Research Topics

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

We have opportunities for MPhil and PhD projects in all of these areas. Please contact the relevant staff member you are interested in for informal discussion and see the sample list of topics available below for further information.

COSMOLOGY AND QUANTUM GRAVITY

Black holes on a microchip

The prediction by Stephen Hawking that black holes radiate particles from the quantum vacuum has had a profound impact on the development of quantum gravity. One aspect of this has been the realisation that quantum gravity theories have non-gravitational equivalents in one less dimension – the holographic principle. This PhD project is concerned with theoretical modelling of one such holographic dual in two dimensions-called the SYK model-as realised in graphene. The project will attack the theory from two sides-numerical modelling electron wave functions on graphene wafers and semi-analytic modelling using the Dirac equation from elementary particle theory. The work will support a possible experiment.

Supervisors: Professor Ian Moss and Dr Alexsey Kozikov.

Cosmology from Rubin Observatory and Euclid

The large-scale structures of our Universe is a rich environment, made of filaments, voids and clusters of matter. This cosmic web encapsulates information about dark matter, dark energy, neutrino masses and the initial conditions that prevailed in the Early Universe. Ongoing surveys such as LSST and Euclid will use weak gravitational lensing and galaxy clustering to map the large-scale structure with unprecedented precision, aiming to extract this information. Maximising the precision of this measurement requires to model the deep non-linear regime of gravitational collapse, which can only be achieved with numerical simulations. 

This PhD project will start with the development of new powerful measurement methods that combine both weak lensing and galaxy clustering, pushing beyond the current mainstream methods. Once validated on mock data, the methods will be applied on the first data releases from LSST and/or Euclid, leading to measurements of the key parameters that describe the dark sector of our cosmological model.

Supervisor: Dr Joachim Harnois-Deraps

QUANTUM MATTER

Quantum mixtures

We know well what happens when two classical systems interact: they can mix (eg milk and water), or phase-separate (eg oil and water). What happens then when two quantum fluids overlap? This depends crucially on their interaction strength, with the quantum nature of the many-body system setting new rules for their coupling – critically also depending on whether the atomic system is composed of bosonic, or fermionic, particles.

Motivated by experiments with a plethora of different mixtures of ultracold quantum gases, at temperatures below micro-Kelvin, the aim of this project is to study the static and dynamic properties of such multi-component systems.

Questions to be studied include:

  • How do such quantum mixtures emerge from their classical systems across the phase transition?
  • What difference does the bosonic, or fermionic, nature of the individual components play, and how does a double superfluid (ie a fluid in which both bosonic and fermionic components of a Bose-Fermi mixture are superfluid) differ from other mixtures?
  • In particular, how does rotation influence the dynamics of quantum mixtures? (a question of indirect relevance to the cores of neutron stars)
  • How does the presence of external (electromagnetic) coupling between different components influence the system’s properties?

Such questions will be addressed in close collaboration with European experimental groups, where such experiments are underway.

Supervisor: Professor Nick Proukakis

Quantum droplets

Landmark experiments with atomic quantum gases since 2017 have demonstrated a new form of quantum matter – a quantum droplet. These droplets have several unusual properties:

  • they are superfluid – meaning that they are free from viscosity and can support eternal flow
  • they are self-supporting – like stars under their own gravity
  • they also have such high particle density that quantum mechanical fluctuations and correlations, normally negligible in the gas phase, become significant

These unique features bring the droplets to the fore for studying exotic physics, such as laboratory analogs of neutron stars and highly-correlated quantum systems, and developing new technologies, such as ultra-precise sensors.

This timely project will develop computational and/or analytical models of the droplets to explore these state-of-the-art opportunities.

Supervisors: Professor Nick Parker and Dr Tom Billam

Stochastic modelling of quantum matter

Understanding 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

Supervisor: Professor Nick Proukakis

ASTROPHYSICAL AND GEOPHYSICAL FLUID DYNAMICS

Interstellar turbulence and galactic winds

Spiral galaxies are among the most spectacular objects in the observable universe. Their visual appearance is dominated by ongoing formation of bright, hot, massive stars. The most massive of the newly formed stars evolve rapidly (in a few million years) and die in a powerful explosion, to produce what is known as a supernova star. Supernova explosions heat and compress the interstellar gas, drive its intense random (turbulent) motions and galactic winds, and produce cosmic rays. In turn, interstellar turbulence amplifies magnetic fields at a wide range of spatial and temporal scales.

