Applied Mathematics MSc Projects 2017-2018 Imperial College London

Applied Mathematics MSc Projects 2017–2018
                        Imperial College London

Last revised: December 4, 2017

Professor 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. Deep learning.
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 algorithms * 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
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

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stem cells, (c) analysis of behavioral time-series of motion of C elegans: Over 10,000 videos of
freely moving 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. Dimensionality reduction and machine learning: a graph-theoretical perspective. The project will
explore the mathematical underpinnings of various dimensionality reduction and machine learning
methods in discrete settings, such as graphs and networks, focusing in particular on connections
between underlying geometry, dynamics on the graph, spectral properties, and graph topology at
multiple scales. Theoretical tools and ideas from statistical physics, spectral graph theory, and
wavelet analysis, will be brought to bear on underlying mathematical questions, as well as on
specific methods of interest, for example the neural network architectures used in deep learning
algorithms. The project will also involve developing better understanding and improvement of
current techniques for clustering/community detection and graph sparsification, and applications
to multiplayer graphs and other hierarchical data. There will also be ample opportunities to work
with real data and implement novel techniques and algorithms.
With Dr Asher Mullokandov.
6.Temporal Graph Signal Analysis - maximal signal compressibility/decomposition:
The dynamics taking place on a network and its structure are intimately connected. It is possible to
use a Fourier-like theory to decompose the signal on the nodes of a network using e.g. the eigenbasis
of the Laplacian of said network and there exists a natural analogy between the eigenvalues of the
Laplacian and the notion of spatial frequencies.
In this project, we are interested in in exploring the effect of the basis used on the characterisation
of the signal. Two possible directions are: i) is it possible to increase signal spectral representation
compressibility, in terms of the frequencies significantly present in the spatial power spectrum,
by altering the underlying network for example by ”increasing its symmetries”? This would be
particularly helpful in the context of understanding network (quasi)-symmetries and their role in
signal characterisation. ii) is there a universal basis , i.e. a (class of) network(s) that possesses
key topological features that allows for an informative decomposition of the signal on any network?
Such a basis would be particularly useful when the network supporting the signal is partially/not
known?
With Dr Paul Expert.
7. Effect of multi-failure in graphs
Graphs are often used to describe complex physical interactions and energy exchanges in a variety
of domains, from power grids to molecules. An important question to tackle in this context is to
precisely understand the effect of affecting, or even destroying a particular connection to the rest
of the graph. Recently, a powerful method has been developed using diffusion on graphs to obtain
this information. The aim of this project is to extend this method to the case where two or more

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connections are affected of destroyed, and apply it to real world example to assess the importance
of such events. For power grid networks, such events can lead to cascading of failures resulting in
a blackout of part of the network.
Requirements: basic notions of graph theory and knowledge of Python programming
With Dr Alexis Arnaudon.
8. Helicity of simplicial complexes
Simplicial complexes are discrete structures composed of a point, lines, faces, etc... with a specific
relationship between them such that they can be seen as discrete manifolds. This implies that
a discrete notion of differential calculus exists on these structures. In fact, simplicial complexes
are often used as discretization of smooth manifolds such that the original differential structure
is approximately preserved. One of the main application of such approach is for simulations of
fluid dynamics. This project will not directly be about fluid dynamics but will be to implement
the continuous notion of helicity of flow lines to simplicial complexes. This notion relates to
how entangled flow lines of a fluid are and can surprisingly be computed only using differential
geometric tools. Helicity provides important topological information about the simplicial complex,
when assumed to be be a discretization of a continuous space but more importantly when it is
obtained by some real world three dimensional datasets, in which case we can learn about the
topology of the data itself.
Requirements: basic notions of differential geometry and graph theory, knowledge of Python pro-
gramming
With Dr Alexis Arnaudon.
9. Metabolic Flux analysis and directed graphs — see Dr Diego Oyarzun
10. Stochastic dynamics of structured populations see Dr Philipp Thomas

                                                  3

Quantum random walks and topological phases – Dr Ryan Barnett
In this project, we will investigate a quantum version of the classical random walk – the quantum
random walk. The quantum walker explores simultaneously all of the possible paths of her classical
counterpart.
A particularly interesting situation, receiving recent attention is as follows. The walker carries with
her a spin (you can think of this as a vector). The walker initially stands at the origin and her spin
points up. The spin is rotated by some specified angle (step 1). Next the walker takes a quantum
step both to the left and right – the probability to go left (right) is proportional to the up (down)
spin component. Step 1 and step 2 are then repeated N times. In [1] it was realised that some of
these random walks (for different initial conditions) can have peculiar features – the walker can be
confined to only a small region of the lattice. This is due to the underlying topological structure
of the equations governing the dynamics.

