# Possible PhD Projects in Applied Mathematics

Suggested projects for postgraduate research in Applied Mathematics.

Applied Mathematics research in the School of Mathematics, Statistics and Physics is concentrated in the following areas:

**Cosmology and Quantum Gravity****Astrophysical and Geophysical MHD****Quantum Matter****Mathematical Biology and Archaeology**

If applying for a PhD project in one of these areas, please provide the titles of up to three projects from the list below, in order of preference. Applicants are invited to apply online.

For further information, please contact the PG tutor/selector in applied mathematics Dr Toby Wood.

#### The Big Bang and inflation

The Big Bang and inflationThe main goal of our cosmology programme is to search for direct observational signatures of early universe cosmological models, paying special attention to the role of quantum processes during a period of cosmological inflation. This PhD project covers topical aspects of the very early universe, including:

- the quantum theory of the big bang and the origin of time
- investigations of inflationary models with quantum gravity and Higgs bosons
- models which seek to explain the origin of primordial magnetic fields

**Supervisor:** Professor Ian Moss

#### The large scale structure of the universe

The large scale structure of the universeObservational cosmology has undergone a revolution in recent years due to the availability of high quality data from satellite observations. This makes it possible to attempt to find out what makes up the dominant form of matter in the universe. It also helps us understand the mysterious dark energy which is driving the universe apart. This PhD project attempts to fit models of dark matter and dark energy to observations of large scale structure using new approaches and new theories of dark energy.

**Supervisors: **Professor Ian Moss

#### The early universe as a hologram

The early universe as a hologramResearch in string theory has shown that gravity is holographic. Certain gravitational theories have an equivalent description in terms of a quantum theory, without gravity, living in one spacetime dimensionless.

Holography represents our sharpest understanding of quantum gravity to date. It sheds new light on the fundamental problems of cosmology, from resolving the big bang singularity to explaining the pattern of perturbations visible in the temperature and polarisation of the cosmic microwave background. According to the theory of inflation, these perturbations began as microscopic quantum fluctuations in the matter density and geometry of spacetime. These fluctuations were subsequently stretched to macroscopic scales by the accelerated expansion of the universe. The dynamics of this process can be constrained through measurements of the cosmic microwave background. But it remains poorly understood from a fundamental theoretical perspective.

This project will use holography to recast the four-dimensional early universe as a three-dimensional quantum field theory. We will develop novel holographic approaches to explore the origin and symmetries of the primordial perturbations, building on recent advances in conformal field theory. Ultimately, the goal is to identify distinctive observational signatures of holography that can be tested through future cosmological observations.

**Supervisor**: Dr Paul McFadden and Prof Ian Moss

#### Convective dynamo action at the solar surface

Convective dynamo action at the solar surfaceSolar magnetism plays a crucial role in determining the Sun’s influence on the environment of the Earth and the rest of the Solar System. Magnetic fields in the Sun, on both large and small scales, are generated and maintained by fluid motions within the solar interior. Whilst certain aspects of this dynamo process are well understood, there are many areas of uncertainty.

In this project, we will investigate the origin of the intermittent magnetic field distribution that is observed in regions of quiet Sun (areas of the solar surface that are devoid of large-scale magnetic features such as sunspots). We will test the hypothesis that these fields are generated locally by the near-surface convective motions.

Much of this project will be based upon high resolution numerical simulations of compressible magnetohydrodynamics, making use of an existing code. The output from this code will be used to address various important issues, including the extent to which the efficiency of the dynamo is influenced by the presence of convective motions on spatial scales that are larger than the small-scale 'granular' convective pattern that covers most of the solar surface.

The existence of one such convective scale, namely supergranulation, is well established. But the reality (or otherwise) of an additional 'mesogranular' scale is the subject of current controversy. By applying the observers' data analysis algorithms to our simulation data, we will attempt to resolve this issue over the course of the project.

Although no prior knowledge of solar physics will be assumed, a good understanding of fluid dynamics is essential.

**Supervisors: **Dr Paul Bushby and Dr Toby Wood

#### Convection in the Earth's core

Convection in the Earth's coreMotions of liquid iron in the Earth’s outer core are driven by thermal and compositional convection due to the cooling of the planet. An on-going debate in the geophysics community is whether the whole outer core is convecting or whether a region near the core-mantle boundary is stably stratified. In this project, we propose to study:

- the mechanisms by which a stratified layer could form at the top of Earth’s core
- the nature of the motions in this layer
- the layer thickness

The project is based on numerical simulations, which will be performed starting from existing numerical codes. A good understanding of fluid dynamics is essential. No prior knowledge of geophysics or computational modelling is required — the necessary training in these areas will be provided during the early stages of the project.

