PhD projects
This page provides a (partial) list of specific (and not so specific) PhD projects currently offered by academics in the School.
Identifying an interesting, worthwhile and doable PhD project is not a trivial task since it depends crucially on the interests and abilities of the student (and the supervisor!). The list below therefore contains a mixture of very specific projects and fairly general descriptions of research interests across the School. In either case you should feel free to contact the potential supervisors to find out more if you're interested.
All projects listed here are eligible for funding via scholarships from the School of Mathematics and/or the University of Manchester; see the Fees and Funding page for details. These scholarships are awarded competitively amongst the eligible applicants by School or University postgraduate funding committees. Members of staff regularly apply for research grants and may therefore be able to offer funding for specific projects without requiring approval from committees. Such projects are listed on a separate page.
Please note that this is not an exhaustive list of all the PhD projects we can offer. You should therefore also explore the School's research groups' pages, and have a look around the homepages of individual members of staff to find out more about their research interests. You may also contact us for general enquiries.
Title 
Quantitative Aspects of Number Theory 

Group  Number Theory 
Supervisor  
Description 
The aim of this project is to investigate problems in number theory that have a quantitative component. Typically, the problems would involve some algebraic objects, such as as algebraic number fields or rational points on algebraic varieties. The questions that we ask are, however, of analytic nature. For example, how many number fields are there of a given type and with given properties, such that their discriminant is bounded by a (large) number B? The answer that we are seeking would then be an asymptotic formula or, if this is too hard, bounds for this number as B tends to infinity. Quantitative results of this kind are of interest by themselves, but moreover they are useful in other proofs, often to establish the existence of objects with certain properties. This holds, for example, for many applications of the HardyLittlewood circle method from analytic number theory. A candidate for this project would have some background in analytic or algebraic number theory, as well as an interest in learning the other field. An interest in algebraic geometry could be useful, but is by no means required. The concrete problems to work on will be chosen to match the student's interests. 
Title 
Fluid Mechanics of Cleaning and Decontamination 

Group  Continuum Mechanics 
Supervisor  
Description 
Cleaning and decontamination processes are important in many applications: from the daily chores of doing the dishes (with or without a dishwasher), to ensuring clean hygiene in hospitals, the food industry, or pharmaceutical companies. Although a lot of research has been done in chemistry and chemical engineering to improve detergents and cleaning devices, much less work has been done on the modelling of the underlying physical and chemical processes. In some cleaning applications, such as the neutralisation of toxic chemicals after a spill, it is crucial to avoid using strong mechanical forces in order to prevent the dispersion of the toxic material in the environment. Instead, a localised dissolution process, aided by chemical reactions neutralising the material, is used. This PhD project will investigate the advection, diffusion and reaction processes involved in this scenario. Through a combination of experiments and modelling work the student will study the influence of flow properties: such as the Reynolds number and the Péclet number; geometry: whether the material is attached to a permeable or impermeable surface; and chemical properties such as solubility, reactivity and diffusivity. This project is directly motivated by industrial applications and will suit candidates interested in using mathematical approaches to solve real challenges. Suitable candidates should have experience in the lab or a keen interest to support theoretical work in fluid dynamics by experimental evidence. Reference: Landel, Thomas, McEvoy & Dalziel (2016). Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 789: 630.

Title 
Fluid Mechanics of Cleaning and Decontamination 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
Cleaning and decontamination processes are important in many applications: from the daily chores of doing the dishes (with or without a dishwasher), to ensuring clean hygiene in hospitals, the food industry, or pharmaceutical companies. Although a lot of research has been done in chemistry and chemical engineering to improve detergents and cleaning devices, much less work has been done on the modelling of the underlying physical and chemical processes. In some cleaning applications, such as the neutralisation of toxic chemicals after a spill, it is crucial to avoid using strong mechanical forces in order to prevent the dispersion of the toxic material in the environment. Instead, a localised dissolution process, aided by chemical reactions neutralising the material, is used. This PhD project will investigate the advection, diffusion and reaction processes involved in this scenario. Through a combination of experiments and modelling work the student will study the influence of flow properties: such as the Reynolds number and the Péclet number; geometry: whether the material is attached to a permeable or impermeable surface; and chemical properties such as solubility, reactivity and diffusivity. This project is directly motivated by industrial applications and will suit candidates interested in using mathematical approaches to solve real challenges. Suitable candidates should have experience in the lab or a keen interest to support theoretical work in fluid dynamics by experimental evidence. Reference: Landel, Thomas, McEvoy & Dalziel (2016). Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 789: 630. 
Title 
Bubble dynamics in confined geometries 