   Initial

   Step 1

   Step 2

After becoming familiar with a few recent results/papers on this topic, the aim of the project is
to find variations of these models that map onto established topological problems in condensed
matter physics. In particular, by adding some additional slow time dependence, the realisation of
the Thouless charge pump [2] will be sought.
[1] Takuya Kitagawa, Mark S. Rudner, Erez Berg, and Eugene Demler, “Exploring topological
phases with quantum walks” Phys. Rev. A 82, 033429 (2010)
[2] F. Duncan M. Haldane, “Nobel Lecture: Topological quantum matter” Rev. Mod. Phys. 89,
040502 (2017)

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Can superfluids rotate rigidly? – Dr Ryan Barnett
A superfluid is a substance that flows with zero viscosity. When mechanically rotated (e.g. when
in a spinning bucket) superfluids typically will form a vortex lattice, in stark contrast to classical
fluids. This is because the velocity of a superfluid is the gradient of a phase having quantum
                           ~
mechanical origins: v = m    ∇θ. As a result, the voticity, ∇ × v, is zero everywhere except the
positions marking the vortex centres (where the phase is ill-defined). Rigid rotation – where ∇ × v
is a non-zero constant everywhere – clearly cannot be obtained for this case.
The above does not apply when the atoms forming the superfluid have internal spin degrees of
freedom. For this case, the vorticity is related to the spin direction, denoted by unit vector n, as
                                                     ~
                                  ∂x vy − ∂y vx =      n · (∂x n × ∂y n)
                                                    2m
(restricting to two spatial dimensions for simplicity). This elegant and geometric equation is known
as the Mermin-Ho relation.
The aim of this project is to investigate if spinor fluids can rotate in ways similar to their classical
counterpart. That is, can such steady state solutions be found in the rotating frame of reference?
A recent affirmative result along these lines was obtained in [1]. Spinor superfluids with spin-orbit
coupling – a topic of considerable recent experimental progress – is likely a crucial ingredient and
will be investigated in this context.
[1] Sandro Stringari, “ Diffused Vorticity and Moment of Inertia of a Spin-Orbit Coupled Bose-
Einstein Condensate” Phys. Rev. Lett. 118, 145302 (2017)

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John W. Barrett

Both projects concern macroscopic fluid models for dilute polymers such as Oldroyd–B or FENE–P.
These models consist of the standard time-dependent Navier–Stokes equations with the divergence
of an extra stress tensor on the righthand-side, which is coupled to an equation for this extra stress
tensor involving the velocity of the fluid. A formal a priori entropy bound is presented for either
model in [1]. Such an estimate is useful in (i) studying the long-term behaviour of the model,
(ii) proving existence of a global-in-time weak solution to the model and (iii) constructing stable
numerical approximations.

                         (1) The Use of the Matrix–Logarithm in
                               Modelling Complex Fluids

The standard entropy estimate for Oldroyd–B in [1] is only meaningful if one can show that the
extra stress tensor remains symmetric positive definite throughout the evolution. Moreover, it
seems one can only prove existence of a global-in-time weak solution to the Oldroyd–B model in
two space dimensions, and convergence of a numerical approximation, if, in addition, a diffusion
term is added to the stress equation. Although this can be achieved, see e.g. the finite element
approximation constructed in [2], it is not straightforward. This positive definite constraint can be
avoided by rewriting the system using the logarithm transform proposed in [3]. Furthermore, the
formal entropy structure in [1] can be adapted to this transformed system, see [4]. This project will
consider the use of this logarithm transformation from both an analysis and practical viewpoint.
                           (2) Finite Element Approximation of
                                   the Oldroyd–B Model

Convergence, as the time step ∆t and spatial discretization parameter h tend to zero, of a finite
element approximation of the Oldroyd–B model in two space dimensions with a diffusion term added
to the stress equation is proved in [2]. A key ingredient is building the finite element approximation
to satisfy a discrete version of the formal entropy bound satisfied by the Oldroyd–B model, see [1].
This numerical approximation at present has three undesirable features: (i) a time step constraint
of the form ∆t ≤ C h2 in order to prove convergence, (ii) at each time level a nonlinear system
involving the approximations of both the fluid velocity and extra stress tensor has to be solved and
(iii) a complicated approximation of the advection term in the extra stress equation in order to
mimic the entropy bound in [1]. This project will consider (a) the use of the characteristic Galerkin
method, see [4], to try to avoid (i) and (iii), and (b) the use of ideas in [5], which are from related
models, to avoid (ii).

                                           Prerequisites
These projects will involve analysis and computation (e.g. Matlab). Knowledge of partial differential

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equations and associated finite element approximations is essential.

                                            References
[1] D. Hu and T. Lelièvre, New entropy estimates for Oldroyd-B and related models, Commun.
Math. Sci., 5, (2007), 909–916.

[2] J. W. Barrett and S. Boyaval, Existence and approximation of a (regularized) Oldroyd-B model,
Math. Models Methods Appl. Sci., 21, (2011), 1783–1837.

[3] R. Fattal and R. Kupferman, Constitutive laws for the matrix logarithm of the conformation
tensor, J. Non-Newton. Fluid Mech., 123, (2004), 281–285.

[4] S. Boyaval, T. Lelièvre and C. Mangoubi, Free-energy-dissiptive schemes for the Oldroyd-B
model, ESAIM: Math. Model. Numer. Anal., 43, (2009), 523–561.

[5] J. Shen and X. Yang Decoupled, energy stable schemes for phase-field models of two-phase
incompressible flows, SIAM J. Numer. Anal., 53, (2015), 279–296.