**Supervisors: **Dr Celine Guervilly and Dr Graeme Sarson

#### Internal waves in the Arctic Ocean

Internal waves in the Arctic OceanInternal waves (IWs) spread along density interfaces within stably-stratified fluids. They are everywhere in the ocean and their properties are strongly influenced by the nature and form of the upper and lower bounding surfaces of the containing basin(s) in which they propagate. As the Arctic Ocean evolves to a seasonally more ice-free state, the IW field will be affected by the change.

The relationship between IW dynamics and ice is important in understanding:

- the general circulation and thermodynamics in the Arctic Ocean
- local mixing processes that supply heat and nutrients from depth into upper layers, especially the photic zone.

This, in turn, has important consequences for sea ice formation processes and the state of local and regional ecosystems.

Despite this, the effect of diminishing sea ice cover on the IW field (and vice versa) is not well established. A better understanding of IW dynamics in the Arctic Ocean is central in understanding how the rapidly changing Arctic will adapt to climate change. In particular, how the IW field is affected by changes in both ice cover and stratification,

In this project, we will carry out laboratory work to investigate the fluid dynamics of IWs under varying surface conditions. We will train the successful candidate in the generation, visualisation and measurement of IWs in a newly built wave flume at Newcastle University.

There may be opportunities for the candidate to undertake:

- numerical simulation of the flow in collaboration with Prof D G Dritschel (St Andrews)
- field work in the Arctic in collaboration with Prof Tom Rippeth (Bangor).

**Supervisors: **Dr Magda Carr

#### Galactic magnetic fields

Galactic magnetic fieldsWe describe Galaxies as islands in the Universe, each containing billions of stars.

Interstellar gas fills space between the stars. This gas is a complex hydrodynamical system involved in intense turbulent motions. It exhibits an exceedingly wide range of temperatures (from a few degrees on the Kelvin scale to several million degrees) and densities. It is permeated by magnetic fields and mixed with relativistic particles: cosmic rays.

The interstellar medium is especially rich in complexity in spiral galaxies, whose notable feature is rapid rotation. The interstellar medium feeds the formation of new stars. This largely controls the optical appearance of the host galaxy, and maintains magnetic fields. Despite the violently turbulent nature of the interstellar gas, these magnetic fields exhibit order at very large scales comparable to the size of the parent galaxy.

This project will focus on the origin of the large-scale magnetic fields in spiral galaxies. Recent developments in hydromagnetic dynamo theory have opened an opportunity to construct models that can be directly compared to astronomical observations, refined and perfected. Such a theoretical development is urgently required to plan and interpret observations with a new generation of powerful radio telescopes.

The work on the project will involve:

- development of the theory of galactic magnetic fields, based on numerical and analytical studies of the dynamo equations
- collection of relevant astronomical data, as required to adapt theoretical models to specific galaxies
- comparison of the theoretical predictions with radio astronomical observations

The work on the project will involve regular international contacts with both theoreticians and observational astronomers. It will involve a modest amount of numerical calculations (eg with Matlab or Fortran).

**Supervisors: **Professor Anvar Shukurov and Dr Andrew Fletcher

#### Buoyant magnetic fields in the Sun

Buoyant magnetic fields in the SunThe complex magnetic structures (coronal loops and prominences) that appear on the Sun's surface are believed to originate at the bottom of the solar convection zone. This is where the magnetic field is stretched and amplified until it becomes buoyant and rises to the surface. However, the details of this magnetic buoyancy process are poorly understood, and are strongly affected by the Sun's rotation and compressibility.

This project will combine analytical theory with high resolution numerical simulations, to determine how magnetic buoyancy operates under the conditions of the solar interior. We will also address the role that the rising magnetic field structures play in the Sun's overall dynamo process.

We will assume no prior knowledge of solar physics. A good understanding of fluid dynamics is essential.

**Supervisors: **Dr Toby Wood and Dr Paul Bushby

#### Neutron star magnetic fields

Neutron star magnetic fieldsNeutron stars are extremely dense and rapidly rotating objects. They have the strongest magnetic fields in the Universe. Regular stars are powered by nuclear reactions. Neutron stars are powered by their vast reservoirs of rotational and magnetic energy.

A neutron star has a solid outer crust surrounding a superfluid core. Within this core the rotation and magnetic field are "quantised" into thin filaments called vortices and fluxtubes.

