrew P.; Blyth, M. G.; Papageorgiou, D. T. 2010 Nonlinear development of two-layer

Couette-Poiseuille flow in the presence of surfactant, it Phys. Fluids, 22, Article Number: 102102.

[2] Kalogirou, A.; Papageorgiou, D. T.; Smyrlis, Y. -S. 2012 Surfactant destabilization and non-

linear phenomena in two-fluid shear flows at small Reynolds numbers, IMA J. Appl. Math., 77,

pp. 351-360.

[3] Bassom, Andrew P.; Blyth, M. G.; Papageorgiou, D. T. 2010 Using surfactants to stabilize

two-phase pipe flows of core-annular type, it J. Fluid. Mech., 704, pp. 333-359.

Structure formation inside channel flows under the action of electric fields – Prof.

D.T. Papageorgiou, Dr. R. Cimpeanu

Project Description. One of the primary goals in modern industrial manufacturing is the re-

duction in size of structures, such as electronic parts, to the order of tens or hundreds of microns

and even sub-micron structures in the case of applications in soft lithography. In this latter exam-

ple, electric forces have been successfully employed to manipulate interfacial instabilities to obtain

periodic pillar-like structures of less than 100 nm in cross-section. Schaffer et al. [1] illustrate

experimentally how highly accurate pattern formation can be obtained to aid manufacturing of

microelectronics (semiconductors and integrated circuit components in general).

This project aims to extend state-of-the-art mathematical models for multi-fluid systems to include

many of the additional effects/complications present in realistic conditions. Previous investigations

have examined the electrically induced instabilities arising in channel flows of stratified fluids mod-

30

elled as either perfect or leaky dielectrics [2,3]. However the inclusion of surface topography (which

assists the structure formation process) has only very recently been considered and is not yet fully

understood. Furthermore, it is also attractive to account for the effect of imposed background

flows, often encountered in the type of small scale devices considered here.

The project provides the interested student with the opportunity for both analytical and computa-

tional progress, as the numerical methods involved in studying the nonlinear features of the flows

are a fascinating topic in itself.

Learning Outcomes. Several learning outcomes are anticipated, which are associated with:

• Mathematical modelling of multi-fluid systems involving the coupling of multi-physics ele-

ments (hydrodynamic and electric), with a particular emphasis on the nonlinear boundary

conditions to be applied across moving interfaces.

• Multi-scale asymptotic analysis applied to nonlinear partial differential equations in order to

derive reduced-dimension evolution PDEs.

• Numerical methods appropriate for such systems, ranging from spectral methods for the

evolution PDEs to direct numerical simulations (DNS) of the full multi-fluid system.

• Result interpretation through rigorous quantitative analysis and critical assessment of the

practical implications of this research.

Relevant courses. The following courses (or equivalent) from the Applied Mathematics M.Sc.

program could prove useful, however not all of them are essential: Fluid Dynamics I, Fluid Dynamics

II, Hydrodynamic Stability, Numerical Solution of Ordinary Differential Equations, Computational

Partial Differential Equations, Asymptotic Analysis, Introduction to Partial Differential Equations.

References:

[1] E. Schaffer, T. Thurn-Albrecht, T.P. Russell and U. Steiner. Electrically induced structure

formation and pattern transfer, Nature 403, 874–877, 2000.

[2] L.F. Pease and W.B. Russel. Electrostatically induced submicron patterning of thin perfect and

leaky dielectric films: A generalized linear stability analysis, Journal of Chemical Physics 118,

3790–3803, 2003.

[3] F. Li, O. Ozen, N. Aubry, D.T. Papageorgiou and P.G. Petropoulos. Linear stability of a

two-fluid interface for electrohydrodynamic mixing in a channel, Journal of Fluid Mechanics 583,

347–377, 2007.

Inferring geometries and energy landscapes from particle trajectory data – Dr. A. B.

Duncan, Prof. G. A. Pavliotis

31

Cell membranes are highly complex structures, comprising of a heterogeneous combination of re-

gions in which membrane-bound proteins can be locally confined, as well as regions in which proteins

are excluded. Determining the parameters that regulate the local physical properties of the proteins

and their environment is often key to understanding important aspects of cell function. Thanks to

super-resolution imaging methods, it is now possible to record trajectories of individual proteins

in a variety of cellular systems, at tens-of-nanometers precision, [1]. Given the abundance of this

trajectory data, the question of whether one can extract any information regarding the structure

of the underlying membrane has been receiving a lot of interest, [2,3].