Modelling inertial particle transport in the ocean – Dr Berloff
Motivated by the modelling of floating plastic debris on the ocean surface, the project is intended to
study the transport of inertial (i.e., those having finite size and being lighter than water) particles
in the ocean. The goal is to investigate a hierarchy of models for inertial particle movement by
turbulent ocean circulation solutions: modelling particle trajectories on the top of fluid-dynamical
ocean eddies and currents. We’ll consider quasigeostrophic approximation for fluid motion, Ekman
boundary layer corrections, and Maxey-Riley equation and its approximations for the inertial par-
ticles. The project will provide an exciting combination of geophysical fluid dynamics, ODEs and
PDEs, numerical modelling and statistical analyses.

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Simulation and control of agent‐based models
                           José A. Carrillo & Dante Kalise
In this project, we will study large‐scale multi‐agent systems modelling animal
and/or human collective behaviour. We will introduce control actions over the
system either via leaders, as in a shepherd’s dog controlling a herd of sheep, or
undercover agents, such as bot accounts in social networks. While agent‐based
modelling is a well‐established tool to study social dynamics, we will explore analytic
and computational techniques for the synthesis of external control actions to induce
different dynamical patterns such as consensus, alignment, or swarming.
1. Wolf‐pack (Canis lupus) hunting strategies emerge from simple rules in computational
   simulations. C. Muro, R. Escobedo, L. Spector, R.P. Coppinger, Behavioural Processes 88
   (2011) 192– 197.
2. Group size, individual role differentiation and effectiveness of cooperation in a
   homogeneous group of hunters. C. Muro, R. Escobedo, L. Spector, R.P. Coppinger, J. Roy.
   Soc. Interface 11 (2014) 20140204.
3. H. Tompkins and T. Kolokolnikov, Swarm shape and its dynamics in a predator‐swarm
   model, to appear, SIAM Undergraduate Research Online.
4. J. A. Carrillo, Y. Huang, S. Martin, Nonlinear stability of flock solutions in second‐order
   swarming models, Nonlinear Analysis: Real World Applications 17, 332–343,
   2014.40204.
5. Invisible control of self‐organizing agents leaving unknown environments. G. Albi, M.
   Bongini, E. Cristiani and D. Kalise, SIAM Journal on Applied Mathematics, 76(4)(2016),
   1683‐‐1710.
6. Modeling and control through leadership of a refined flocking system. A. Borzì and S.
   Wongkaew, Math. Models Methods Appl. Sci. 25, 255 (2015).

Aggregation patches for a model of laser traps
                        José A. Carrillo and Matias Delgadino
We will study a model proposed in [3,4] 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. This equation has similar properties to the Keller‐Segel model for which
aggregation patches have been obtained. The first part of the proposed topic will be to do a
similar existence‐uniqueness results for solutions of these equations in the case without
diffusion like in [1]. Explicit solutions will be sought for both in the repulsive and attractive
cases. The structure of the skeleton obtained for this new equation will be explored by
numerical methods as in [1].

1‐ Bertozzi, T. Laurent, and F. Léger. Aggregation via the Newtonian potential and
   aggregation patches. M3AS, vol. 22, Supp. 1, 2012.
2‐ https://cims.nyu.edu/~leger/aggregation.html
3‐ Long‐range one‐dimensional gravitational‐like interaction in a neutral atomic cold gas,
   M. Chalony, J. Barré, B. Marcos, A. Olivetti, and D. Wilkowski. Phys. Rev. A 87, 013401
   https://hal.archives‐ouvertes.fr/hal‐01141171/document
4‐ Non‐equilibrium Phase Transition with Gravitational‐like Interaction in a cloud of Cold
   Atoms, J. Barré, B. Marcos, and D. Wilkowski, Phys. Rev. Lett. 112, 133001.
   http://arxiv.org/pdf/1312.2436.pdf;

Geometric integrators for fluid in a hosepipe – Dr Cotter Geometric integrators are numeri-
cal algorithms that are designed to preserve fundamental conservation properties of the ODE/PDE.
Recently, Gay-Balmaz and Putkaradze (Gay-Balmaz, Franois, and Vakhtang Putkaradze. ”Exact
geometric theory for flexible, fluid-conducting tubes.” Comptes Rendus Mcanique 342.2 (2014):
79-84) produced a geometric theory for a fluid moving in a flexible pipe. This describes, for ex-
ample, the motion of a pipe when you switch on a tap, and the pipe flexes all over the place. In
this project, we will take this geometric theory, and build a discretised version of it, using a finite
element discretisation; then we’ll investigate its properties in numerical experiments, facilitated
using the Python code generation library Firedrake (firedrakeproject.org).
Advanced numerical algorithms for the simulation of weather fronts – Dr Cotter The
phenomenon of frontogenesis (the process of front formation in weather systems) was first given
a mathematical description by Hoskins and Bretherton (1972), by proposing a model called the
semigeostrophic equations, and providing a coordinate transformation that exposes the structure
in these equations. In the 1990s, this transformation was interpreted as the solution of an optimal
transport problem in the setting of Kantorovich, and this led to algorithms for solving these equa-
tions which showed the process of front formation. However, these algorithms are very slow and so
have not found their way into use by meteorologists. This is a shame, because these SG solutions
have exposed some issues with operational weather forecasting models which, if solved, could dra-
matically improve the accuracy of weather forecasts. Recently, a faster algorithm has been invented
for these transport problems (Mrigot, Quentin, Jocelyn Meyron, and Boris Thibert. ”An algorithm
for optimal transport between a simplex soup and a point cloud.” arXiv preprint arXiv:1707.01337
(2017)) together with open source software for implementing them. In this project we will adapt
this code to solve the frontogenesis problem.

