Sample thesis topics

Below are sample topics available for prospective postgraduate research students. These sample topics do not contain every possible project; they are aimed at giving an impression of the breadth of different topics available. Most prospective supervisors would be more than happy to discuss projects not listed below.

Funded projects are projects with project-specific funding. Funding for other projects is usally available on a competitive basis.

Mathematics

Algebra sample thesis topics

Quantum spin-chains and exactly solvable lattice models (PhD)

Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight-vertex models remain an active area of research with many surprising connections to other areas of mathematics.

Some of the algebra underlying these models deals with quantum and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute Gromov-Witten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).

Integrable quantum field theory and Y-systems (PhD)

Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.

In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to so-called Y-systems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.

Operator algebras associated to self-similar actions. (PhD)

Supervisors: Mike Whittaker
Relevant research groups: Algebra, Analysis, Geometry and Topology

This project will focus on self-similar groups and their operator algebras. The primary aim will be to examine a new class of groups that act self-similarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these self-similar groups.

Self-similar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*-algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*-algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.

Analysis sample thesis topics

Operator algebras associated to self-similar actions. (PhD)

Supervisors: Mike Whittaker
Relevant research groups: Algebra, Analysis, Geometry and Topology

This project will focus on self-similar groups and their operator algebras. The primary aim will be to examine a new class of groups that act self-similarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these self-similar groups.

Self-similar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*-algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*-algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.

Aperiodic substitution tilings and their C*-algebras. (PhD)

Supervisors: Mike Whittaker
Relevant research groups: Analysis, Geometry and Topology

A tiling is a collection of subsets of the plane, called tiles, for which any intersection of the interiors of two distinct tiles is empty and whose union is all of the plane. A tiling said to be aperiodic if it lacks translational periodicity. The most common method of producing aperiodic tilings is to use a substitution rule; a method for breaking each tile into smaller pieces, each of which is a scaled down copy of one of the original tiles, and then expanding so that each tile is congruent to one of the original tiles.

This project will focus on a natural class of operator algebras associated with an aperiodic substitution tiling. These algebras were first considered by Kellendonk and reflect the symmetries of a tiling in an algebraic object that allows up to consider invariants in a noncommutative framework. A key area of study are spectral triples associated with aperiodic tilings, which allow us to think of tilings as noncommutative geometric objects.

Continuum Mechanics sample thesis topics

Continuous production of solid metal foams (PhD)

Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems

Porous metallic solids, or solid metal foams, are exceedingly useful in many engineering applications, as they can be manufactured to be strong yet exceedingly lightweight. However, industrial processing methods for producing such foams are problematic and unreliable, and it is not currently possible to control the porosity distribution of the final product a priori.

This project will consider a new method of solid foam production, where bubbles of gas are introduced continuously into a molten metal flowing through a heat exchanger; foaming and solidification then occur almost simulatanously, allowing the foam structure to be controlled pointwise. The aim of this project is to construct a simple mathematical model for a gas bubble moving in a liquid filled channel ahead of a solidification front, to predict optimal conditions whereby the gas bubble is drawn toward the phase boundary, hence forming a porous solid.

This project will require some background in fluid mechanics and a combination of analytical and numerical techniques for solving partial differential equations.

Radial foam fracture (PhD)

Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems

Gas-liquid foams are a useful analgoue of crystalline atomic solids. 2D foam fracture has been used to study the mechanisms of fracture in metals. A two-dimenisonal network model (formed from a large system of differential equations) has recently been produced to study foam fracture in a rectangular channel which is pressurised along one edge. This model has helped to explain the origin of the velocity gap (a range of inadmissable steady fracture velocities), observed both in foam fracture experiments and in atomistic simulations of brittle fracture. This project will apply this network modelling approach to study radial foam fracture in a Hele-Shaw cell, to mimick recent experiments. This system has strong similarity to radial Saffmann-Taylor fingering, where fingering has been observed when a less viscous fluid displaces a more viscous fluid in a confined geometry. This project will involve studying systems of ordinary and partial differential equations using both numerical and analytical methods.

Numerical simulations of planetary and stellar dynamos (PhD)

Supervisors: Radostin Simitev
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Using Fluid Dynamics and Magnetohydrodynamics to model the magnetic fields of the Earth, planets, the Sun and stars. Involves high-performance computing.

Mathematical models of vasculogenesis (PhD)

Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems, Mathematical Biology

Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusion-limited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusion-limited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for in vitro tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMechMP project.