This project will develop a model for the dynamics of the vortices in the star's core, and their interaction with the strong magnetic field. We will base our model on suitably modified fluid equations that take account of the superfluid nature of the core.

Basic knowledge of fluid dynamics is required, as well as interest in developing computational skills.

**Supervisors: **Dr Toby Wood and Professor Carlo F Barenghi

#### Stellar dynamics and evolution

Stellar dynamics and evolutionThe evolution of stars and their ultimate demise is affected by hydrodynamic processes occurring within their interiors throughout their lifetime.

Dynamical processes such as convection, rotation, waves and magnetism all greatly impact how these stars explode, chemically enrich the galactic environment and the properties of the stellar remnant.

This project will involve using multi-dimensional hydrodynamic processes to understand these dynamical processes and how they contribute to stellar evolution. Using this understanding, combined with observational constraints, we will develop one-dimensional prescriptions for use in standard stellar evolution models.

**Supervisor: ** Dr Tamara Rogers

#### Stochastic modelling of quantum matter

Stochastic modelling of quantum matterUnderstanding the behaviour of matter often requires the use of stochastic methods. We add random noise to numerical equations in a controlled way. This mimics the physical response of a system.

This arises across all aspects of modelling, from biological, chemical to physical systems. An obvious example is the random jitter of particles. Here, a random displacing noise is added to the otherwise stationary particle evolution. In the physical setting, the noise usually arises from the interaction of the object with a so-called “heat bath”. The object can exchange energy and particle number with the "heat bath".

Major advances in the last decades have led to a system of appropriate equations to model multi-particle quantum systems confined in appropriate geometries. These have been linked to many recent Nobel Prizes. Beyond a curiosity, such systems are accessible in the lab and promise to revolutionise our future quantum technologies.

The aims of this project are to:

- become familiar with the mathematical background, physical origin, and numerical implementation of such stochastic approaches
- use this knowledge to model cutting-edge experiments in at least two different physical systems which exhibit quantum effects on a macroscopic scale.

**Supervisors**: Professor Nick Proukakis

#### Ultracold quantum gases: from the fundamental to the technological

Ultracold quantum gases: from the fundamental to the technologicalIn laboratories world-wide and at temperatures less than a millionth of a degree above absolute zero, quantum gases of atoms are being generated, probed and studied. These gases are ruled by quantum mechanics on a macroscopic scale. They represent waves of matter at the limit of Heisenberg's uncertainty principle. Moreover, these gases may be controlled and manipulated at will by sophisticated experimental techniques.

On one hand these gases provide an ideal playground to study fundamental quantum physics. Examples include:

- the phenomena of superfluidity (flow without viscosity)
- quantum entanglement (Einstein's "spooky action at a distance")
- the quantum phases of matter (the fundamental forms of matter at zero temperature)
- the nature of the phase transitions between them

On the other hand, these gases are pursued for applications in quantum technologies, such as:

- next-generation interferometers (devices to measure forces, eg for use in navigation and exploration)
- novel microscopes
- quantum computers
- encryption

This project will use state-of-the-art simulation techniques to model quantum gases. We will explore their fundamental quantum properties and/or technological possibilities, based on one or more of the above examples.

We will liaise with some of our international collaborators across Europe, the US and Australia. It will take place within the Joint Quantum Centre (JQC) Durham-Newcastle, a world-leading centre for research into quantum matter. The lead supervisory team will be discussed with the student when they apply.

**Supervisors: **Dr Andrew Baggaley, Professor Carlo Barenghi, Dr Tom Billam, Dr Nick Parker, Professor Nick Proukakis

#### Is turbulence knotted?

Is turbulence knotted?Mathematicians, physicists and engineers have studied turbulence for more than a century. Almost all investigations into this fundamental problem of the natural sciences have concentrated the attention on two aspects:

- the geometry
- the dynamics of turbulence

Little attention has been paid, in comparison, to a third equally important aspect: the topology.

This is despite the fact that 19th century pioneers of fluid dynamics such as Kelvin and Helmholtz were already aware of the possibility of vortices becoming twisted, linked and knotted. Unfortunately, until recently, the only vortex structures which could be created in the laboratory were either very simple (such as vortex rings) or utterly complex (such as turbulence): the 'hydrogen atom' of topological complexity was missing. This situation suddenly changed in 2013, when Kleckner and Irvine, at the University of Chicago, showed that it is possible to create trefoil vortex knots under controlled and reproducible laboratory conditions. This breakthrough is now driving a great interest in the study of the topology of vortices and turbulence.