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Vertical coordinate mappings for numerical
                  weather prediction
                Dr Colin Cotter and Dr Jemma Shipton
                             November 30, 2017

    Finite element methods are attractive for numerical weather prediction and
climate modelling due to the flexibility of the choice of the underlying grid. In
the horizontal direction, this means that unstructured grids can be used, thereby
avoiding the parallel scalability problem encountered when using structured
grids that inevitably have points clustered in the polar regions. In the vertical
direction, there is freedom to deform the mesh to fit the shape of the Earth’s
surface, i.e. to follow mountain ranges. However, this deformation can lead to
unwanted numerical effects such as decreased accuracy in the representation of
the horizontal pressure gradient and noisy tracer transport. These effects will
only be exacerbated as model resolution increases, enabling representation of
steeper slopes.
    Accurate modelling of waves generated by mountains is important due to
the effect they have on both the local and global dynamics. Local effects in-
clude strong downslope winds, enhanced precipitation and clear-air turbulence
generated in the lee of mountains. Global effects are due to the transport of
momentum by vertically propagating waves. New numerical schemes for solv-
ing the equations used in numerical weather prediction have to prove that they
are capable of modelling these effects. The first step is to introduce a small
mountain and subsequent mesh deformation and observe the effect this has on
a simulation of a resting atmosphere. Tests then increase in difficulty, incorpo-
rating flow past the mountain in linear and nonlinear regimes, more complex
mountain range profiles and more complex vertical stratification profiles.
    Recently, in collaboration with the Met Office, we have developed a finite
element model for numerical weather prediction that exhibits the required con-
servation and large scale wave propagation properties for accurate modelling of
the atmosphere [?]. The schemes developed in this model will provide the basis
for schemes used in the next generation Met Office model. This project would
involve using our model, Gusto, to simulate 2D flow over idealised mountains
using a variety of different vertical coordinate mappings based on the ideas pre-
sented in [?], which suggests using a mapping that smooths coordinate surfaces
with height.

                                       1

References
[1] Cotter, Colin J and Shipton, Jemma, Mixed finite elements for numerical
    weather prediction, Journal of Computational Physics, 231(21): 7076–7091,
    2012
[2] Klemp, Joseph B, A terrain-following coordinate with smoothed coordinate
    surfaces, Monthly weather review, 139(7): 2163–2169, 2011

                                     2

MAXWELL’S EQUATIONS AND NEGATIVE INDEX MATERIALS

                         RICHARD CRASTER AND HARSHA HUTRIDURGA

   Many modern-day technological developments have their roots in electromagnetic theory; radio,
radar, microwaves amongst many others. Maxwell’s equations for the electric and magnetic fields
are considered amongst the most influential equations in all of science. These equations are quite
challenging, for instance, the time harmonic Maxwell’s equations in three dimensions take the form
                                        
                                          Curl E = iωµH
(1)
                                          Curl H = −iωεE
where E and H are vectors. Here µ and ε are material parameters representing permeability and
permittivity respectively. The frequency ω ∈ R comes from the time dependence e−iωt in the
time-harmonic wave.
   The permittivity differs for different materials and is often complex-valued. It can have negative
real part but non-negative imaginary part. Essentially, the medium is transparent if the real part
of ε is positive and is opaque if the real part of ε is negative (which is the case for many metals).
   Veselago [Ves68] studied the implications of having both µ and ε negative. Materials with
negative parameters are termed negative index materials. When both µ and ε negative, we obtain
something called left-handed materials which yield some interesting effects on the wave propagation.
   In the last two decades, there has been renewed interest in the investigation of negative index
materials. Construction of such materials, both theoretically and experimentally, have been made
possible with the help of meta-materials. Meta-materials are composite materials which are nothing
but the assemblies of smaller components consisting of ordinary materials which behave effectively
in a way that is not known for the naturally occurring materials. Mathematically speaking, this
question of studying effective behaviour of composites falls under the theory of homogenization for
partial differential equations.
   The mathematical theory of homogenization [BLP78], loosely speaking, corresponds to the study
of differential equations with highly oscillatory coefficients. For example, take the solution uη (x)
to a differential equation
                                           Aη uη (x) = f (x),
where Aη is a differential operator with η-periodic coefficients. A typical result in homogenization
theory is to show that uη ≈ uhom in the regime η  1, i.e. in the regime where the coefficients in
the operator Aη oscillate more and more. Furthermore, one shows that the approximation uhom
solves the homogenized equation
                                       Ahom uhom (x) = f (x),
with Ahom an effective differential operator whose coefficients are η-independent.
   In the context of time-harmonic Maxwell’s equations (1), periodic homogenization problem trans-
lates to studying the solution family (E η , H η ) to the system
                                       
                                         Curl E η = iωµη H η
(2)
                                         Curl H η = −iωεη E η
                                                  1