This project will investigate methods for using particle trajectory data to infer the structure of the

cell membrane. Given a model for the particle motion and a parametrization of the cell membrane,

this problem can be naturally formulated as a Bayesian inverse problem [4]. The solution of this

inverse problem is a probability distribution over the space of admissible membrane geometries,

which characterises how well any given surface configuration explains the observed trajectories.

Once an expression for this posterior distribution has been formulated, one can then use MCMC

methods to sample from this distribution, or a PDE-based approach to compute the most likely

configuration. Once this milestone has been reached, the student is open to investigate a number

of computational and/or analytical questions, such as:

1) The use of alternative models for particle transport beyond the Langevin model.

2) Dealing with the intractibility of computing the likelihood as the number of trajectories increases.

3) Dealing with low time resolution of the trajectory samples.

Clearly this project involves a strong computational component, and familiarity with a program-

ming language is essential (preferably C/C++). However, the project will also involve an amount

of analysis, and good knowledge of stochastic processes is strongly recommended.

References

[1] Giannone, Gregory, et al. “Dynamic superresolution imaging of endogenous proteins on living

cells at ultra-high density.” Biophysical journal 99.4 (2010): 1303-1310.

[2] Holcman, D., N. Hoze, and Z. Schuss. “Analysis and Interpretation of Superresolution Single-

Particle Trajectories.” Biophysical journal 109.9 (2015): 1761-1771.

[3] Masson, Jean-Baptiste, et al. “Mapping the energy and diffusion landscapes of membrane

proteins at the cell surface using high-density single-molecule imaging and Bayesian inference:

application to the multiscale dynamics of glycine receptors in the neuronal membrane.” Biophysical

journal 106.1 (2014): 74-83.

[4] Stuart, Andrew M. “Inverse problems: a Bayesian perspective.” Acta Numerica 19 (2010):

451-559.

Energy transport in anharmonic chains of oscillators and Fourier’s law – Prof Greg

Pavliotis

32

Anharmonic chains of oscillators, possibly in contact with thermal reservoirs, serve as prototype

models in statistical mechanics. A well known example of such a system is the Fermi-Pasta-Ulam

(FPU) lattice that has been used to investigate relaxation to equilibrium and equipartition of

energy. Such systems can be used via a combination of the theoretical analysis and numerical

simulations.

A very interesting question is whether Fourier’s law for heat conduction holds for chains of an-

harmonic oscillators. The goal of this project is to study the transport of energy along a chain

of anharmonic oscillators, coupled to thermal reservoirs at different temperatures at the two end.

The plan will be to derive a kinetic equation for the distribution of energy along the chain and to

compare the theoretical results obtained from kinetic theory with numerical simulations.

References

K. Aoki, J. Lukkarinen, H. Spohn Energy transport in weakly anharmonic chains. J. Stat. Phys.

124 (2006), no. 5, 11051129.

H. Spohn Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154 (2014),

no. 5, 11911227

Bonetto, F.; Lebowitz, J. L.; Rey-Bellet, L. Fourier’s law: a challenge to theorists. Mathematical

physics 2000, 128150, Imp. Coll. Press, London, 2000.

Power-law correlations and power-law distributions – Dr Gunnar Pruessner

Heavy-tailed statistics and power laws are eagerly observed across very many disciplines, such as

traditional condensed matter theory, particle physics, engineering, medicine and even the social

sciences. While they often have a deep-rooted (physical) explanation, in many cases it is difficult

to gain an understanding for the causes of the power law. They are often seen as the hallmark of

effective long-range interactions, but what these are and how they manifest (and can be measured)

possibly more directly remains unclear.

In this project I would like to determine the necessary conditions for power law statistics in “neu-

ronal avalanche” (e.g. Beggs and Plenz (2003), J Neurosci 23 11167 or Priesemann et al (2013),

PLoS Comp Bio 9 e1002985). It is clear that the data analysis has an effect on the results obtained.