Group  Continuum Mechanics 
Supervisors  
Description 
The motion of deformable bubbles, drops, capsules and cells surrounded by viscous fluid and confined in a narrow channel has applications in microfluidic devices and labonachip design. When viewed from above, the motion often appears two dimensional, suggesting that we might be able to use two dimensional, depthaveraged models to predict, design and control system behaviour. Depthaveraged models are known to be accurate in some situations, such as when the bubble is very flattened and far from any walls, and where thin films are relatively passive. However, the typical assumptions break down if two bubbles are very close to each other (e.g. coalescence or breakup), or to walls, constrictions or obstacles, when it highly likely that threedimensional effects come into play. Furthermore, the typical depthaveraged model is of lower order and dimension than the Stokes equations and hence neglects certain free boundary effects. The aim of this project is to establish the limits of 2D modelling, and to determine how we can extend our models to be valid in these extreme scenarios. This project will involve asymptotic analysis, model development, finite element computations of two or three dimensional systems in oomphlib, and likely also collaborations with experimentalists in the Manchester Centre for Nonlinear Dynamics. 
Title 
Thermodynamic Quantum Chaos and large networks 

Group  Analysis and Dynamical Systems 
Supervisor  
Description 
Quantum chaos is a field that is aimed to study the properties of eigenfunctions of the Laplacian (stationary quantum states) on a Riemannian manifold using the chaotic properties of the underlying geodesic flow in high energy, that is, in the large eigenvalue limit. In this sense the field is connecting quantum mechanics to classical mechanics. A key result in the field is the Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière, which is an equidistribution result of the eigenfunctions for large eigenvalues when the geodesic flow is ergodic. In our recent work we have been attempting to study the theory for a problem of Thermodynamic Quantum Ergodicity (TQE), where instead of large energy, we fix an energy window and vary the geometric properties of the manifold such as volume or genus. The project would aim to develop TQE, in particular for the context of Lie groups and variable curvature manifolds. Moreover, we would attempt to find connect the ideas from TQE to discrete analogues such as spectral theory of large networks, which are welldeveloped by the recent works of Anantharaman, Brooks, Le Masson, Lindenstrauss, Sabri, and others. 
Title 
Additive combinatorics in dynamics and spectral theory 

Group  Analysis and Dynamical Systems 
Supervisor  
Description 
The highfrequency asymptotics of Fourier coefficients of functions and measures describe their local structure. For example, they can be used to yield geometric or arithmetic features of the object under study such as on dimension, curvature, equidistribution or combinatoric structure. In my recent work I have been working on finding conditions based on ergodic theory, dynamical systems and stochastics which yield efficient estimates for Fourier transforms of dynamically or randomly constructed objects. Furthermore, we aim to use these estimates to obtain new arithmetic/geometric applications for them. Recent revolutions on applications of additive combinatorics to dynamics by Bourgain, Dyatlov, Hochman, Shmerkin et al. have presented many interesting problems that we will attempt to now solve in the context of thermodynamical formalism. Specific projects include establishing connections between nonlinearity and Fourier transforms, and Fractal Uncertainty Principle for Gibbs measures of Kleinian group actions, which would yield to new essential spectral gap estimates for Laplacian on hyperbolic manifolds. 
Title 
Scenery flow and fine structure of fractals 