2                           RICHARD CRASTER AND HARSHA HUTRIDURGA

where the permeability-permittivity pair µη , εη are η-periodic functions of the spatial variable. It
so happens that the effective equation in this scenario is again a system similar in structure to (2)
with effective permeability-permittivity pair µhom (ω), εhom (ω). Note that the effective coefficients
are frequency dependent.
   At first this project will be concerned with the understanding of
     • constructing thin wire structures with extreme values to obtain negative effective permit-
       tivity εhom (ω) (some references of interest are [PHSY96, FB97])
     • obtaining negative effective permeability µhom (ω) from periodic split ring structures (refer-
       ences of interest are [PHRS99, BBF2009, BS2010])
     • combining the construction in wire structures and that in periodic split ring structures
       to obtain both the effective parameters µhom (ω), εhom (ω) to be negative simultaneously
       (references of interest are [SPVNS96, LS2016]).
   With a good understanding of the above mentioned constructions, the following finer (and more
difficult) objectives can be set:
     • Exploring the range of applicability of the aforementioned constructions in the frequency
       variable (both theoretically and numerically).
     • Lifting the lossy-layer construction in [LS2016] to arrive at the same results.
  Accomplishing the above objectives will require the student to learn some basic notions from the
theory of homogenization – preferably, by working on the electrostatic problem. In fact, studying
the periodic homogenization problem for (2) is a classic example for homogenization with resonance
which is quite difficult compared to homogenization problems in electrostatics.
  This project is quite challenging as it blends both the mathematical analysis of differential
equations and some associated computational techniques. A successful completion of this project
should result in a journal publication. This project could involve collaboration with a group from
the department of Physics and there will be opportunities to interact with them, attend group
meetings, and work within an active group of PhD students and post-docs. It is an exciting and a
very topical area of research with plenty of applications and opportunities.

Pre-requisites: Student taking on this project is required to have some basic understanding of
differential equations and functional analysis. It is also preferable to have some working knowledge
of computational environments such as MATLAB (Note that this project might also give the student
a possibility to get acquainted with COMSOL).

References:
   [Ves68] v.veselago, The electrodynamics of substances with simultaneously negative values of
ε and µ, Soviet Phys. Uspekhi, Vol 10, pp.509–514, 1968.
   [BLP78] a.bensoussan, j.-l.lions, g.papanicolaou, Asymptotic analysis of periodic struc-
tures, 1978.
  [PHSY96] j.b.pendry, a.j.holden, w.j.stewart, i.youngs, Extremely low frequency plas-
mons in metallic mesostructures, Phys. Rev. Lett., Vol 76, 1996.
 [FB97] d.felbacq, g.bouchitté, Homogenization of a set of parallel fibres, Waves Random
Media, Vol 7, pp.245–256, 1997.

MAXWELL’S EQUATIONS AND NEGATIVE INDEX MATERIALS                            3

  [PHRS99] j.b.pendry, a.j.holden, d.j.robbins, w.j.stewart, Magnetism from conductors
and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Tech., Vol 47, 1999.
  [BBF2009] g.bouchitté, c.bourel, d.felbacq, Homogenization of the 3D Maxwell system
near resonances and artificial magnetism, C. R. Math. Acad. Sci., Vol 347, pp.571–576, 2009.
  [BS2010] g.bouchitté, b.schweizer, Homogenization of Maxwell?s equations in a split ring
geometry, Multiscale Model. Simul., Vol 8, pp.717–750 2010.
  [SPVNS96] d.r.smith, w.j.padilla, d.c.vier, s.c.nemat-nasser, s.schultz, Composite
medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., Vol 84, 2000.

  [LS2016] a.lamacz, b.schweizer, A negative index meta-material for Maxwell?s equations,
SIAM J. Math. Anal., Vol 48, pp.4155–4174 2016.
  R.C.: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.
  E-mail address: r.craster@imperial.ac.uk

  H.H.: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.
  E-mail address: h.hutridurga-ramaiah@imperial.ac.uk

Characterization of spatially graded metamaterial – Prof Richard Craster and Dr
Andrea Colombi
Elastic metamaterials are an exciting and novel branch of wave physics devoted to the study of
heterogeneous and complex media able to control the propagation of mechanical waves at different
scales and in various applications. Wave control is made possible by a microstructure made of
subwavelength resonators, randomly arranged in the media, leading to extraordinary propagation
properties. The result is that waves can be stopped, slowed down, rerouted around an obstacle
or directed to a target. The potential for applications is high, involving several fields such as
electromagnetism, optics, acoustics and elasticity. On large scales, we can aim at controlling seismic
waves and ground vibrations. In ultrasonic (kHz-MHz) situations, strongly attenuating media can
be developed starting from metamaterials, while in hypersonics (GHz) we aim at developing of very
sensitive nanosensors. Despite the differences in lengthscale, the underlying physics remain very
similar across the applications and scales, with the main complexity given by the accurate modeling
of the resonant microstructure.
This project consists of the characterization of a recently developed type of metamaterial (resonant
metawedge), where the size and properties (i.e. the resonant frequency) of the resonators vary
spatially, giving rise to a material with a spatially graded refractive index. The main investigation
methods involve spectral element simulations and modal analysis (based on finite elements). De-
pending on the student interests, the project may include an analytical study using a simplified
metamaterial model made of masses and springs attached to a plate.
The student will join a multidisciplinary team where mathematicians, engineers and acousticians
work together on wave physics and metamaterials. In the course of the project there will be the
possibility of participating in a laboratory experiment where the spatially graded metamaterial will
be tested.