What does that mean for the underlying data? Concretely, what are the necessary conditions for

the correlation function in the underlying temporal signal to produce the surprising power laws

found in this field? The project aims to clarify to what extent power laws obtained in Nature are

real and to what extent they are artefacts of the data analysis (see also Font-Clos et al (2015),

New J Phys 17 043066). I expect this project to have a strong analytical element, but numerical

simulations and data analysis will be important as well.

Statistics of topologically associating domains – Dr Gunnar Pruessner

Lots of statistics is available on the distance distribution of contact points along strands of DNA,

33

also known as TADs (topologically associating domains, see Lieberman-Aiden et al (2009), Science

326, 289). One may think of this as a bowl of spaghetti with individual strands heavily curled up into

balls and thus mostly touching themselves. The mathematics of such randomly folded polymers

is pretty well understood but the question remains which interaction is at work to explain the

statistics. What are the underlying correlation functions, what determines the characteristic sizes

observed? Can we develop tools to determine which TADs are deterministic?

This is primarily an exploratory project, which requires a solid literature review of the existing

mathematical models. Solid numerical estimates from the experimental data are available, but have

not been fully explained (e.g. https://biobabel.wordpress.com/tag/hi-c/). Numerical simulations

will probably be necessary. It would be very interesting to formulate the problem field theoretically.

Field theory for the cover times of random searches – Dr Gunnar Pruessner

Recently, it was demonstrated (Chupeau et al (2015), Nat Phys 11, 844) that the time it takes to

visit all sites of a given domain (known as the cover time) follows a universal law for non-compact

searches. In other words, up to a pre-factor, the time it takes to search all of space is independent

of the strategy used. The underlying argument, namely that visiting the last node is essentially a

matter of the tail of the first passage time distribution is very straight forward. These results hinge

crucially on the non-compactness and are thus related to the critical dimension of the walk used.

The aim of this project is to re-formulate the process field-theoretically, reproduce the results above

and explore a larger variety of searches. This is of particular interest for random walks on networks.

Some numerics will be useful, but the focus of this project is firmly on the theoretical aspects.

Approximation of nonautonomous invariant manifolds – Dr Martin Rasmussen

The project is based on a numerical scheme to compute invariant manifolds for time-variant dis-

crete dynamical systems, which was developed in [1]. Here a truncation of the Lyapunov-Perron

operator, used for the construction of invariant manifolds, results in a system of nonlinear alge-

braic equations which can be solved both locally using Newton, and globally using continuation

algorithms, yielding both local and global approximations of the desired invariant manifold. The

project aims in particular at using continuation techniques to study approximations of one- and

two-dimensional invariant manifolds.

References:

[1] C. Poetzsche and M. Rasmussen, Computation of nonautonomous invariant and inertial mani-

folds, Numerische Mathematik 112 (2009), no. 3, 449-483.

Period-doubling cascade of the randomly perturbed logistic map – Dr Martin Ras-

mussen

This project deals with bifurcations in the logistic map perturbed by bounded additive noise. It

is well known that the logistic map exhibits an infinite cascade of period-doubling bifurcations

34

before the onset of chaos. The aim of this project is to study this situation in the presence of

noise, both numerically and analytically. Preliminary results suggest that adding noise gives rise

to new bifurcation phenomena, which need to be understood better. The problem will be studied

on different levels of detail, given by an analysis of both the involved set-valued dynamical system

and random dynamical system.

References:

[1] J.S.W. Lamb, M. Rasmussen and C.S. Rodrigues, Topological bifurcations of minimal invariant

sets for set-valued dynamical systems, Proceedings of the American Mathematical Society 143, 9

(2015), 39273937.

[2] K.R. Schenk-Hoppe, Bifurcations of the randomly perturbed logistic map, Discussion Paper No.

353, Department of Economics, University of Bielefeld, 1997.

[3] H. Zmarrou and A.J. Homburg, Bifurcations of stationary measures of random diffeomorphisms,

Ergodic Theory and Dynamical Systems 27 (2007), 5, 16511692.