Properties of the resulting complex system is a subject of ongoing active research, both theoretical and observational. The goal of this project is to advance and use realistic numerical simulations of the interstellar gas driven by supernova explosions. Such simulations are being carried out in the Newcastle Applied Mathematics group, so this work will be done in close collaboration with other researchers who work on various aspects of this system. There are many interesting topic to study, including the effects of star formation and spiral arms on the turbulence and galactic winds, properties of turbulence in the galactic halos, its effects on the interstellar magnetic fields, etc.

This work will involve close contact with international groups of computational scientists and radio astronomers and will be based on intensive, state-of-the-art numerical simulations. Particular topic of research in this very wide and fascinating area can be selected according to the background and preferences of the student.

Supervisor: Professor Anvar Shukurov

Stellar dynamics & evolution

The 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: Professor Tamara Rogers

Neutron star magnetic fields

Neutron 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.

Supervisor: Dr Toby Wood 

Instabilities in the solar interior

The surface of the Sun is characterised by a broad range of complex magnetic structures. The most prominent of these are sunspots, which form within (so-called) active regions. The distribution of active regions waxes and wanes, following a cyclic pattern with a period of approximately 22 years. It is believed that such regions are the surface manifestations of an underlying large-scale magnetic field that is buried deep within the solar interior, probably localised around the solar tachocline, where the magnetic field is subject to strong shearing motions. 

To understand the formation of active regions, we need to understand the evolution of the magnetic field in the solar tachocline. In particular, we need to understand the various competing magnetohydrodynamic instabilities that may be playing a role in this evolution. It is generally accepted that magnetic buoyancy plays a crucial role in this regard; it has been suggested more recently that the magnetorotational instability (which is driven by shearing motions) could also be an important factor, particularly at high latitudes.

This project will combine analytical theory with high-resolution numerical simulations, to determine how these instabilities shape the evolution of the solar interior.

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

Supervisors: Professor Paul Bushby and Dr Toby Wood

Laboratory modelling of internal solitary waves

The world’s oceans are stratified since variations in seawater temperature and salinity lead to layers of different densities. This density stratification allows for internal waves to exist and propagate on density interfaces within the water column. These waves are barely visible at the surface of the ocean but can reach amplitudes of up to 240 meters in its interior. Internal waves play a major role in the mixing of different layers in the oceans, they transport energy over long distances, and they are an important dissipation mechanism for tidal motions. Therefore, understanding these waves leads to better climate models and better predictions of the future climate. Internal waves can have trapped cores which transport material (e.g. pollutants or biological matter) over long distances and are thus important in the biogeochemical cycle in the ocean. They can present a danger to offshore industries, such as floating wind farms and oil rigs. Internal solitary waves (ISWs) are a particular form of internal waves that are composed of a single or very few pulses and can travel large distances without significantly changing their shape or amplitude. This project will model ISWs in a bespoke wave flume at Newcastle University designed to study stratified flows. The velocity field will be, measured using particle image velocimetry (PIV) and density measured via micro-conductivity sensors and/or synthetic schlieren. Avenues of potential study include studying ISWs in the context of (i) floating offshore wind, (ii) ice-calving in polar oceans, (iii) shoaling and (iv) mode-2 ISWs. There will be scope to run numerical simulations to support and advance the understanding from the laboratory investigation and there will be opportunity to collaborate with physical oceanographers and offshore engineers.

Supervisor: Dr Magda Carr

MATHEMATICS OF LIFE AND ENVIRONMENTAL SCIENCES

Resilient treescapes for a changing climate: a mathematical approach

Climate change is rapidly increasing the risk to our native treescapes and agricultural ecosystems from a variety of ecological hazards, including invasive pests, diseases and fires. Such outbreaks harm plant and crop health and reduce biodiversity, resulting in profound environmental, economic and social consequences. Strategic forest and agricultural planning and management can help build resilience to these outbreaks and contain them when they occur. Therefore, enhancing biosecurity is a key priority for the UK government, through the control of existing risks, and by building resilience against emerging concerns. The Department for Environmental, Food and Rural Affairs (Defra) has highlighted the particular importance of computational modelling in developing robust plans and management policies for minimising the impacts of these threats.