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Waves in Photonic and Phononic Crystals
           Prof. Richard Craster and Dr. Mehul Makwana
                             November 18, 2017

Project Definition
This project is about extraordinary wave transport properties within contin-
uum electronic and acoustic systems, which have various applications to opto-
electronic devices. Quite remarkably it is becoming possible to design materials
with properties that are not possible in nature, examples being materials with
negative effective mass, or negative refractive index; these are being used in de-
vices. Our aim is to model these exotic materials, design new ones, and interact
with groups that build devices. This can be approached from several different
angles, each could be a project in its own right:
   • Engineering mathematics: There are numerous modelling problems involv-
     ing waves propagating through finite “crystals” of microstructured mate-
     rials and these can be approached analytically, using asymptotic methods
     or numerically.

   • Scientific computation: We are developing software to solve these problems
     systematically in a general manner. The numerical algorithms are based
     on applied mathematical methods and there is scope to generalise the
     methods and implement them.
   • Mathematical modelling: Concepts often seen in a mathematics degree,
     such as group theory, are very useful when dealing with periodic media
     on a lattice. This does not require a deep Pure mathematics knowledge
     of group theory, but is an application of it to an area of physics. By
     manipulating the symmetries of the lattice structure unidirectional edge
     states can be produced and their effects amplified. So there is scope for a
     mathematics project that draws upon these ideas and blends it with the
     physics application.
   • Mathematical physics: The area of topological insulators in solid state
     physics and condensed matter theory is vibrant and many exciting ideas
     are emerging, one of which “topologically protected edge states” has been
     very influential. The ideas behind this are now moving into other areas of
     physics such as the photonic and phononic crystals. There are differences
     when dealing with the continuum cases and there is scope here for moving

                                        1

ideas from quantum mechanics to continua and we want to explore this
     aspect.
    There will be opportunities to interact with a vibrant research group and
attend weekly group meetings. The project could also involve collaboration with
the Physics department or a company that designs these materials.

                                      2

Numerical simulations of a pedestrian model including congestion effects – Prof De-
gond
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. In the AR model, this velocity offset is taken as a direct function of the local
car density. In the proposed new 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.
In previous Msc projects [3, 4], the mathematical analysis of the equations were devloped. The
present project aims at the development of numerical simulations and testing of the model in
practical situations. More precisely, the project has two different aspects: (i) The realisation of
numerical simulations in one and two dimensions to compare the behaviour of this model with that
of the AR model and to assess the performances of the model in classical benchmark situations
(circulation in lanes, exit scenario, etc) (ii) The study of the transition between free and congested
traffic.
This project requires knowledge some experience of programmation of scientific computing prob-
lems. The student will be supervised by Prof. Pierre Degond (pdegond@imperial.ac.uk) and
Research Associate Pedro Aceves-Sanchez (p.aceves-sanchez@imperial.ac.uk). He/she will benefit
of the team environment. The simulations will be run on the teams dedicated computer server.
References:
[1] A. Aw and M. Rascle, Resurrection of “Second Order” Models of Traffic Flow, SIAM J. Appl.
Math., 60 (2000) 916938.
[2] N. Brigouleix, A new model for pedestrian traffic, Msc report, 2016.
[3] B. Cavin, A new pedestrian model with congestion, Msc report, 2017
Contact:
Pierre Degond : pdegond@imperial.ac.uk
Pedro Aceves-Sanchez : p.aceves-sanchez@imperial.ac.uk

Numerical simulations of active suspensions in fluids – Prof Degond

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‘Active particles’ is a generic name for agents producing their own motion, such as micro-organisms.
Active particles immersed in a fluid are ubiquitous in nature from bacteria suspended in water or
sperm. There is considerable effort from the scientific community to produce realistic models of
active particle suspensions in fluids (see review [1]). Generically, the models are either individual-
based models, which consist of differential equations for the motion of each individual particle,
or continuum models, which are partial differential equations for mean quantities such as the lo-
cal density or mean particle velocity. Macroscopic models are obtained by coarse-graining the
individual-based model. Through this procedure, macroscopic observations can inform us on indi-
vidual agents behavior.
The coarse graining of active particle models is difficult but the Imperial College team has realized
it for the Vicsek model, which describes self-propelled particles locally aligning with their neighbors
[2]. The coarse-graining of the Vicsek model leads to compressible fluid-like equations called the
Self-Organized Hydrodynamic model (SOH). However, for suspensions of active particles in a fluid,
it is necessary to include the influence of the fluid. In [3], both the coupling of the Vicsek individual-
based model and that of the continuum SOH model with an incompressible fluid are performed.
In this project we will focus on a description of the fluid through Darcys law, which is valid for
instance in a very shallow fluid on which the bottom exerts a strong friction.
The goal of this project is to develop simulations of the coupled Vicsek-Darcy system on the one
hand, and of the coupled SOH-Darcy system on the other hand, in two dimensions and in a simple
geometry. The goal is to document what differences the coupling with the fluid bring to the system
compared to the case where the particle motion is uncoupled and how these differences translate
at the individual-based model level and at the continuum model level.
The student will benefit from the availability of a library of programmes developed in the team.
In particular, two dimensional codes for the Vicsek model, for the SOH model and for the Darcy
law already exist. The student will have to enrich the current Vicsek and SOH models with the
coupling with the fluid on the one hand, and the current Darcy model with the driving force due
to the presence of the active particles. Among possible applications of this work, a connection
with biologists working on collective sperm cell swimming will be made through the EPSRC grant
modelling sperm-mucus interaction across scales that involves E. Keaveny from the Department.
Research Associates Sara Merino and Pedro Aceves-Sanchez will be involved in the mentoring of
the project.
References:
[1] D. Saintillan and M. J. Shelley, Theory of active suspensions, in Complex Fluids in Biological
Systems, Biological and Medical Physics, Biomedical Engineering, S.E. Spagnolie (ed.), Springer,
2015.
[2] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,
Math. Models Methods Appl. Sci., 18, Suppl. (2008), pp. 1193-1215.
[3] P. Degond, S. Merino-Aceituno, F. Vergnet, H. Yu, Coupled Self-Organized Hydrodynamics and