35

2 Projects with Prof. A. Ruban

Criss-Cross Interaction in 3D Boundary Layers

This project is aimed at analysing three-dimensional perturbations in otherwise two-dimensional boundary layer on a concave surface of a rigid body; see Figure 1.

It will be assumed that the perturbations are caused by a narrow hump on the surfacealigned with the x-axis. It is expected that the centrifugal effects will make the boundarylayer three-dimensional in a region of width ∆z ∼ Re−1/4. Here Re is the Reynoldsnumber. The project will consist of the following steps:

1. Derivation of the boundary-layer equations with centrifugal effects. This should bedone by applying the method of matched asymptotic equations to the incompressibleNavier–Stokes equations.

2. Identification of possible self-similar solutions of the boundary-layer equations.

3. Linear analysis of self-similar solutions for a small hump hight.

4. Numerical solution of non-linear self-similar flow regimes.

x

y

zV∞

Figure 1: Flow layout.

1

Upstream Influence in Hypersonic Boundary Layers

This project is aimed at analysing two-dimensional boundary layer on the surface ofa flat plate at large values of the free-stream Mach number M∞; see Figure 1. It will beassumed that the pressure perturbations (induced in the inviscid flow by the boundary-layer thickness) are strong enough to influence the boundary-layer flow itself. The mainobjective of this project is to demonstrate that in these conditions, the boundary-layerflow can no longer be treated as parabolic. Instead it allows for perturbations to travelupstream.

To perform this task two approaches will be used:

• In the first one the plate will be substituted by a thin body, and the analysis of theboundary-layer equations will concentrate on a small region near the leading edge.It is expected that the solution could be constructed in an analytic form using themethod of matched asymptotic expansions with coordinate x playing the role of asmall parameter.

• An alternative way is to construct (numerically) a self-similar solution to the boundary-layer equations, and then consider small perturbations to this solution.

l x

y

O

V∞

front shock

inviscid region

outer edge of the boundary layer

Figure 1: Flow layout.

1

Low-dimensional modelling of fluid flows – Dr. Ubaid Qadri and Prof. Peter Schmid

Previous studies have found that the dynamics of some classes of fluid flows can be modelled

reasonably well using simple equations. These studies have shown that, for example, the nonlinear

dynamics of jets and flames are very similar to the nonlinear behaviour of a van der Pol oscillator,

and that the dynamics of the flow behind a cylinder can be modelled using a Ginzburg-Landau

equation. This offers the possiblity of understanding the dynamics of complex flows by using

these simple equations/models as building blocks. In this project, we will be investigating this by

studying the dynamics of coupled Ginzburg-Landau equations numerically.

The first step would be to study the simple case of one oscillator and gain an understanding of how

its linear and nonlinear behaviour and its response to periodic forcing compare to an actual flow.

Two possibilities exist for the next step:

1. One project will investigate two (and more) coupled oscillators and investigate how the linear and

nonlinear behaviour of the coupled system is related to the properties of the individual oscillators.

The response of the coupled system to small amounts of periodic forcing will also be investigated.

This can then be compared to numerical simulations of the flow behind multiple cylinders to assess

the validity of the model.

2. Another project will investigate the effect of adding more complicated terms to the Ginzburg-

Landau equation. For example, a feedback term can be used to model the effect of acoustics in a

flow. The feasibility and physical validity of this approach will be assessed.

Prerequisites: Students should be familiar with numerical methods for the solution of ODEs and

PDEs, and have experience of using Matlab/Python. Some background and interest in fluid dy-

namics will be beneficial. This project would suit up to 2 students.

Singularly perturbed plasmonic eigenvalue problems – Dr Ory Schnitzer

Surface plasmons, collective electron-density oscillations at an interface between a dielectric and a

metal, exist as dispersive surface plasmon polariton waves, or, at specific frequencies, as localised

modes of metal nano-particles [1,2]. Their resonant excitation enables numerous optical-sensing

applications, and light localisation and manipulation on nanometric scales, below the diffraction

limit of standard optical apparatuses. Nano-metallic configurations featuring multiple geometric

scales are of particular interest. The latter support tuneable plasmon-resonance frequencies over a

wide spectrum and giant near-field enhancements; promising applications include nano-antennas,

plasmon rulers, plasmonic lasers, second-harmonic generation, and light trapping for photovoltaics.