The project aims to:

- perform a numerical investigation of turbulent flows by solving the governing Euler or Navier-Stokes equations
- look for evidence of knotted structures.

The objectives are to:

- define and quantify the topological complexity of turbulence
- to explore the possibility of scaling laws

You should have an interests in fluid dynamics and methods of computational mathematics. You should be willing to learn tools from other relevant disciplines such as knot theory.

**Supervisors: **Professor Carlo F Barenghi and Dr Andrew Baggaley

#### Vortex lines as topological defects at phase transitions: micro big-bangs in superfluid helium

Vortex lines as topological defects at phase transitions: micro big-bangs in superfluid heliumThe progress of cosmology is limited as it cannot be studied in the laboratory. This remark motivates attempts to find physical systems which model some of the fundamental properties of the universe but are also experimentally accessible. One of such systems is the formation of topological defects at phase transitions. Below the temperature of approximately a milliKelvin, superfluid helium 3 exhibits a phase transition to a superfluid state characterised by the breaking of various symmetries that are good analogues to those broken after the Big Bang.

The experimental set up is the following. Superfluid helium 3 is locally heated by neutrons, creating a hot spot of normal liquid which expands and quickly cools back down to the superfluid state. Coherent superfluid regions of the liquid grow simultaneously, and, when they come in contact, the mismatch of the order parameter creates linear topological defects called vortex lines. It is thought that this process, called the Kibble-Zurek mechanism, in the context of cosmology, may be responsible for the formation of inhomogeneous large-scale structures, such as super clusters of galaxies. In liquid helium, the Kibble-Zurek mechanism has been observed experimentally at Helsinki and Grenoble.

To make better connection with theory, it is necessary to understand how the small region of vortex lines, a turbulent spot, diffuses in space and time: this is what we plan to do numerically. In particular, we want to find how strongly nonlinear processes such as vortex reconnections, which occur when two vortex lines collide with each other, affect the diffusion. Preliminary numerical experiment suggest that vortex rings evaporate away from the turbulent spot.

**Supervisor: **Professor Carlo F Barenghi

#### Vortex motion in atomic condensates

Vortex motion in atomic condensatesCurrent experiments with atomic condensates are concerned with the motion of vortices confined in a small geometry (typically discs or spheres). The small size of these systems means that this motion is affected by the boundaries.

The aim of this project is to gain more understanding of the behaviour of vortices and vortex clusters and possibly make connections with experiments. In particular, we shall use the point vortex method of classical Euler fluid dynamics and the nonlinear Gross-Pitaevksii equation to determine vortex trajectories and to study the chaotic properties of these vortex systems.

**Supervisor: **Professor Carlo F Barenghi

#### Atomtronic devices

Atomtronic devicesExperimental advances over the last decade are beginning to usher in an age of quantum devices, such as sensors and interferometers. They have the potential to surpass their classical predecessors in terms of:

- sensitivity
- reliability
- miniaturization

One paradigm for constructing such devices involves using ultracold atoms as the quantum element. This has led to the creation of "atomtronic" analogs of electronic (and optical) devices in which the electrons (or light) are replaced with a superfluid current of ultracold atoms. This atomic superfluid can be made to flow without viscosity through carefully shaped channels in a similar way to electricity flowing through circuits, and light travelling through photonic media such as optical fibres or nonlinear crystals. Owing to many-body quantum effects inherent in the atomic superfluid, atomtronics has possible applications in making ultra-sensitive, quantum-enhanced sensors and interferometers.

This project will develop computational and analytical models of novel atomtronic quantum interference devices. In particular, we will connect with recent developments in quantum electronics to generate and develop new proposals for atomtronic devices that exhibit quantum-enhanced performance.

S**upervisors:** Dr Tom Billam and Dr Clive Emary

#### Collective behaviour from cells to animals

Collective behaviour from cells to animalsLarge aggregates of living entities, from biological cells to animals, can exhibit rich and complex collective behaviour. This behaviour often arises from:

- simple actions of the individual members
- their interaction with their immediate neighbours and environment.

Striking examples of this in nature are bird flocks (most spectacularly the aerial display of huge numbers of starlings at dusk) and fish schools.

Collective behaviour also plays a key role on the microscopic scale of biological cells. In particular, in-vitro stem cells undergo complex dynamics as they evolve to form colonies and tissues. This process underlies future medical applications of stem cells for the controlled regeneration of biological tissue.