A coupled cardiovascular-respiration model for mechanical ventilation (PhD)

Supervisors: Peter Stewart, Nicholas A Hill
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Mathematical Biology

Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted by-products. However, over-inflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilation-perfusion mis-match. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patient-specific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a cross-disciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.

Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD) (PhD)

Supervisors: Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Solar magnetic fields are produced by a dynamo process in the Solar convection zone by turbulent motions acting against Ohmic dissipation. Solar magnetic activity affects nearEarth space environment and can harm modern technology and endanger human health. Further, Solar magnetism poses fundamental physical and mathematical problems, e.g. about the nature of plasma turbulence and the topology of magnetic field generation. Current models of the global Solar dynamo fall in two classes (a) mean-field dynamos (b) convection-driven dynamos. The mean-field models are only phenomenological as they replace turbulent interactions by ad-hoc source and quenching terms. On the other hand, spherical convection-driven dynamo models are derived from basic principles with minimal assumptions and potentially offer true predictive power; these can also be extended to other stars and giant planets. However, at present, convection driven dynamo models operate in a wrong dynamical regime and have limited success in reproducing a number of important 1 observations including (a) the sunspot cycle period, polarity reversals and the sunspot butterfly diagram, (b) the poleward migration of diffuse surface magnetic fields, (c) the polar field strength and phase relationships between poloidal/toroidal components. The aims of this project are to (a) develop a three-dimensional convection-driven Solar dynamo model constrained by assimilation of helioseismic data, and (b) start to use the model to estimate turbulent properties that determine the internal dynamics and activity cycles of the Sun.

Force balances in planetary cores and atmospheres (PhD)

Supervisors: Robert Teed, Radostin Simitev
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Current dynamo simulations are run, not under the conditions of planetary cores and atmospheres, but in a regime idealised for computations. To forecast changes in planetary magnetic fields such as reversals and dynamo collapse, it is vital to understand the actual fluid dynamics of these regions.

The aim of this project is to produce simulations of planetary cores and atmospheres with realistic force balances and, in doing so, understand how such force balances arise and affect the dynamics of the flow. Force balances control many aspects of the fluid dynamics, and hence the dynamo process itself, including the size of flow structure, the buoyancy flux and zonal flows, so an understanding of the force balance available in various planetary cores and atmospheres is vital for understanding their dynamo processes. To achieve this the project will use a different technique to that typically used in dynamo simulations. The approach is to perform global simulations in a spherical shell with a magnetic field imposed by explicitly setting a component (or components) of the field at one of the boundaries. Within the interior the field is evaluated as normal using the induction equation. This set-up amounts to a model of magnetoconvection where the dynamics of the flow and magnetic field can be studied independently of the dynamo mechanism.

Magnetic helicity as the key to dynamo bistability (PhD)

Supervisors: David MacTaggart, Radostin Simitev
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

The planets in the solar system exhibit very different types of large-scale magnetic field.The Earth has a strongly dipolar field, whereas the fields of other solar system planets, such as Uranus and Neptune, are far more irregular. Although the different physical compositions of the planets of the solar system will influence the behaviour of the large-scale magnetic fields that they generate, the morphology of planetary magnetic fields can depend on properties of dynamos common to all planets. Here, we refer to an important and recent discovery from dynamo simulations. Remarkably, two very different types of chaotic dipolar dynamo solutions have been found to exist over identical values of the basic parameters of a generic model of convection-driven dynamos in rotating spherical shells. The two solutions mentioned above can be characterised as ‘mean dynamos’, MD, where a strong poloidal field dominates and ‘fluctuating dynamos’, FD, where the poloidal component is weaker and the large-scale field can be described as multipolar. Although these two states have been shown to be bistable (co-exist) for a wide range of identical parameters, it is not clear how a particular state, MD or FD, is chosen and how/when one state can change to the other. Some of the bifurcations of such states has been investigated, but a deep understanding of the dynamics that cause the bifurcations remains to be developed. Since the magnetic topology of MD and FD states are fundamentally different, an important part of this project will be to probe the nature of MD and FD states by studying magnetic helicity, a magnetohydrodynamic invariant that combines information on the topology of the magnetic field with the magnetic flux. The role of magnetic helicity and other helicities (e.g. cross helicity) is currently not well understood in relation to MD and FD states, but these quantities are conjectured to be very important in the development of MD and FD states.  