Group  Analysis and Dynamical Systems 
Supervisor  
Description 
After the recent influential works of Furstenberg and HochmanShmerkin techniques based on magnifying measures (and taking their tangent measures) have been essential in the study of arithmetic and geometric features of sets and measures in new settings. For example, these approaches work well with notions that involve questions on the entropy or dimensions of a measure, projections and distance sets, or features related to equidistribution. The key ideas are based on the dynamics or stochastics of the process of magnification and applying classical tools from ergodic theory and Markov chains. I have been recently working on developing these techniques with new arithmetic and geometric applications in mind. The project would try to attempt develop these in the setting of nonconformal dynamics, in particular selfaffine fractals such as Baranski carpets and also fractals arising from nonsmooth dynamics such as quasiregular geometry. 
Title 
Open Maps 

Group  Analysis and Dynamical Systems 
Supervisor  
Description 
The study of open maps is an exciting relatively new area of the theory of dynamical systems. The characterisation of the holes involves geometry (‘shape’) and measure theory (‘size’), so their study involves a complex interplay between dynamics, geometry and analysis. The standard dynamical approach is to assume that the survivor set (= the points whose orbits do not fall into a hole) is “sufficiently large” and investigate ergodic and geometric properties of the induced map. The novelty of the proposed project is to take a step back and give sufficient conditions that the survivor set is indeed "sufficiently large" (uncountable, say) for the induced map to be meaningful. This involves a detailed analysis of the class of holes under investigation. The classes of maps under investigation include  but are not limited to  expanding maps of the interval, algebraic toral automorphisms and subshifts. 
Title 
Analytic Number Theory and mean values of Lfunctions 

Group  Number Theory 
Supervisor  
Description 
The Riemann zetafunction and other Lfunctions play a central role in analytic number theory and in mathematics in general. For example, the Riemann zetafunction satisfies an Euler product, which underlines a connection between the natural numbers and the prime numbers. The problem of determining the properties of prime numbers has a long history, from the ancient theorem of Euclid that there are infinitely many primes, to the celebrated eight page paper of Riemann on the zetafunction in the midnineteenth century. Since that time, several important problems in analytic number theory have been solved, and Riemann's ideas have been the inspiration behind much of this progress. Investigating the properties of the Riemann zetafunction and Lfunctions in various contexts leads to many other interesting problems, which now represent major challenges in modern mathematics. In fact both the Riemann Hypothesis, which asserts that all the nontrivial zeros of the Riemann zetafunction lie on a particular line, and the Birch and SwinnertonDyer Conjecture, which concerns some properties of the Lfunctions associated to elliptic curves, have been included in the seven Millennium Prize Problems. The aim of the project is to study various questions related to the moments of the Riemann zetafunction and Lfunctions, which are the mean values over certain families of these functions. These questions have applications to the distribution of zeros of the Riemann zetafunction (partial answers to the Riemann Hypothesis), the order of magnitude of Lfunctions (partial answers to the Lindelof Hypothesis), order of vanishing of Lfunctions at the central point (analytic progress towards the Birch and SwinnertonDyer Conjecture), and many others. There is a remarkable connection between the subject and Random Matrix Theory, an area of Mathematical Physics used to describe complex quantum systems. 
Title 
Distributional approximation by Stein's method 

Group  Probability and Stochastic Analysis 
Supervisor  
Description 
Stein's method is a powerful (and elegant) technique for deriving bounds on the distance between two probability distributions with respect to a probability metric. Such bounds are of interest, for example, in statistical inference when samples sizes are small; indeed, obtaining bounds on the rate of convergence of the central limit theorem was one of the most important problems in probability theory in the first half of the 20th century. The method is based on differential or difference equations that in a sense characterise the limit distribution and coupling techniques that allow one to derive approximations whilst retaining the probabilistic intuition. There is an active area of research concerning the development of Stein's method as a probabilistic tool and its application in areas as diverse as random graph theory, statistical mechanics and queuing theory. There is an excellent survey of Stein's method (see below) and, given a strong background in probability, the basic method can be learnt quite quickly, so it would be possible for the interested student to make progress on new problems relatively shortly into their PhD. Possible directions for research (although not limited) include: extend Stein's method to new limit distributions; generalisations of the central limit theorem; investigate `faster than would be expected' convergence rates and establish necessary and sufficient conditions under which they occur; applications of Stein's method to problems from, for example, statistical inference. Literature: Ross, N. Fundamentals of Stein's method. Probability Surveys 8 (2011), pp. 210293. 
Title 
Generalized Flame Balls and their Stability 