In this project we shall obtain the surface-plasmon frequencies and modes of multiple-scale metallic

nano-structures by applying singular perturbation methods to the governing boundary eigenvalue

problem [4,5]. We shall focus on the configuration of a sphere nearly touching a substrate, which

is the one most common in experiments. The projects revolves around asymptotic analysis, but

there will also be a small amount of numerical work.

38

[1] S. A. Maier, Plasmonics: fundamentals and applications, Springer Science & Business Media,

2007

[2] I. D. Mayergoyz, R. F. Donald, Z. Zhenyu, Electrostatic (plasmon) resonances in nanoparticles,

Physical Review B, 72.15 155412. 2005

[3] R. T. Hill et al., Leveraging nanoscale plasmonic modes to achieve reproducible enhancement

of light, Nano letters, 10 4150-4154 2010

[4] O. Schnitzer, Singular perturbations approach to localised surface-plasmon resonance: nearly

touching metal nano-spheres, arXiv preprint arXiv:1508.04947

[5] O. Schnitzer, V. Giannini, R. V. Craster and S. A. Maier, Asymptotics of surface-plasmon

redshift saturation at sub-nanometric separations, arXiv preprint arXiv:1511.04895

Multiple-scale WKBJ analysis of wave propagation through micro-structured media –

Dr Ory Schnitzer and Professor Richard V. Craster

There is great current interest in the creation of artificial micro-structured metamaterials, which

can manipulate waves in ways unfamiliar in nature [1]. This endeavour was triggered around the

beginning of the 21st century, when outlandish ideas in electromagnetism came together with micro-

fabrication techniques, resulting in startling experimental demonstrations of such phenomena as

negative refraction, super-concentration, and cloaking; analogous implementations with acoustic,

elastic, and seismic waves quickly followed. These first demonstrations were carried out with sub-

wavelegnth metamaterials (SWM), where small-scale resonant structures endow the material with

negative effective properties. As the name implies, SWM are designed to operate at wavelengths

large relative to the constituent micro-scale elements; as a consequence, effective properties can be

obtained via standard techniques of asymptotic homogenisation.

An alternative approach for achieving anomalous wave propagation, at operating wavelengths on

the order of the microstructure dimensions, has its origins in ideas from solid-state physics and pho-

tonic band-gap materials [2]. Such structures are easier to manufacture, and avoid the resonantly

enhanced losses characteristic of SWM. The design task is complicated however by the fact that ef-

fective homogenised properties cannot in general be defined. This project explores a new approach

for analysing such structures. Exploiting the disparity between the macro- and micro- scales of

a given structure allows deriving a geometric-optics-like description, with periodically modulated

plane waves (Bloch waves) taking the place of plane waves as the the approximate local wave form.

The present project will focus on demonstrating this approach through an in-depth analysis of

a one-dimensional gradient photonic crystal, with the intent of describing transmission through

finite slabs, trapped modes, and resolution of certain Bloch-wave singularities by matching with

local analyses. The project entails learning the basics of wave propagation in periodic media,

gaining experience in asymptotic multiple-scale and WKBJ expansions [3], and Transfer-Matrix

and spectral numerical methods. There will be opportunities to interact with an active group of

Maths and Physics PhD students and postdocs working on related topics.

39

[1] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Metamaterials and negative refractive index,

Science 305.5685 788792 2004

[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic crystals: Molding the

flow of light, Princeton university press, 2011

[3] E. J. Hinch, Perturbation methods, Cambridge university press, 1991

Nonlinear electrohydrodynamics of fluid Quincke rotators – Dr Ory Schnitzer and

Professor Demetrios Papageorgiou

Symmetry suggests that an uncharged spherical particle exposed to a uniform electric field will

remain motionless. Under certain conditions and above a critical field magnitude, however, sym-

metry breaking occurs whereby the particle adopts a state of steady Quincke rotation. In the

inertial rgime, the governing dynamical system can be reduced to the Lorenz equations, whereby

at sufficiently strong fields a transition to chaos occurs. In recent experiments, oil drops subjected

to an electric field were also shown to exhibit a bifurcation to steady rotation, and a plethora

of unsteady states leading to chaotic motion; this is remarkable given the very small Reynolds

numbers in these experiments. In this project we shall try to make headway in the analysis of a

fluidic Quincke rotator. After formulating the governing equations, which at the fluid-fluid interface

nonlinearly couple hydrodynamics, electromagnetics, and surface-charge transport, we shall investi-

gate the dynamics and instabilities of the system using a combination of numerical and asymptotic

methods.