Bistability is also related to a very important phenomenon in dynamos - global field reversal. A strongly dipolar (MD) field can change to a transitional multipolar (FD) state before a reversal and then settle into another dipolar equilibrium (of opposite polarity) again after the reversal.This project aims to develop a coherent picture of how bistability operates in spherical dynamos. Since bistability is a fundamental property of dynamos, a characterisation of how bistable solutions form and develop is key for any deep understanding of planetary dynamos and, in particular, could be crucial for understanding magnetic field reversals.

Stellar atmospheres and their magnetic helicity fluxes (PhD)

Supervisors: Simon Candelaresi, Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Our Sun and many other stars have a strong large-scale magnetic field with a characteristic time variation. We know that this field is being generated via a dynamo mechanism driven by the turbulent convective motions inside the stars. The magnetic helicity, a quantifier of the field’s topology, is and essential ingredient in this process. In turbulent environments it is responsible for the inverse cascade that leads to the large-scale field, while the build up of its small-scale component can quench the dynamo.
In this project, the student will study the effects of magnetic helicity fluxes that happen below the stellar surface (photosphere), within the stellar atmosphere (chromosphere and corona) and between these two layers. This will be done using two-dimensional mean field simulations that allow parameter studies for different physical parameters. A fully three-dimensional model of a convective stellar wedge will then be used to provide a more detailed picture of the helicity fluxes and their effect on the dynamo. Using recent advancements that allow us to extract surface helicity fluxes from solar observations, the student will make use of observations to verify the simulation results. Other recent observational results on the stellar magnetic helicity will be used to benchmark the findings.

Geometry and Topology sample thesis topics

Operator algebras associated to self-similar actions. (PhD)

Supervisors: Mike Whittaker
Relevant research groups: Algebra, Analysis, Geometry and Topology

This project will focus on self-similar groups and their operator algebras. The primary aim will be to examine a new class of groups that act self-similarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these self-similar groups.

Self-similar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*-algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*-algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.

Aperiodic substitution tilings and their C*-algebras. (PhD)

Supervisors: Mike Whittaker
Relevant research groups: Analysis, Geometry and Topology

A tiling is a collection of subsets of the plane, called tiles, for which any intersection of the interiors of two distinct tiles is empty and whose union is all of the plane. A tiling said to be aperiodic if it lacks translational periodicity. The most common method of producing aperiodic tilings is to use a substitution rule; a method for breaking each tile into smaller pieces, each of which is a scaled down copy of one of the original tiles, and then expanding so that each tile is congruent to one of the original tiles.

This project will focus on a natural class of operator algebras associated with an aperiodic substitution tiling. These algebras were first considered by Kellendonk and reflect the symmetries of a tiling in an algebraic object that allows up to consider invariants in a noncommutative framework. A key area of study are spectral triples associated with aperiodic tilings, which allow us to think of tilings as noncommutative geometric objects.

Geophysical & Astrophysical Fluid Dynamics sample thesis topics

Postdoctoral and PhD projects with specific funding will appear here when available. However, there is also the possibility to apply for funding via various schemes, so please get in touch with one of the group members if you are interested.

Integrable Systems and Mathematical Physics sample thesis topics

Quantum spin-chains and exactly solvable lattice models (PhD)

Supervisors: Christian Korff

Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight-vertex models remain an active area of research with many surprising connections to other areas of mathematics.

Some of the algebra underlying these models deals with quantum groups and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute Gromov-Witten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).

Integrable quantum field theory and Y-systems (PhD)

Supervisors: Christian Korff

Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.

In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to so-called Y-systems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.

Cherednik Algebras and related topics (PhD)

Supervisors: Misha Feigin

Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

The project is aimed at clarifying certain questions related to Cherednik algebras. These questions may include study of homomorphisms between rational Cherednik algebras for particular Coxeter groups and special multiplicity parameters, defining and studying of new partial spherical Cherednik algebras and their representations related to quasi-invariant polynomials, study of differential operators on quasi-invariants related to non-Coxeter arrangements. Relations with quantum integrable systems of Calogero-Moser type may be explored as well. Some other possible topics may include study of quasi-invariants for non-Coxeter arrangements in relation to theory of free arrangements of hyperplanes.

qDT invariants and deformations of hyperKahler geometry (PhD)

Supervisor: Ian Strachan

Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics

The project seeks to understand and exploit the integrable structure behind quantum Donaldson-Thomas invariants in terms of deformation of hyperKahler geometry and quantum-Riemann-Hilbert problems.