Group  Continuum Mechanics 
Supervisor  
Description 
Flame balls are balls of burnt gas in a reactive mixture, which constitute stationary solutions to nonlinear Poisson's equations. These were first described by the famous Russian physicist Zeldovich (the father of Combustion Theory) about 70 years ago. The fact that these solutions are typically unstable provides a powerful fundamental criterion for successful ignition, i.e. determines the minimum energy (of the spark) required to generate propagating flames. Several projects are available to extend the study of these fascinating flames (mainly their existence and stability) to take into account realistic effects such as the presence of flowfield, nonuniformity of the reactive mixture, proximity of walls, etc. Methodology: The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs), depending on the preference of the candidate. 
Title 
Laminar aspects of turbulent combustion/ Flame propagation in a multiscale flow 

Group  Continuum Mechanics 
Supervisor  
Description 
The idea is to ask if the fundamental questions of turbulent combustion can be answered for simple laminar flows. Since the answer is often no, we shall formulate and study problems to answer these questions in simpler laminarflow situations. An exciting topic! Methodology: The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs), depending on the preference of the candidate. 
Title 
Taylor dispersion and hydrodynamic lubrication theory in premixed combustion 

Group  Continuum Mechanics 
Supervisor  
Description 
In 1953, the British physicist G.I. Taylor published an influential paper describing the enhancement of diffusion processes by a (shear) flow, a phenomenon later termed Taylor dispersion. This has generated to date thousands of publications in various areas involving transport phenomena, none of which, surprisingly, in the field of combustion. In 1940, the German chemist G. Damköhler postulated two hypotheses which have largely shaped current views on the propagation of premixed flames in turbulent flow fields. The project consists of pioneering investigations linking Taylor dispersion and Damköhler’s hypotheses, and is expected to provide significant insight into turbulent combustion. The work will be carried out in the framework of lubrication theory, generalized to combustion situations, and will include interesting stability problem such as the SaffmanTaylor instability in a reactive mixture. Methodology: The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs), depending on the preference of the candidate. 
Title 
Mathematical Combustion and Flame Instabilities 

Group  Continuum Mechanics 
Supervisor  
Description 
Several projects are available related to the mathematical theory of flame propagation, a fascinating multidisciplinary area of applied mathematics involving ordinary and partial differential equations; combustion basics will be introduced to candidate. The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs). The multidisciplinary experience in combustion involved will be useful for tackling research problems in other fields of application, and will constitute a valuable asset for jobs in industry (such as the automobile or the aeronautics industry). Depending on the preference of the candidate, each of the projects can be tailored in its scope and the methodology of study. Suggested sample projects:

Title 
Additive combinatorics and Diophantine problems 

Group  Analysis and Dynamical Systems 
Supervisor  
Description 
The study of Diophantine equations encompasses a diverse portion of modern number theory. Recent years have seen spectacular progress on solving linear Diophantine equations in certain sets of interest, such as dense sets or the set of primes. Much of this progress has been achieved by breaking the problem down into a structure versus randomness dichotomy, using tools from additive combinatorics. One tackles the structured problem using techniques from classical analytic number theory and dynamical systems, whilst the ‘random' problem is handled using ideas informed by probabilistic combinatorics and Fourier analysis. The consequences of this rapidly developing theory for nonlinear Diophantine equations have yet to be fully explored. Some possible research topics include (but are not limited to) the following:  Existence of solutions to systems of Diophantine equations in dense sets. To what extent can Szemerédi’s theorem be generalised to nonlinear systems of equations?  Quantitative bounds for sets lacking Diophantine configurations. Can one obtain good quantitative bounds in the polynomial Szemerédi theorem? What about sets lacking progressions with common difference equal to a prime minus one?  Partition regularity of Diophantine equations. Can one generalise a Ramseytheoretic criterion of Rado to systems of degree greater than one?  Higher order Fourier analysis of nonlinear equations. Is it possible to count solutions to hitherto intractable Diophantine equations by developing the Hardy—Littlewood method along the lines of Green and Tao? What are the obstructions to uniformity for such equations? 
Title 
Additive combinatorics and Diophantine problems 