Modelling molecule accumulation and cellular economics during cell growth –Dr Vahid

Shahrezaei and Dr Samuel Marguerat (MRC)

Growth is the process by which cells create mass. Increase in mass goes together with an increase in

molecule numbers inside the cell. RNA and protein molecules coming from most genes accumulate

proportionally with size. Intriguingly, a single genome, that is a single set of genes, can produce

only a limited amount of molecules. This quantitative property of genomes restrict how large a cell

can become [1]. What determines the maximal number of molecules that a single set of genes can

produce remains unknown. Our collaborator on this project the group of Samuel Marguerat at MRC

Clinical Sciences Centre, have have lots of data in the unicellular eukaryote Schizosaccharomyces

pombe to study this question. They have measured number of RNA and protein molecules for

each gene in cells while they grow until they reach a maximal size. Coarse-grained mathematical

models of cellular physiology are quite useful. In this project we aim to make simple ODE models

of cellular economy as the cells grows. The model builds up on existing models of cells economy

[2] to include accumulation of molecules as a function of cell size in a systematic way. We will then

look at our model predictions and see if it is consistent with the available data. We will then refine

our models to explain the cellular resources allocation changes when cells become larger as it is

observed in the data. Ultimately, we would like to produce stochastic version of our model so that

40

in addition to average molecular levels it explains variability in molecular levels.

[1] Marguerat, S. and Bhler, J. (2012) Coordinating genome expression with cell size. Trends

Genet., 28, 5605.

[2] Scott, M., Klumpp,S., Mateescu, E.M. and Hwa, T. (2014) Emergence of robust growth laws

from optimal regulation of ribosome synthesis. Mol. Syst. Biol., 10, 747..

The role of diffusion and system size in intrinsic and extrinsic noise in genetic networks

– Dr Vahid Shahrezaei

Biochemical reactions involve small number of participating molecules. Therefore, the timing of

these reactions are stochastic. Mathematical modelling of stochastic dynamics of genetic networks

usually ignore spatial and diffusion effects. In this project we will use computational and analytical

approaches to assess the role of diffusion and spatio-temporal correlations in stochastic behaviour

of simple gene regulatory networks. In particular, we look at the effect of diffusion of extrinsic

factors on the intrinsic component of gene expression.

reference: J.S. van Zon, M.J. Morelli, S. Tanase-Nicola and P.R. ten Wolde, Diffusion of tran-

scription factors can drastically enhance the noise in gene expression, Biophys. J. 91, 43504367

(2006).

Investigation of the Instability Properties of Thread-Annular Flow – Dr A. Walton

The prediction of the behaviour of a fluid when it is injected into the body using a needle or

syringe is a challenging problem for theoretical fluid dynamicists. A plastic surgeon may not

only inject fluid into the body but also specially designed surgical threads that consist of synthetic

biocompatible materials. Through modelling the procedure mathematically, a better understanding

can be obtained of the way in which key parameters, such as the speed of injection, affect the fluid

flow characteristics within the syringe. Ultimately, it is hoped that the entire procedure can be

carried out more proficiently, with a minimum of surgical trauma.

A simplified version of the injection process can be modelled mathematically by considering the

axial flow between concentric cylinders with the inner cylindrical core (representing the thread of

material) moving at a constant velocity. The thread itself ideally should be modelled as a form of

compliant surface. A simple model for this would be one which takes into account the elasticity

properties of the thread and the deformation of its surface. This is a new problem that has not

been studied in any detail before. We will assume that any perturbations of the thread surface are

small, so that their effect on the basic flow will be a linear one. The aim of the project is then to

investigate the linear stability properties of the ensuing flow at high Reynolds number and/or at

finite Reynolds number. The former analysis requires a knowledge of asymptotic methods, while

the latter would involve computation with Matlab and require the student to write their own codes.

