# 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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 SofTMech^{MP} 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.

### Multiscale modelling of liquid crystal-filled porous media (PhD)

**Supervisors:** Raimondo Penta, Nigel Mottram

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

Liquid crystals are scientifically fascinating and visually beautiful liquids that are all around us, forming an integral part of the liquid crystal display (LCD) used in almost every smart phone and computer display, and contained in both the cell wall and internal cytoplasm of all biological cells. The theory of liquid crystal materials that has been developed over the last fifty years has helped to bring about a revolution in display technology and increase the fundamental understanding of this phase of matter. The delicate nature of this phase, which can be disturbed by piconewton forces (about a trillionth of the force I am using to type these words on my keyboard) means that, to be made useful, they often need to be contained between rigid boundaries. In an LCD this is achieved by sandwiching the liquid crystal between to flat plates. However, more complicated structures to contain the liquid crystal have been proposed in recent years, including a polymer matrix or porous solids. By tailoring the porous medium in which the liquid crystal is contained, the optical and electrical properties and the flow of the liquid crystal can be controlled.

This PhD project will be focussed on developing a completely new multiscale continuum models of these liquid crystal-solid composites in order to understand and predict the microscopic behaviour of the liquid crystal within the pores, as well as the macroscopic properties of the whole composite system. The new modelling framework, which will be obtained by applying suitable homogenisation techniques (see for instance the reference below), will provide a connection between the composite’s response at the micro- and macro-scale. It is the feedback between the effects at different scales which we aim to understand using this new theory.

Applicants should have an undergraduate degree in mathematics/applied mathematics and experience in one or more of the following is desirable: continuum mechanics and elasticity; numerical methods for solving differential equations; scientific programming in Matlab; excellent writing and presentation skills.

During the project the student will benefit from training through the Scottish Mathematical Sciences Training Centre and develop expertise in multiscale continuum models, liquid crystal theory, partial differential equations, and finite element software to perform multi-dimensional numerical simulations related to the implementation of the modelling framework.

The project will be supervised by Dr Raimondo Penta and Prof. Nigel Mottram, experts in homogenisation theory of porous media and liquid crystals respectively, and the student will join a research group of around fifteen postgraduate and postdoctoral researchers working on liquid crystals and/or porous media and will join the group of over one hundred postgraduate students, in the School of Mathematics and Statistics at Glasgow.

Competitive scholarships, for UK and International student, are available and further information can be obtained by contacting Dr Penta (Raimondo.Penta@glasgow.ac.uk) and Prof. Mottram (Nigel.Mottram@glasgow.ac.uk).

Penta, R., Ramírez-Torres, A., Merodio, J., & Rodríguez-Ramos, R. (2021). Effective governing equations for heterogenous porous media subject to inhomogeneous body forces. Mathematics in Engineering, Vol. 3, No. 4, pp. 1-17, Open Access https://www.aimspress.com/article/id/5584.

### Interactions between groups, topological dynamics and operator algebras (PhD)

**Supervisors:** Xin Li

**Relevant research groups:** Analysis

The goal of this project is to develop a better understanding of the concept of continuous orbit equivalence for topological dynamical systems. This new notion has not been studied in detail before, and there are many interesting and important questions which are not well-understood, for instance rigidity phenomena. Apart from being interesting on its own right from the point of view of dynamics, continuous orbit equivalence is also closely related to the concepts of quasi-isometry in geometric group theory and Cartan subalgebras in C*-algebras. Hence we expect that progress made in the context of this project will have an important impact on establishing a fruitful interplay between operator algebras, topological dynamics and the geometry of groups.

The theme of this research project has the potential of shedding some light on long-standing open problems. At the same time, it leads to many interesting and feasible research problems.

### 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.

### 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

### Forecasting Local Net-electricity Demand at Scale (PhD)

**Supervisors:** Jethro Browell, Duncan Lee

**Relevant research groups:** Statistics and Data Analytics

Electricity supply and demand must balance in real-time, which is increasingly challenging as low-carbon technologies revolutionise energy production (wind, solar) and consumption (electric vehicles, heat pumps). Short-term forecasts are therefore essential to maintain an economic and reliable supply of electricity. Such forecasts are widely used in the energy sector, but forecasters face emerging challenges from new consumer behaviours, small scale generation and storage, as well as data quality, privacy, and security issues. This PhD project will give you the opportunity to develop statistical models to forecast electricity demand at regional and local levels of our continuously evolving energy system. Research themes include:

- Computationally efficient modelling and forecasting of 100s or 1000s of regions (or potentially millions of smart meters!).
- Adaptive modelling and forecasting in the presence of structural breaks.
- Probabilistic forecasting accounting for spatial and temporal dependencies and hierarchies.

The project provides an excellent opportunity to conduct cutting edge methodological development complemented by a practical application of societal importance. The successful candidate will need to be comfortable with interfacing with other disciplines and industry partners and be passionate about their research.** **