Almost-duality for arbitrary genus Hurwitz spaces (PhD)

Supervisor: Ian Strachan

Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics

The space of rational functions (interpreted as the space of holomorphic maps from the Riemann sphere to itself) may be endowed with the structure of a Frobenius manifolds, and hence there also exists an almost-dual Frobenius manifold structure. The class of examples include Coxeter and Extended-Affine-Weyl orbit group spaces. This extends to spaces of holomorphic maps between the torus and the sphere, where one can proved stronger results than just existence results. The project will seek to extend this to the explicit study of the space of holomorphic maps from an arbitrary genus Riemann surface to the Riemann sphere.

Mathematical Biology sample thesis topics

Efficient asymptotic-numerical methods for cardiac electrophysiology (PhD)

Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology

The mechanical activity of the heart is controlled by electrical impulses propagating regularly within the cardiac tissue during one's entire lifespan. A large number of very detailed ionic current models of cardiac electrical excitability are available.These realistic models are rather difficult for numerical simulations. This is due not only to their functional complexity but primarily to the significant stiffness of the equations.The goal of the proposed project is to develop fast and efficient numerical methods for solution of the equations of cardiac electrical excitation with the help and in the light of newly-developed methods for asymptotic analysis of the structure of cardiac equations (Simitev & Biktashev (2006) Biophys J; Biktashev et al. (2008), Bull Math Biol; Simitev & Biktashev (2011) Bull Math Biol)

The student will gain considerable experience with the theory of ordinary and partial differential equations, dynamical systems and bifurcation theory, asymptotic and perturbation methods,numerical methods. The applicant will also gain experience in computerprogramming, scientific computing and some statistical methods for comparison with experimental data.

Fast-slow asymptotic analysis of cardiac excitation models (PhD)

Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology

Mathematical models of cardiac electrical excitation describe processess ocurring on a wide range of time and length scales. 

Mathematical models of vasculogenesis (PhD)

Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems, Mathematical Biology

Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusion-limited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusion-limited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for in vitro tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMechMP project.

A coupled cardiovascular-respiration model for mechanical ventilation (PhD)

Supervisors: Peter Stewart, Nicholas A Hill
Relevant research groups: Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Mathematical Biology

Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted by-products. However, over-inflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilation-perfusion mis-match. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patient-specific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a cross-disciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.

Number Theory sample thesis topics

Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.

Statistics

Statistics and Data Analytics sample thesis topics

Modelling genetic variation (MSc/PhD)

Supervisors: Vincent Macaulay
Relevant research groups: Statistics and Data Analytics

Variation in the distribution of different DNA sequences across individuals has been shaped by many processes which can be modelled probabilistically, processes such as demographic factors like prehistoric population movements, or natural selection. This project involves developing new techniques for teasing out information on those processes from the wealth of raw data that is now being generated by high-throughput genetic assays, and is likely to involve computationally-intensive sampling techniques to approximate the posterior distribution of parameters of interest. The characterization of the amount of population structure on different geographical scales will influence the design of experiments to identify the genetic variants that increase risk of complex diseases, such as diabetes or heart disease.

The evolution of shape (PhD)

Supervisors: Vincent Macaulay
Relevant research groups: Statistics and Data Analytics

Shapes of objects change in time. Organisms evolve and in the process change form: humans and chimpanzees derive from some common ancestor presumably different from either in shape. Designed objects are no different: an Art Deco tea pot from the 1920s might share some features with one from Ikea in 2010, but they are different. Mathematical models of evolution for certain data types, like the strings of As, Gs , Cs and Ts in our evolving DNA, are quite mature and allow us to learn about the relationships of the objects (their phylogeny or family tree), about the changes that happen to them in time (the evolutionary process) and about the ways objects were configured in the past (the ancestral states), by statistical techniques like phylogenetic analysis. Such techniques for shape data are still in their infancy. This project will develop novel statistical inference approaches (in a Bayesian context) for complex data objects, like functions, surfaces and shapes, using Gaussian-process models, with potential application in fields as diverse as language evolution, morphometrics and industrial design.

Estimating the effects of air pollution on human health (PhD)

Supervisors: Duncan Lee
Relevant research groups: Statistics and Data Analytics

The health impact of exposure to air pollution is thought to reduce average life expectancy by six months, with an estimated equivalent health cost of 19 billion each year (from DEFRA). These effects have been estimated using statistical models, which quantify the impact on human health of exposure in both the short and the long term. However, the estimation of such effects is challenging, because individual level measures of health and pollution exposure are not available. Therefore, the majority of studies are conducted at the population level, and the resulting inference can only be made about the effects of pollution on overall population health. However, the data used in such studies are spatially misaligned, as the health data relate to extended areas such as cities or electoral wards, while the pollution concentrations are measured at individual locations. Furthermore, pollution monitors are typically located where concentrations are thought to be highest, known as preferential sampling, which is likely to result in overly high measurements being recorded. This project aims to develop statistical methodology to address these problems, and thus provide a less biased estimate of the effects of pollution on health than are currently produced.