Here we will address the question: What are effective and economical strategies to build climate-resilient treescapes? This topical and open problem will be tackled through scenario-testing with sophisticated spatio-temporal models of the spread of disease, invasive pest species, or wildfire. This approach will be applied to existing hazards and hypothetical future ones, and will incorporate effects such as seasonal weather patterns, year-on-year climate trends, and regional variability. Importantly, we will use the models to test the effectiveness of a variety of scenarios designed to build resilience against these outbreaks.

Our group has close links with experts across the School of Natural and Environmental Sciences, providing expertise on ecology, remote sensing and climate change. Our collaboration with Defra, Forestry Commission and Fera, will facilitate knowledge exchange, data sharing and give direct experience of working with government policymakers, ultimately providing an opportunity to support national environmental policy.

Supervisors: Dr Laura Wadkin & Professor Nick Parker

Mathematical tools for medical imaging and diagnostics

Advances in medical imaging enables clinicians to probe the body with remarkable precision, gathering a range of extensive images and datasets. Such advances demand sophisticated mathematical techniques to extract clinically relevant information from these datasets. Our researchers work in collaboration with clinicians and biologists, applying a range of analytical and computational methods to develop diagnostic and monitoring tools. 

One of our key areas of interests is monitoring the recovery of the corneal epithelium post limbal stem cell transplant in patients with limbal stem cell deficiency (LSCD). Following disease or trauma, the natural cellular structure of the corneal epithelium is damaged, which can cause chronic inflammation, persistence ocular surface pain, scarring and loss of vision. In LSCD and other ocular diseases, the differences between healthy, diseased, and pre- and post-operative corneal cellular microscopy are often subtle and not easily discernible, and clinical manifestations of different corneal diseases may appear similar. Currently, diagnosis is based on slit-lamp eye examination, corneal impression cytology and the manual assessment of many in-vivo confocal microscopy (IVCM) images. In collaboration with Prof Francisco Figueiredo, leading consultant ophthalmologist at Newcastle upon Tyne Hospitals NHS Foundation Trust, we are utilising a rich library of LSCD patient IVCM images from the clinic and modern machine learning techniques to identify rigorous quantitative markers of corneal wound healing. 

The opportunities for mathematical modelling are vast and diverse across both 2D and 3D applications, describing cellular structure, cell migration, intra-cellular properties and wound healing mechanisms. These mathematical models help us to deepen our understanding of the biological system and to develop prediction and monitoring tools.

Supervisors: Dr Laura Wadkin & Professor Anvar Shukurov

Modelling swimming microbes for the environment and green biotechnologies

Microbes, such as bacteria and microalgae, inhabit almost every habitat on Earth, from oceans to snow fields. As agents of biochemical transformation, they play critical roles in global biogeochemical cycles. For example, microalgae fix roughly half the planet’s atmospheric carbon, helping climate regulation and coupling to climate change. Microbes are also critical to green biotechnologies, where they can be used to treat waste or produce bioproducts in an environmentally friendly way.

Many microbes swim, and bias their swimming in response to environmental cues, such as light, gravity, chemical gradients and fluid flow. The mathematical study of swimming and its bias at the individual level, and the wonderful patterns arising at the collective level, is a topic of great topical interest in mathematical biology and biophysics. Current models do reasonably well in predicting the patterns that swimming activity and bias swimming cause in biological fluids in the laboratory. However, these predictions are often only qualitative and models have not been adapted to industrial or agricultural conditions outside the lab.

In this project, current models of swimmers will be developed, and if necessary substantially reformulated, so that they can be tested for their usefulness in industrial and agricultural settings. Specifically, interested students will be able to study one of the following research topics:

  • Biofluid dynamics of swimming algae in photobioreactors and harvesting systems
  • Response of swimming algae to toxic chemicals produced by other microbes in the ocean
  • Degradation of pollutants by swimming soil bacteria
  • Movement of soil bacteria near plant roots

All of these projects will involve interaction with our collaborators in biology, physics, engineering and industry. For the photobioreactor project, students with an interest in carrying out experimental work will be able to do experiments with our photobioreactors located in the labs of our collaborator Dr Gary Caldwell in School of Natural and Environmental Sciences.