Group  Number Theory 
Supervisor  
Description 
The study of Diophantine equations encompasses a diverse portion of modern number theory. Recent years have seen spectacular progress on solving linear Diophantine equations in certain sets of interest, such as dense sets or the set of primes. Much of this progress has been achieved by breaking the problem down into a structure versus randomness dichotomy, using tools from additive combinatorics. One tackles the structured problem using techniques from classical analytic number theory and dynamical systems, whilst the ‘random' problem is handled using ideas informed by probabilistic combinatorics and Fourier analysis. The consequences of this rapidly developing theory for nonlinear Diophantine equations have yet to be fully explored. Some possible research topics include (but are not limited to) the following:  Existence of solutions to systems of Diophantine equations in dense sets. To what extent can Szemerédi’s theorem be generalised to nonlinear systems of equations?  Quantitative bounds for sets lacking Diophantine configurations. Can one obtain good quantitative bounds in the polynomial Szemerédi theorem? What about sets lacking progressions with common difference equal to a prime minus one?  Partition regularity of Diophantine equations. Can one generalise a Ramseytheoretic criterion of Rado to systems of degree greater than one?  Higher order Fourier analysis of nonlinear equations. Is it possible to count solutions to hitherto intractable Diophantine equations by developing the Hardy—Littlewood method along the lines of Green and Tao? What are the obstructions to uniformity for such equations? 
Title 
Algebraic differential equations and model theory 

Group  Mathematical Logic 
Supervisor  
Description 
Generally speaking this area is currently my main focus of research. Differential rings and algebraic differential equations have been a crucial source of examples for model theory (more specifically, geometric stability theory), and have had numerous application in number theory, algebraic geometry, and combinatorics (to name a few). In this project we propose to establish and analyse deep structural results on the model theory of (partial) differential fields. It has been known, for quite some time now, that while the classical notions of 'dimension' differ for differential fields, there is a strong relationship between them. We aim to tackle the following foundational (still open) question of this theory: are there infinite dimensional types that are also strongly minimal? This is somewhat related to the understanding of regular types, which interestingly are quite far from being fully classified. A weak version of Zilber's dichotomy have been established for such types, but is the full dichotomy true? The above is also connected to the understanding of differentialalgebraic groups (or definable groups in differentially closed fields). While the notions of dimension agree for these objects in the 'ordinary' case, the question is still open in the 'partial' case. We expect that once progress is made in the direction of the above problems, we will also be closer to the answer of this question. There are classical references for all of the above concepts and standing problems, so the interested student should have no problem in learning the background material (and start making progress) in a relatively short period of time.

Title 
Algebraic differential equations and model theory 

Group  Algebra 
Supervisor  
Description 
Generally speaking this area is currently my main focus of research. Differential rings and algebraic differential equations have been a crucial source of examples for model theory (more specifically, geometric stability theory), and have had numerous application in number theory, algebraic geometry, and combinatorics (to name a few). In this project we propose to establish and analyse deep structural results on the model theory of (partial) differential fields. It has been known, for quite some time now, that while the classical notions of 'dimension' differ for differential fields, there is a strong relationship between them. We aim to tackle the following foundational (still open) question of this theory: are there infinite dimensional types that are also strongly minimal? This is somewhat related to the understanding of regular types, which interestingly are quite far from being fully classified. A weak version of Zilber's dichotomy have been established for such types, but is the full dichotomy true? The above is also connected to the understanding of differentialalgebraic groups (or definable groups in differentially closed fields). While the notions of dimension agree for these objects in the 'ordinary' case, the question is still open in the 'partial' case. We expect that once progress is made in the direction of the above problems, we will also be closer to the answer of this question. There are classical references for all of the above concepts and standing problems, so the interested student should have no problem in learning the background material (and start making progress) in a relatively short period of time.