Bayesian variable selection for genetic and genomic studies (PhD)

Supervisors: Mayetri Gupta
Relevant research groups: Statistics and Data Analytics

An important issue in high-dimensional regression problems is the accurate and efficient estimation of models when, compared to the number of data points, a substantially larger number of potential predictors are present. Further complications arise with correlated predictors, leading to the breakdown of standard statistical models for inference; and the uncertain definition of the outcome variable, which is often a varying composition of several different observable traits. Examples of such problems arise in many scenarios in genomics- in determining expression patterns of genes that may be responsible for a type of cancer; and in determining which genetic mutations lead to higher risks for occurrence of a disease. This project involves developing broad and improved Bayesian methodologies for efficient inference in high-dimensional regression-type problems with complex multivariate outcomes, with a focus on genetic data applications.

The successful candidate should have a strong background in methodological and applied Statistics, expert skills in relevant statistical software or programming languages (such as R, C/C++/Python), and also have a deep interest in developing knowledge in cross-disciplinary topics in genomics. The candidate will be expected to consolidate and master an extensive range of topics in modern Statistical theory and applications during their PhD, including advanced Bayesian modelling and computation, latent variable models, machine learning, and methods for Big Data. The successful candidate will be considered for funding to cover domestic tuition fees, as well as paying a stipend at the Research Council rate for four years.

Analysis of Spatially correlated functional data objects. (PhD)

Supervisors: Surajit Ray
Relevant research groups: Statistics and Data Analytics

Historically, functional data analysis techniques have widely been used to analyze traditional time series data, albeit from a different perspective. Of late, FDA techniques are increasingly being used in domains such as environmental science, where the data are spatio-temporal in nature and hence is it typical to consider such data as functional data where the functions are correlated in time or space. An example where modeling the dependencies is crucial is in analyzing remotely sensed data observed over a number of years across the surface of the earth, where each year forms a single functional data object. One might be interested in decomposing the overall variation across space and time and attribute it to covariates of interest. Another interesting class of data with dependence structure consists of weather data on several variables collected from balloons where the domain of the functions is a vertical strip in the atmosphere, and the data are spatially correlated. One of the challenges in such type of data is the problem of missingness, to address which one needs develop appropriate spatial smoothing techniques for spatially dependent functional data. There are also interesting design of experiment issues, as well as questions of data calibration to account for the variability in sensing instruments. Inspite of the research initiative in analyzing dependent functional data there are several unresolved problems, which the student will work on:

  • robust statistical models for incorporating temporal and spatial dependencies in functional data
  • developing reliable prediction and interpolation techniques for dependent functional data
  • developing inferential framework for testing hypotheses related to simplified dependent structures
  • analysing sparsely observed functional data by borrowing information from neighbours
  • visualisation of data summaries associated with dependent functional data
  • Clustering of functional data

Mapping disease risk in space and time (PhD)

Supervisors: Duncan Lee
Relevant research groups: Statistics and Data Analytics

Disease risk varies over space and time, due to similar variation in environmental exposures such as air pollution and risk inducing behaviours such as smoking.  Modelling the spatio-temporal pattern in disease risk is known as disease mapping, and the aims are to: quantify the spatial pattern in disease risk to determine the extent of health inequalities,  determine whether there has been any increase or reduction in the risk over time, identify the locations of clusters of areas at elevated risk, and quantify the impact of exposures, such as air pollution, on disease risk. I am working on all these related problems at present, and I have PhD projects in all these areas.