                                                   20

Stokes models for suspensions of active particles. https://arxiv.org/abs/1706.05666
Contact:
Pierre Degond : pdegond@imperial.ac.uk
Pedro Aceves-Sanchez : p.aceves-sanchez@imperial.ac.uk
Sara Merino-Aceituno : s.merino-aceituno@imperial.ac.uk

                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 Schrödinger 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
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 understand the the quantum theory for integer N . This can be through the complex Schrödinger
equation or alternatives such as the Feynman path integral and resurgence theory.

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).

                                                 21

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
’t Hooft’s solutions. For the gauge group SU (2), investigate the properties of charge 21 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).

Projects in quantum theory (Dr E. M. Graefe)
In my research group we devise and investigate models to describe all sorts of quantum phenomena.
We are in particular interested in quantum dynamics, and in the properties of open quantum
systems (with losses and gains of particles or systems coupled to heat baths). Another focus of
our research is the correspondence between quantum systems and their classical counterparts, in
particular in the realm of chaos.
The projects below offer the opportunity to learn more about quantum mechanics, one of the most
successful theories of modern physics, while applying and developing tools from different branches

                                                 22

of mathematics, such as spectral theory, dynamical systems, and geometry. These projects are
“real” research problems linked to other research currently on-going in the group, and offer a great
opportunity to obtain original new results.
While the projects do not require a big amount of background knowledge, a minimum working
knowlegde of quantum mechanics and a willingness to use numerical tools are prerequistes.
Project 1: Quantum rates - Quantum dynamics in a double well potential at finite
temperatures
Background. The understanding of the diffusion properties and mechanisms of hydrogen in metals
is an important scientific challenge with applications in many industries and technologies. The
nature of the problem is such that quantum effects become important even at typical room tem-
peratures. This links the problem to a research topic of a more fundamental type, regarding the
description and simulation of transport phenomena in quantum systems at finite temperature. A
main challenge here is the provision of a quantum theory of transition rates, that counts the typ-
ical number of transitions between different states in activated processes. In classical physics, the
macroscopic transition rate can be connected to the microscopic dynamics. Here we are interested
in formulating a quantum transition rate theory along similar lines, incorporating activated quan-
tum dynamics.
Project. In this project you will focus on a toy-model of a quantum particle in a one-dimensional
double-well potential under the influence of a heat bath, which is effectively modeled by a Lind-
blad equation, or effectively an ensemble of stochastic Schrödinger equation. You will analyse the
behaviour of the system in detail and compare it to its classical counterpart. For this purpose you
will use a combination of numerical and analytical methods.
Project 2: Dissipation and losses in atom-molecule conversion in Bose-Einstein con-
densates
Background. The experimental realisation of Bose-Einstein condensates (BECs) was one of the ma-
jor achievements in physics in the late 20th century. Recent progress in confining and manipulating
BECs in optical potentials has led to a variety of spectacular results. One interesting problem
is the combination of condensed atoms into molecules, which themselves form condensates, and
the inverse process of splitting molecules into atoms. The mathematical description of interacting
atoms and molecules, which can be converted into one another, and possible dynamical schemes
for an effective conversion, is challenging and interesting at the same time. In particular the influ-
ence of noise and particle losses, which is of huge experimental relevance, is theoretically not well
understood.
Project. In this project you will familiarise yourself with different approximations for the descrip-
tion of interacting atoms and molecules in the ground state of an external potential, leading to
classical dynamics on unnusual phase spaces. You will analyse the approximation and the resulting
dynamics in detail, and then explore the influence of noise and dissipation described by Lindblad
equations.

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Project 3: PT-symmetric quantum chaos
Background. The striking difference between quantum and classical behaviour becomes most appar-
ent in the realm of chaos, an extreme sensitivity to initial conditions, which is common in classical
systems but impossible under quantum laws. The investigation of characteristic features of quan-
tum systems whose classical counterparts are chaotic lies at the heart of the flourishing research
area of quantum chaos. One important result in this field is that the statistical fluctuations of the
eigenvalues of quantum systems with classically chaotic counterparts are similar to those of the
eigenvalues of certain Hermitian random matrices (matrices whose elements are random numbers).
The surprising properties of quantum systems with balanced gain and loss (non-Hermitian, but
PT-symmetric systems) have sparked much interest recently, and new experimental areas (involv-
ing for example optical wave guides, cold atoms, and meta materials) are rapidly emerging. Here
we are interested in the hitherto nearly unexplored interplay of chaos and PT-symmetry.
Project a. PT-symmetric quantum maps. In this project you will investigate the quantum and
classical features of PT-symmetric generalisations of so-called quantum maps, described by time-
evolution operators that map the state of the system from one discrete point in time to the next.
The classical counterparts, maps on phase-space, are standard examples in the theory of chaos.
We will investigate the dynamical features of the quantum and classical maps, such as the fixed
points and their stability as well as the occurence of chaos and dissipative attractors. We will fur-
ther investigate the spectral features of the quantum systems and try to identify possible universal
behaviour.
Project b. Non-Hermitian random matrix ensembles. In this project you will investigate the eigen-
value statistics of two types of non-Hermitian random matrices known as the real and complex
Ginibre ensembles. These are very simple in structure, comprising of matrices whose elements
are identically and independently normal distributed complex or real numbers. The properties of
their eigenvalues, however, are still not well understood. You will combine a mixture of numerical
and analytical tools to investigate the eigenvalues of these matrices, with a particular focus on
identifying universal features that might be observed as well in physical model systems.