Title 
Rational points on algebraic varieties 

Group  Number Theory 
Supervisor  
Description 
Diophantine equations are a classical object of study in number theory. During the course of the 20th century, it was realised that one obtains a more powerful conceptual framework by considering them through a more geometric lens, namely viewing a solution to a Diophantine equation as a rational point on the associated algebraic variety. Given an algebraic variety over a number field, natural questions are: Is there is a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin’s conjecture). A popular current research theme is to consider these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve these problems one usually uses a combination of techniques from algebraic geometry and analytic number theory, but the project could be tailored towards the preferences of the student (e.g. for a student without much knowledge of algebraic geometry). 
Title 
Thermoviscoacoustic metamaterials for underwater applications 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
The ability to control underwater noise has been of practical interest for decades. Such noise, radiating from e.g. offshore wind farms, turbines, and merchant vessels, frequently needs to be attenuated artificially given the close proximity of its generation to sensitive marine environments for example. Over the last century a number of materials have been designed to assist with underwater noise attenuation. However, recently there has been an explosion of interest in the topic of acoustic metamaterials and metasurfaces. Such media have special microstructures, designed to provide overall (dynamic) material properties that natural materials can never hope to attain and lead to the potential of negative refraction, wave redirection and the holy grail of cloaking. Many of the mechanisms to create these artificial materials rely on the notion of resonance, which in turn gives rise to the possibility of low frequency sound attenuation. This is extremely difficult to achieve with normal materials. The mechanisms of sound attenuation, i.e. thermal and viscous, have not yet been properly understood for the many metamaterials under study, particularly in an underwater context. The aim of this project is to study this aspect via mathematical analysis and then to optimize designs in order to design and employ metamaterials for use in underwater noise reduction applications. Although there has been some initial interest over the last few years in the “inair” context, the parameter regime underwater gives rise to new effects that need to be explored and understood thoroughly. Initially canonical geometries such as simple apertures and infinite and semiinfinite ducts shall be considered before moving on to more complex, realistic scenarios and geometries where resonance plays a key role. Mathematical modelling using the method of matched asymptotics shall be employed. This is ideally suited to the scenarios considered given the low frequency regime. Comparisons shall be drawn with direct numerical simulations using finite element methods in e.g. COMSOL. 
Title 
Thermoviscoacoustic metamaterials for underwater applications 

Group  Continuum Mechanics 
Supervisor  
Description 
The ability to control underwater noise has been of practical interest for decades. Such noise, radiating from e.g. offshore wind farms, turbines, and merchant vessels, frequently needs to be attenuated artificially given the close proximity of its generation to sensitive marine environments for example. Over the last century a number of materials have been designed to assist with underwater noise attenuation. However, recently there has been an explosion of interest in the topic of acoustic metamaterials and metasurfaces. Such media have special microstructures, designed to provide overall (dynamic) material properties that natural materials can never hope to attain and lead to the potential of negative refraction, wave redirection and the holy grail of cloaking. Many of the mechanisms to create these artificial materials rely on the notion of resonance, which in turn gives rise to the possibility of low frequency sound attenuation. This is extremely difficult to achieve with normal materials. The mechanisms of sound attenuation, i.e. thermal and viscous, have not yet been properly understood for the many metamaterials under study, particularly in an underwater context. The aim of this project is to study this aspect via mathematical analysis and then to optimize designs in order to design and employ metamaterials for use in underwater noise reduction applications. Although there has been some initial interest over the last few years in the “inair” context, the parameter regime underwater gives rise to new effects that need to be explored and understood thoroughly. Initially canonical geometries such as simple apertures and infinite and semiinfinite ducts shall be considered before moving on to more complex, realistic scenarios and geometries where resonance plays a key role. Mathematical modelling using the method of matched asymptotics shall be employed. This is ideally suited to the scenarios considered given the low frequency regime. Comparisons shall be drawn with direct numerical simulations using finite element methods in e.g. COMSOL. 
Title 
Rational points on algebraic varieties 