Modality of mixtures of distributions (PhD)

Supervisors: Surajit Ray
Relevant research groups: Statistics and Data Analytics

Finite mixtures provide a flexible and powerful tool for fitting univariate and multivariate distributions that cannot be captured by standard statistical distributions. In particular, multivariate mixtures have been widely used to perform modeling and cluster analysis of high-dimensional data in a wide range of applications. Modes of mixture densities have been used with great success for organizing mixture components into homogenous groups. But the results are limited to normal mixtures. Beyond the clustering application existing research in this area has provided fundamental results regarding the upper bound of the number of modes, but they too are limited to normal mixtures. In this project, we wish to explore the modality of non-normal distributions and their application to real life problems

Bayesian statistical data integration of single-cell and bulk “OMICS” datasets with clinical parameters for accurate prediction of treatment outcomes in Rheumatoid Arthritis (PhD)

Supervisors: Mayetri Gupta
Relevant research groups: Statistics and Data Analytics

In recent years, many different computational methods to analyse biological data have been established: including DNA (Genomics), RNA (Transcriptomics), Proteins (proteomics) and Metabolomics, that captures more dynamic events. These methods were refined by the advent of single cell technology, where it is now possible to capture the transcriptomics profile of single cells, spatial arrangements of cells from flow methods or imaging methods like functional magnetic resonance imaging. At the same time, these OMICS data can be complemented with clinical data – measurement of patients, like age, smoking status, phenotype of disease or drug treatment. It is an interesting and important open statistical question how to combine data from different “modalities” (like transcriptome with clinical data or imaging data) in a statistically valid way, to compare different datasets and make justifiable statistical inferences. This PhD project will be jointly supervised with Dr. Thomas Otto and Prof. Stefan Siebert from the Institute of Infection, Immunity & Inflammation), you will explore how to combine different datasets using Bayesian latent variable modelling, focusing on clinical datasets from Rheumatoid Arthritis.

Funding Notes

The successful candidate will be considered for funding to cover domestic tuition fees, as well as paying a stipend at the Research Council rate for four years.

New methods for analysis of migratory navigation (PhD)

Supervisors: Janine Illian
Relevant research groups: Statistics and Data Analytics

Joint project with Dr Urška Demšar (University of St Andrews)

Migratory birds travel annually across vast expanses of oceans and continents to reach their destination with incredible accuracy. How they are able to do this using only locally available cues is still not fully understood. Migratory navigation consists of two processes: birds either identify the direction in which to fly (compass orientation) or the location where they are at a specific moment in time (geographic positioning). One of the possible ways they do this is to use information from the Earth’s magnetic field in the so-called geomagnetic navigation (Mouritsen 2018). While there is substantial evidence (both physiological and behavioural) that they do sense magnetic field (Deutschlander and Beason 2014), we however still do not know exactly which of the components of the field they use for orientation or positioning. We also do not understand how rapid changes in the field affect movement behaviour.

There is a possibility that birds can sense these rapid large changes and that this may affect their navigational process. To study this, we need to link accurate data on Earth’s magnetic field with animal tracking data. This has only become possible very recently through new spatial data science advances:  we developed the MagGeo tool, which links contemporaneous geomagnetic data from Swarm satellites of the European Space Agency with animal tracking data (Benitez Paez et al. 2021).

Linking geomagnetic data to animal tracking data however creates a highly-dimensional data set, which is difficult to explore. Typical analyses of contextual environmental information in ecology include representing contextual variables as co-variates in relatively simple statistical models (Brum Bastos et al. 2021), but this is not sufficient for studying detailed navigational behaviour. This project will analyse complex spatio-temporal data using computationally efficient statistical model fitting approches in a Bayesian context.

This project is fully based on open data to support reproducibility and open science. We will test our new methods by annotating publicly available bird tracking data (e.g. from repositories such as Movebank.org), using the open MagGeo tool and implementing our new methods as Free and Open Source Software (R/Python).

References

Benitez Paez F, Brum Bastos VdS, Beggan CD, Long JA and Demšar U, 2021. Fusion of wildlife tracking and satellite geomagnetic data for the study of animal migration. Movement Ecology, 9:31. https://doi.org/10.1186/s40462-021-00268-4

Brum Bastos VdS, Łos M, Long JA, Nelson T and Demšar U, 2021, Context-aware movement analysis in ecology: a systematic review. International Journal of Geographic Information Science, https://doi.org/10.1080/13658816.2021.1962528

Deutschlander ME and Beason RC, 2014. Avian navigation and geographic positioning. Journal of Field Ornithology, 85(2):111–133. https://doi.org/10.1111/jofo.12055

Scalable Bayesian Models for Inferring Evolutionary Traits of Plants (PhD)

Supervisors: Vinny Davies
Relevant research groups: Statistics and Data Analytics

The functional traits and environmental preferences of plant species determine how they will react to changes resulting from global warming. The main global biodiversity repositories, such as the Global Biodiversity Information Facility (GBIF), contain hundreds of millions of records from hundreds of thousands of species in the plant kingdom alone, and the spatiotemporal data in these records can be associated with soil, climate or other environmental data from other databases. Combining these records allow us to identify environmental preferences, especially for common species where many records exist. Furthermore, in a previous PhD studentship we showed that these traits are highly evolutionarily conserved (Harris et al., 2022), so it is possible to impute the preferences for rare species where little data exists using phylogenetic inference techniques.