41

The procedures will be similar to those for a solid inner cylinder but there is likely to be new and

interesting effects due to the interaction with the compliant surface.

References:

P. G. Drazin: Introduction to Hydrodynamic Stability (C.U.P.)

A. G. Walton: Stability of circular Poiseuille-Couette flow to axisymmetric disturbances Journal

of Fluid Mechanics (2004) Vol. 500, p.169–210.

Useful course to take: M4A30 Hydrodynamic stability.

Roll/streak/wave interaction in shear flows and turbulent spot formation – Dr A.

Walton

In recent times a dynamical systems picture of laminar-turbulent has emerged in which equilibrium

solutions of the Navier-Stokes equations play a key role in transition and turbulent dynamics. These

equilibrium solutions consist of three crucial components: a roll flow in the cross-stream plane, a

streamwise streak and a three-dimensional wave. These three components interact in a mutually

sustaining manner in which the roll flow drives a spanwise-modulated streak which is itself unstable

to the wave. The wave then self-interacts nonlinearly to reinforce and re-energize the roll flow. At

high Reynolds number this interplay can be expressed in terms of an asymptotic theory known as

vortex-wave interaction. It is possible to formulate and solve the interaction equations for a wide

variety of viscous shear flows. In this project we concentrate on one of the well-known properties

of the solution of such systems: the appearance of solutions which localize as the amplitude of the

motion is increased. This localization is thought to be connected to the experimental observation

that at relatively small disturbance levels viscous travelling waves cause an instability of the flow

which leads to the formation of turbulent spots.

We propose to study the interaction between a roll, streak and viscous wave in an asymptotic

suction boundary layer. It can be shown that the main part of such an interaction is governed

by the nonlinear system:∂v

∂y+∂w

∂z= 0, v

∂w

∂y+ w

∂w

∂z=∂2w

∂y2,

subject to the boundary conditions

w = A sin z, v = −1 on y = 0, w → 0 as y →∞.Here (v, w) are the normal and spanwise components of the roll flow and the condition on w at

the wall represents the wave forcing with amplitude A. It can be shown that this system has a

nonlinear exact solution for small values of the spanwise coordinate z up to a critical amplitude

for A. We propose to carry out numerical calculations of the full equations to investigate how the

localization in the system is related to the singularity in the local exact solution.

References

Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex-wave inter-

action states in plane Couette flow. J. Fluid Mech. 721, 58–85.

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Dempsey, L.,J., Deguchi, K., Hall, P. & Walton, A. G. 2015 Localized vortex/Tollmien-

Schlichting wave interaction states in plane Poiseuille flow. Preprint available on request.

Useful course to take: M4A30 Hydrodynamic stability.

Wave-flow interaction: reflection, absorbtion and radiation – Prof. X. Wu

Interactions of waves with shear flows is a fundamental process that takes place in many techno-

logical applications as well as in nature. For example, the problem of sound waves impinging on

the boundary layer over an wing is of importance for supersonic flights, while interaction of gravity

waves with a shear flows in the atmosphere or ocean is of me- teorological relevance. As a wave

impinges on, and then propagates through, a shear flow, it may be reflected and absorbed by the

latter, and exhibit several interesting behaviours. In certain circumstances, the reflected wave may

be stronger than the incident wave. This is referred to as over-reflection, in which case the shear

layer acts as a wave amplifier (or soundboard). The wave activity may concentrate in a thin region

known as the critical layer, where the shear-flow velocity coincides with the phase speed of the

wave. A large amount of momentum and energy may be trapped/deposited in the critical layer,

and non- linear effects may become important there (Haberman 1972). Furthermore, a shear flow

may support so-called eigen modes/waves (i.e. characteristic wave motions of the system) that

radiate to the far field. In such a case, the flow exhibits an extreme sensitive response to external

incident waves in the sense that the reflected wave becomes unbounded, a phe- nomenon referred

to as resonant over-reflection (e.g. McIntyre & Weissman 1978). Extra physical effects have to be

reinstated in order to regularize the problem.

This project involves two aspects, which may be be investigated independently by two students.