Supervisors: Dr Otti Croze 

Prehistoric population dynamics

Population 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

Resolved mechanics in continuum models for bio-films and flocs

Experimental techniques such as DNA sequencing and genome editing enable elegant studies into the inner workings of ‘simple’ single-cell organisms. Designer bugs that can eat plastics, digest toxic chemicals and produce bio-fuels are being extensively researched. In parallel with such lab based work there is also a growing body of mathematical and computational research directed at furthering our understanding of how micro-organisms organize their world, and how we might use this knowledge to better our own. This PhD project aligns with the mathematical/computational approach, and seeks to develop hi-fidelity models that accurately describe the collective behaviour and emergent properties of colonies of micro-organisms.

Many species of bacteria have evolved the ability to manufacture and secrete a sort of ‘glue’, commonly referred to as extra-polymeric substance (EPS). This substance serves a variety of purposes, not least it enables bacteria to ‘stick together’ and form colonies. These may adhere to surfaces as bio-films, or be suspended in fluids as bio-flocs. The EPS forms a protective matrix in which the cells can grow and divide. It also acts as a medium through which nutrients and cell metabolites can be transported, and by which cells may exchange chemical signals. In building mathematical models for the growth and behaviour of microbial colonies it is therefore important to take into account the role played by this EPS matrix.

A widely used modelling approach is that of agent-based descriptions; the individual cells in a colony are represented as discrete entities (agents) that grow, divide and interact with each other (and the EPS) through imposed biochemical and mechanical rules. Conceptually simple, and allowing detailed interactions to be prescribed relatively easily, this approach is designed for computer simulation. It has proved very effective for simulating the behaviour of colony formation and growth at small spatial scales, but as colony size increases the large number of cells is computationally prohibitive. (A 1mm square patch of biofilm will contain millions of individual cells). At larger scales, therefore, alternative modelling strategies are needed.

The main focus of this project will be on the development and application of population-based continuum models. Continuum models for bio-films, whilst not new, are perhaps less well developed and studied than agent-based models. In particular, the inclusion of microscale mechanical properties of EPS within continuum descriptions is an area where there is considerable modelling work to be done. Established modelling techniques developed in the context of multicomponent and granular continua will be adapted and applied to the type of biological media central to this project. Analysis of resulting models will involve both theoretical and numerical methods, and will require the development of some research codes.

The project will also have access to a recently developed, and mechanically detailed, agent-based simulation code, offering further reference data against which new continuum type models can be assessed.

The project would suit a mathematics or physics graduate with some background in continuum (fluid and/or solid) mechanics. Numerical work will require the development and use of computer codes, and programming experience/interest is necessary.

Supervisor: Dr David Swailes

Probabilistic models for particle transport in turbulent flows

Fluid flows often transport material in the form of small solid particles, liquid droplets or gas bubbles. Sometimes all three at once; sand grains, oil drops and air bubbles in water for example. The particulate material may be present by design (spray atomization is an integral part of many engineering processes), or be unwelcome (micro-plastics in water systems, volcanic emissions etc.). In many of these multi-phase systems the underlying flow is turbulent, and the way in which the dispersed particulates interact with this flow is crucial to the overall transport process. Aerosols, for example, tend to cluster in high-strain/low-vorticity regions, which influences the rate at which these droplets coalesce.

One way to study particle dynamics in turbulent flows is through computer simulations: Based on an underlying particle equation of motion we can simulate the trajectories of many hundreds of thousands of individual particles, and thereby build up a statistical picture of the collective behaviour of the disperse phase, described in terms particle concentrations, average velocities, kinetic energies etc. These will inevitably depend on (and perhaps influence) the statistical properties of the turbulence.

A second approach is to develop models that govern directly how these statistical measures evolve in both space and time. A model that allows us to compute directly the statistical distribution of particles obviates the need to perform time-consuming particle tracking simulations (other than to test that the models are correct!). It is this second approach that forms the basis for the research in this project.