The aim of this PhD project is to investigate the application of Bayesian variable selection methods to identify these evolutionarily conserved traits more effectively, and to quantify these traits and their associated uncertainty for all plant species for use in a plant ecosystem digital twin that we are developing separately to forecast the impact of climate change on biodiversity. In another PhD studentship, we previously developed similar methods for trait inference in viral evolution (Davies et al., 2017; Davies et al., 2019), but due to the scale of the data here, these methods will need to be significantly enhanced. We therefore propose a project to investigate extensions to methods for phylogenetic trait inference to handle datasets involving hundreds of millions of records in phylogenies with hundreds of thousands of tips, potentially through either sub-sampling (Quiroz et al, 2018) or modelling splitting and recombination (Nemeth & Sherlock, 2018).

Metabolomics DIA Resolver (PhD)

Supervisors: Vinny Davies
Relevant research groups: Statistics and Data Analytics

In metabolomics we take a sample (blood, urine, etc) and put it through a mass spectrometer. The mass spectrometer scans the sample in multiple ways to help us work out what metabolites can be found in the sample. Identifying these metabolites can be useful for clinical trials, disease diagnosis and progression and various other medical applications. There are various way of choosing the scans, but in one particular method (DIA) we often see multiple fragments from multiple metabolites in a single scan. In order to identify the metabolites we need to work out which fragments belong to which metabolites. The project will use our recently developed virtual mass spectrometer, ViMMS (Wandy et al., 2019Wandy et al., 2022), to continue the development of our new metabolomics DIA resolver, MSdeconvolve. We will expand MSdeconvole to work across multiple repeated samples collected in different ways and then extended it to work for completely different samples. Initially this will be done using standard statistical and machine learning methods, but we will look to extend this into a Bayesian modelling framework.

Integrated spatio-temporal modelling for environmental data (PhD)

Supervisors: Janine Illian
Relevant research groups: Statistics and Data Analytics

(Jointly supervised by Peter Henrys, CEH)

The last decade has seen a proliferation of environmental data with vast quantities of information available from various sources. This has been due to a number of different factors including: the advent of sensor technologies; the provision of remotely sensed data from both drones and satellites; and the explosion in citizen science initiatives. These data represent a step change in the resolution of available data across space and time - sensors can be streaming data at a resolution of seconds whereas citizen science observations can be in the hundreds of thousands.  

Over the same period, the resources available for traditional field surveys have decreased dramatically whilst logistical issues (such as access to sites, ) have increased. This has severely impacted the ability for field survey campaigns to collect data at high spatial and temporal resolutions. It is exactly this sort of information that is required to fit models that can quantify and predict the spread of invasive species, for example. 

Whilst we have seen an explosion of data across various sources, there is no single source that provides both the spatial and temporal intensity that may be required when fitting complex spatio-temporal models (cf invasive species example) - each has its own advantages and benefits in terms of information content. There is therefore potentially huge benefit in beginning together data from these different sources within a consistent framework to exploit the benefits each offers and to understand processes at unprecedented resolutions/scales that would be impossible to monitor. 

Current approaches to combining data in this way are typically very bespoke and involve complex model structures that are not reusable outside of the particular application area. What is needed is an overarching generic methodological framework and associated software solutions to implement such analyses. Not only would such a framework provide the methodological basis to enable researchers to benefit from this big data revolution, but also the capability to change such analyses from being stand alone research projects in their own right, to more operational, standard analytical routines. 

FInally, such dynamic, integrated analyses could feedback into data collection initiatives to ensure optimal allocation of effort for traditional surveys or optimal power management for sensor networks. The major step change being that this optimal allocation of effort is conditional on other data that is available. So, for example, given the coverage and intensity of the citizen science data, where should we optimally send our paid surveyors? The idea is that information is collected at times and locations that provide the greatest benefit in understanding the underpinning stochastic processes. These two major issues - integrated analyses and adaptive sampling - ensure that environmental monitoring is fit for purpose and scientists, policy and industry can benefit from the big data revolution. 