This project will develop a model for such emergent collective behaviour. You may look at the macroscopic domain of birds (Dr Baggaley / Dr Fletcher / Dr Gillespie) or the microscopic realm of stem cells (Dr Parker / Prof Shukurov). Comparison to experimental observations (for birds, this will be through live imaging taken as part of the PhD project; for stem cells, this will be obtained through a collaboration with state-of-the-art experiments at the Institute of Genetic Medicine) will help deduce the biological, physical and geometrical processes which govern these dynamics, and can be expected to shed new light on collective behaviour in these systems.

**Supervisors:** Dr Andrew Baggaley, Dr Andrew Fletcher, Dr Colin Gillespie, Dr Nick Parker and Professor Anvar Shukurov

#### Modelling of ecological systems

Modelling of ecological systemsWe live in an era of marked global change from climate change, deforestation and urbanisation. This has major implications for natural ecological systems such as plants and trees, insects, animals and coral reefs.

Drs Baggaley and Parker are working with government to understand the spread of tree disease from invasive species. Topical examples of this are the dieback fungus and borer beetle affected UK ash trees. Our work focusses on understanding the key factors governing the spread of the disease and how the damage might be mitigated.

Meanwhile Dr Emary is studying ecological networks – abstract representations of the interactions between species in an ecosystem. Key questions include the response of such networks to species loss and environmental change. This work is performed in in collaboration with field ecologists with applications in agriculture and climate-change mitigation.

Moreover, engineering ecological systems, such as bacterial colonies, may help us meet future challenges in energy provision and waste management. We are developing mathematical models for the bacterial colonies, so as to highlight conditions and strategies to optimising the efficiency of these processes.

**Supervisors:** Dr Andrew Baggaley, Dr Clive Emary and Dr Nick Parker

#### Improving medical imaging and diagnostics

Improving medical imaging and diagnosticsAdvances in medical imaging enable clinicians to probe the body with remarkable precision. Clinicians can gather extensive images and datasets. Such advances demand sophisticated mathematical techniques to extract clinically-relevant information. Our researchers are working with clinicians to contribute to these challenges. We work in a range of settings and use a range of analytical and computational methods.

One example is our work in monitoring the recovery of the cornea to stem cell therapies (Prof Shukurov). Following major trauma, the natural cellular structure of the cornea is destroyed. Stem cell therapies can lead to a recovery of this structure. Working with clinicians and microscope images of the eye, we are developing advanced methods to quantitatively characterise the cell structure. This means we can assess levels of damage and monitor the recovery process.

Another example is our work in studying medical ultrasound imaging within the body, in collaboration with clinical medical physics (Dr Parker). The refraction of the ultrasound (eg at tissue boundaries) leads to a geometrical distortion of the images which is not accounted for. This mathematical modelling may lead to the development of corrective strategies which may be translated into clinical devices.

**Supervisor:** Dr Andrew Baggaley, Dr Nick Parker and Professor Anvar Shukurov

#### Prehistoric population dynamics

Prehistoric population dynamicsPopulation dynamics is a well-established field of applied mathematics. It has a wide range of applications to biological and social systems. It has been especially successful in applications to prehistory where the fundamental features of the evolution of human populations were free from the unmanageable complications of politics, long-distance travel, etc. One of the most fascinating ages in the human prehistory was the Neolithic, the last period of the Stone Age. The defining feature of the Neolithic was the birth of agriculture and food production (as opposed to food-gathering and hunting). This resulted in:

- a more sedentary lifestyle
- the emergence of urbanism
- human societies as we now know them.

The Neolithic first appeared in the Near East and China (perhaps apart from other relatively minor sources) about 12-10 thousand years ago. It then spread across Europe and Asia. There are well-developed mathematical models of this process, but they suffer from several shortcomings:

- virtually all of them focus on a limited geographical region (eg Western Europe) and neglect any connections, spatial and temporal, with other regions
- it remains unclear how such environmental factors as topography, climate, soil quality, etc. affected the spread of the agriculturalists and their technologies.

This project aims to develop comprehensive mathematical models of the spread (and subsequent development) of the Neolithic in Eurasia. It will allow for the environmental effects, and take account of the multiple centres where agriculture was independently introduced.

Mathematical modelling, mostly based on numerical simulations, will be constrained by the archaeological and other evidence available, which we will interpret using statistical tools. The project will involve close contacts not only with other mathematicians but also with archaeologists. It may include participation in archaeological field trips and excavations, if desired.

**Supervisors:** Professor Anvar Shukurov and Dr Graeme Sarson