By treating particle equations of motion as stochastic ordinary differential equations (SDEs) we can formulate transport equations for probability density functions (pdfs) that describe the resulting, ensemble-based distributions of particle properties such as position and velocity. The SDEs are non-standard in that they incorporate stochastic processes and fields that are correlated both in time and in space. This feature reflects correlation structures inherent in turbulent flow, and has a profound effect on the form of the resulting pdf transport equations; previous mathematical analysis has identified a number of subtle challenges associated with both theoretical and numerical treatment of these pdf models. In this work, we will consider how some of these issues may be addressed. Extended phase-space models (generalized Langevin equations) have been proposed. These eliminate non-Markovian features in the pdf models, but at the expense of higher-dimensionalities. Moreover, these extended models, as they now stand, are not capable of reproducing some key features of particle-phase transport associated with preferential sampling and drift. We will aim to develop and assess strategies that address these important issues.

The project would suit a mathematics or physics graduate with a background in fluid dynamics modelling and/or stochastic analysis. Numerical work will require the development and use of computer codes, and programming experience/interest is highly desirable.

Supervisor: Dr David Swailes

Artificial Biomolecular Condensates for Sustainability

Over the past decade, the scientific world has become increasingly fascinated with biomolecular condensates (BCs) — membrane-less compartments within cells that play a vital role in biological processes by bringing together molecules like proteins and nucleic acids. Our project is at the forefront of this exciting field, aiming to engineer artificial BCs to develop advanced membranes for cutting-edge applications in healthcare diagnostics and biomedical research.

The field of artificial BCs is still emerging, with significant challenges in creating stable and functional condensates. A key issue is controlling when and where the artificial BCs form. Our project tackles this challenge by using a technique called diffusiophoresis (DP), which moves particles by concentration gradients. By applying DP, we aim to precisely control the formation and spatial organization of artificial BCs. This is crucial for to develop advanced, bio-inspired membranes.

This research is a collaborative effort combining the expertise of Newcastle University and the University of Surrey. We will focus on three core objectives:

  • Exploring DP to control BC formation, monitored using advanced microscopy.
  • Scaling up the BC formation process.
  • Creating functionalised membranes for integration into biosensing devices.

This project offers an exceptional opportunity for students passionate about physics, bioscience, and nanoscience. The students will engage in experimental and computational approaches, mastering advanced techniques such as nano/biomaterial synthesis, optical microscopy, and computational modelling, while contributing to groundbreaking innovations in bio-inspired membranes with the potential to drive sustainability in healthcare. All this training will prepare the students for diverse career paths in academia and industry, while fostering leadership skills.

Supervisor: Dr Vasu Kalangi

Nature-inspired Nanofluidics

The surfaces of living cells feature exquisitely engineered pores, less than a nanometre in size (about one ten-thousandth the diameter of a human hair). These natural pores efficiently harness intermolecular forces and surface effects to achieve remarkable water and ion selectivity, even against concentration gradients. They are crucial in enabling neurons to generate electrical signals for brain function.

Replicating nature’s precision at the nanoscale presents significant challenges. The physics of fluid motion at this scale is not well understood, and artificial pores struggle to differentiate between ions with similar properties like sodium and potassium. Our project aims to overcome these limitations by investigating how ions and water interact with the walls of nanometer-sized pores, improving fluidic transport and ion selectivity. Further details: http://gtr.ukri.org/projects?ref=MR%2FX03660X%2F1

This research integrates experimental and theoretical approaches, focusing on three core objectives:

  • Fabrication of nanochannels and investigation of the microscopic flows they generate.
  • Quantification of liquid-solid friction using electrokinetic techniques and spectroscopy.
  • Design and scale-up of fluidic devices for sustainability and energy harvesting.

This project offers an exciting opportunity for students passionate about physics, materials science, and nanoscience. The students will gain hands-on experience with cutting-edge techniques such as nanochannel fabrication, advanced microscopy, spectroscopy, and computational modelling. These skills will prepare the students for a range of career paths in both academia and industry, while contributing to innovative technologies addressing global challenges in energy, water, and health.

Supervisor: Dr Vasu Kalangi