This project will develop an integrated statistical modelling strategy that provides a single modelling framework for enabling quantification of ecosystem goods and services while accounting for the fundamental differences in different data streams. Data collected at different spatial resolutions can be used within the same model through projecting it into continuous space and projecting it back into the landscape level of interest.  As a result, decisions can be made at the relevant spatial scale and uncertainty is propagated through, facilitating appropriate decision making. 

Evaluating probabilistic forecasts in high-dimensional settings (PhD)

Supervisors: Jethro Browell
Relevant research groups: Statistics and Data Analytics

Many decisions are informed by forecasts, and almost all forecasts are uncertain to some degree. Probabilistic forecasts quantify uncertainty to help improve decision-making and are playing an important role in fields including weather forecasting, economics, energy, and public policy. Evaluating the quality of past forecasts is essential to give forecasters and forecast users confidence in their current predictions, and to compare the performance of forecasting systems.

While the principles of probabilistic forecast evaluation have been established over the past 15 years, most notably that of “sharpness subject to calibration/reliability”, we lack a complete toolkit for applying these principles in many situations, especially those that arise in high-dimensional settings. Furthermore, forecast evaluation must be interpretable by forecast users as well as expert forecasts, and assigning value to marginal improvements in forecast quality remains a challenge in many sectors.

This PhD will develop new statistical methods for probabilistic forecast evaluation considering some of the following issues:

  • Verifying probabilistic calibration conditional on relevant covariates
  • Skill scores for multivariate probabilistic forecasts where “ideal” performance is unknowable
  • Assigning value to marginal forecast improvement though the convolution of utility functions and Murphey Diagrams
  • Development of the concept of “anticipated verification” and “predicting the of uncertainty of future forecasts”
  • Decomposing forecast misspecification (e.g. into spatial and temporal components)
  • Evaluation of Conformal Predictions

Good knowledge of multivariate statistics is essential, prior knowledge of probabilistic forecasting and forecast evaluation would be an advantage.

Adaptive Probabilistic Forecasting (PhD)

Supervisors: Jethro Browell
Relevant research groups: Statistics and Data Analytics

Data-driven predictive models depend on the representativeness of data used in model selection and estimation. However, many processes change over time meaning that recent data is more representative than old data. In this situation, predictive models should track these changes, which is the aim of “online” or “adaptive” algorithms. Furthermore, many users of forecasts require probabilistic forecasts, which quantify uncertainty, to inform their decision-making. Existing adaptive methods such as Recursive Least Squares, the Kalman Filter have been very successful for adaptive point forecasting, but adaptive probabilistic forecasting has received little attention. This PhD will develop methods for adaptive probabilistic forecasting from a theoretical perspective and with a view to apply these methods to problems in at least one application area to be determined.

In the context of adaptive probabilistic forecasting, this PhD may consider:

  • Online estimation of Generalised Additive Models for Location Scale and Shape
  • Online/adaptive (multivariate) time series prediction
  • Online aggregation (of experts, or hierarchies)

A good knowledge of methods for time series analysis and regression is essential, familiarity with flexible regression (GAMs) and distributional regression (GAMLSS/quantile regression) would be an advantage.

Statistical methodology for Assessing the impacts of offshore renewable developments on marine wildlife (PhD)

Supervisors: Janine Illian
Relevant research groups: Statistics and Data Analytics

(jointly supervised by Esther Jones and Adam Butler, BIOSS)

Assessing the impacts of offshore renewable developments on marine wildlife is a critical component of the consenting process. A NERC-funded project, ECOWINGS, will provide a step-change in analysing predator-prey dynamics in the marine environment, collecting data across trophic levels against a backdrop of developing wind farms and climate change. Aerial survey and GPS data from multiple species of seabirds will be collected contemporaneously alongside prey data available over the whole water column from an automated surface vehicle and underwater drone.

These methods of data collection will generate 3D space and time profiles of predators and prey, creating a rich source of information and enormous potential for modelling and interrogation. The data present a unique opportunity for experimental design across a dynamic and changing marine ecosystem, which is heavily influenced by local and global anthropogenic activities. However, these data have complex intrinsic spatio-temporal properties, which are challenging to analyse. Significant statistical methods development could be achieved using this system as a case study, contributing to the scientific knowledge base not only in offshore renewables but more generally in the many circumstances where patchy ecological spatio-temporal data are available. 

This PhD project will develop spatio-temporal modelling methodology that will allow user to anaylse these exciting - and complex - data sets and help inform our knowledge on the impact of off-shore renewable on wildlife.