Projects in automated numerical PDE methods. - Dr David Ham

The numerical solution of PDEs is a key problem of mathematical computing. Many of the largest
supercomputers are dedicated to this task for applications including weather forecasting, engineer-
ing design, and financial instrument pricing. As simulation scale and sophistication has increased,
the combination of numerical analysis, parallel algorithms and complex software engineering re-
quired has frustrated advances in this field.
However, in recent years a radical new approach has emerged. The PDE to be solved, along with its
discretisation, are specified in a high level symbolic mathematics language. High performance paral-
lel implementations are then created using specialised compilers, which combine domain knowledge

                                                 24

with cutting edge advances in parallel code generation.
Students choosing these projects have the opportunity to work on complex mathematical problems
while gaining the experience of contributing to professionally engineered open source mathematical
software as an integral part of the Firedrake development team (http://firedrakeproject.org).
The results of their work will be incorporated into released software in production use at institutions
around the world. All of the projects detailed below have the potential, if executed well, to produce
results publishable as journal papers.
Each project combines a core of numerical mathematics with significant programming, so some
level of knowledge of a language such as Python or C is a requirement.

Post-processing techniques for mixed and discontinuous Galerkin methods - with
Thomas Gibson

This project focuses on studying the effects of elementwise post-processing techniques for discontin-
uous Galerkin and mixed finite element methods. Procedures of this type are able to produce new
approximations, which superconverge at accelerated rates and possess better conservation proper-
ties. We restrict our focus to second order elliptic problems and possibly hyperbolic problems.
Software using the Firedrake project will be a core part of this study. We explore the effects of
different post-processing techniques of model problems and automate its implementation using a
domain-specific language (DSL) for linear algebra. These post-processing techniques may also be
incorporated in preconditioners to accelerate the convergence of iterative solvers as well.
Prerequisites: PDEs, linear algebra, functional analysis (familiarity with basic concepts), program-
ming experience (useful)

Automated differentiation for inverse problems

Inverse problems are pervasive in science and engineering: the forward simulation answers the
question “what happens if?” while the inverse problem ask “what was the cause?”. In fields as
diverse as climate science and financial mathematics, we need to invert simulations to find the
causes of phenomena. In engineering, optimal design requires inverse simulations to design the
system which best produces a desired outcome.
A key requirement in inverse simulation is to differentiate the model. For an automated system
such as Firedrake, this requires the symbolic mathematics code that Firedrake programs are writ-
ten in to be differentiated automatically using techniques from computer algebra. Many parts of
Firedrake are already automatically differentiable, but important holes remain. This project will
enable a student to learn about inverse simulation techniques while contributing new automatic
differentiation capabilities to the Firedrake system.

                                                  25

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

Cell diversity in tumour growth

Supervisor: Professor Henrik Jeldtoft Jensen

As a cancer tumour grows, evolutionary dynamics involving mutations may lead to a divers collec-
tion of cell types. See e.g. [1] The project will use the framework of the spatial Tangled Nature
model of evolutionary ecology, see [2], to investigate the co-evolutionary aspects of tumour het-
erogeneity. Phenomenological insights will be obtained by further development and simulation of
existing C-codes. Some analytical analysis may also be possible at mean field level.

References
[1] https://www.quantamagazine.org/studies-reveal-extreme-diversity-of-cancer-cells-20131113/
[2] D. Lawson and H.J. Jensen, The species-area relationship and evolution. J. Theor. Biol., 241,
590-600 (2006).

Information theoretic analysis of EEG time series

Supervisor: Professor Henrik Jeldtoft Jensen

The project will use information theoretic causal measures to investigate EEG time series from
musicians and possibly also from schizophrenic patients. First the information theoretic formalism
(see e.g. [1]) will be studied, this will be followed by data analysis of various data sets. In particular
we’ll try to relate the analysis to the current suggestion that schizophrenia is related to an increase
in fluctuations of the dynamics of the brain, see e.g. [2].

References
[1] X. Wan, B Cruts and H.J. Jensen, The causal inference of cortical neural networks during music
improvisations. PLoS ONE DOI:10.1371/journal.pone.0112776 December 9, 2014. arXiv:1402.5956.
[2] Front Psychiatry. 2015; 6: 92. Published online 2015 Jun 24. doi: 10.3389/fpsyt.2015.00092
Simulation Based Inference for Molecular Machines – Dr David Rueda (MRC), Dr
Tom Ouldridge, Dr Nick Jones

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,

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