The first one concerns reflection and absorbtion of a sound wave by a boundary layer (or a free

shear layer) in the linear and nonlinear regimes. The role of reflection in instability (Wu 2014) may

be investigated as well depending on the interest of the student.

The second is concerned with existence of resonant over-reflection, and with the cou- pling of

external waves and internal eigen modes at the point of resonant over-reflection. Appropriate regu-

larization effects are to be introduced so that physically meaningful so- lutions can be constructed.

References:

1. Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths. 51, 139-161.

2. McIntyre, M.E. & Weissman, M.A. On radiating instability and resonant overreflec- tion. J.

Atmos. Sci. 35, 1190-1196.

3. Wu, X. 2014 On the role of acoustic feedback in boundary-layer instability. Phil. Trans. R. Soc.

Lond. 372 (2020), 20130347.

Vortex roll-up and paring in shear layers – Prof. X. Wu

Shear flows are often unstable. As a result, instability waves in these flows amplify and develop

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into concentrated vortices, which then undergo pairing. Such vortex roll-up and pairing represent

some of most remarkable features observable in nature and laboratory. They are recognized to

play a crucial role in momentum and mass transport, entrainment and mixing of species and noise

production.

A famous theory explaining vortex pairing was proposed by Kelly (1968), and it in- volves the sub-

harmonic parametric resonance. However, neither vortex roll-up nor pairing was actually demon-

strated and the critical layer, the key region controlling vortex roll-up and pairing, was not analysed.

This project is going to revisit this theory. By investigating in detail the critical-layer dynamics

(Goldstein & Leib 1988), a more complete theory will be presented. A set of evolution equations

governing the spatial nonlinear development of interacting vortices will be derived using the matched

asymptotic expansion technique in junction with the multiple-scale method. This project involves

some analysis, but the main task is numerical, that is, to solve the evolution system numerically in

order to predict how vortices roll up and merge (i.e. pair up).

References:

1. Kelly, R. 1968 On the resonant interaction of neutral disturbances in the inviscid shear flows. J.

Fluid Mech. 31, 789-799.

2. Goldstein, M.E. & Leib, S.J. 1988 Nonlinear roll-up of externally excited free shear layers. J.

Fluid Mech. 191, 481-515.

Two velocity formulation of the compressible Navier-Stokes-Fourier equations – Dr

Ewelina Zatorska, room 6M22

The purpose of this project is to analyze the compressible Navier-Stokes-Fourier system. This is

a system of partial differential equations of mixed hyperbolic-parabolic type that describes the

evolution of the density, the velocity and the temperature of the compressible viscous fluid [2].

When the viscosity coefficients in such system depend of the density it is possible to introduce

another formulation of these equations based on artificial velocity vector field [1]. In addition to

the second velocity this formulation involves a generalization of the temperature. This system

was already known in physics [3] but relatively weakly studied from the mathematical point of

view. It turns out that the information encoded in the new formulation is closely related to the

hypocoercivity of nonlinear systems of PDEs [5]. The new two-velocity and in fact two-density

formulation of Navier-Stokes system is also somehow similar to the equations of two phase flows.

The purpose of the project is to ivestigate these connections. The starting point could be analysis

of the one-dimensional system: stability, existence, uniqueness of solutions. It can also be focused

on analysis of the linealized system and existence of solutions close to equilibrium. Finally an

interesting point is to check when both of the formulations: the classical and the two-velocity one

are equivalent.

[1] D. Bresch, B. Desjardins, and E. Zatorska: Two-velocity hydrodynamics in fluid mechanics: Part

II Existence of global entropy solutions to the compressible Navier-Stokes systems with degenerate

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viscosities, JMPA, Vol. 104, no. 4, 801836 (2015).

[2] A. Novotny and I. Straskraba: Introduction to the mathematical theory of compressible flow,

Oxford University Press, Oxford, (2004).

[3] S.M. Shugrin: Two-velocity hydrodynamics and thermodynamics, J. Appl. Mech. Tech. Phys.

39, p. 522537, (1994).

[4] A. Vasseur and C. Yu: Existence of global weak solutions for 3D degenerate compressible

NavierStokes equations, arXiv:1501.06803, (2015).

[5] C. Villani: Hypocoercivity, Mem. Am. Math. Soc., vol.202:950, (2009).

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