Group  Geometry and Topology 
Supervisor  
Description 
Diophantine equations are a classical object of study in number theory. During the course of the 20th century, it was realised that one obtains a more powerful conceptual framework by considering them through a more geometric lens, namely viewing a solution to a Diophantine equation as a rational point on the associated algebraic variety. Given an algebraic variety over a number field, natural questions are: Is there is a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin’s conjecture). A popular current research theme is to consider these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve these problems one usually uses a combination of techniques from algebraic geometry and analytic number theory, but the project could be tailored towards the preferences of the student (e.g. for a student without much knowledge of algebraic geometry). 
Title 
Rational points on algebraic varieties 

Group  Algebra 
Supervisor  
Description 
Diophantine equations are a classical object of study in number theory. During the course of the 20th century, it was realised that one obtains a more powerful conceptual framework by considering them through a more geometric lens, namely viewing a solution to a Diophantine equation as a rational point on the associated algebraic variety. Given an algebraic variety over a number field, natural questions are: Is there is a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin’s conjecture). A popular current research theme is to consider these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve these problems one usually uses a combination of techniques from algebraic geometry and analytic number theory, but the project could be tailored towards the preferences of the student (e.g. for a student without much knowledge of algebraic geometry). 
Title 
Turbulent particleladen jets 

Group  Continuum Mechanics 
Supervisors  
Description 
Turbulent particleladen jets are relevant to many geophysical and industrial applications: from volcanic eruptions, to sediment resuspension, fluidisation processes and chemical reactors. Much work has been done on the dilute regime of these twophase flows, where the particles have a small impact on the fluid and can often be considered as passive tracers. In this experimental project, we focus on the poorly understood dense regime, where the coupling between the solid particles and the fluid is more complex. Many fundamental questions, of high relevance to the applications mentioned above, are still unresolved. This project will explore the impact of the particle density on turbulent entrainment processes. Entrainment processes during an explosive volcanic eruption have a considerable impact on the extent of the damages. They determine whether the eruption will collapse and form a pyroclastic flow, with local implications, or whether the eruption column will rise and form an ash cloud spreading over extended regions, such as in the case of the 2010 eruption of the Icelandic volcano Eyjafjallajökul. This project will also explore the effect on mixing processes, which are very important for instance in chemical reactors where the efficiency of the reaction depends strongly on the efficiency of the mixing. These dense particleladen jets are still poorly understood due to the considerable challenges faced analytically and numerically. Technical difficulties have also prevented progress on the experimental side for a long time. New experimental techniques, based on novel experimental design and imaging techniques, recently developed in the laboratory have allowed to probe much further into the complex dynamics of these dense particle laden jet. The main goal of this project is to pursue the development of these techniques in order to address the questions on entrainment and mixing described above. The project is suitable for an enthusiastic and creative candidate who has some experience in experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary. The motivation and readiness of the candidate to learn new techniques and develop them to explore fundamental scientific questions will be key to the success of this project. 
Title 
Turbulent particleladen jets 

Group  Industrial and Applied Mathematics 
Supervisors  
Description 
Turbulent particleladen jets are relevant to many geophysical and industrial applications: from volcanic eruptions, to sediment resuspension, fluidisation processes and chemical reactors. Much work has been done on the dilute regime of these twophase flows, where the particles have a small impact on the fluid and can often be considered as passive tracers. In this experimental project, we focus on the poorly understood dense regime, where the coupling between the solid particles and the fluid is more complex. Many fundamental questions, of high relevance to the applications mentioned above, are still unresolved. This project will explore the impact of the particle density on turbulent entrainment processes. Entrainment processes during an explosive volcanic eruption have a considerable impact on the extent of the damages. They determine whether the eruption will collapse and form a pyroclastic flow, with local implications, or whether the eruption column will rise and form an ash cloud spreading over extended regions, such as in the case of the 2010 eruption of the Icelandic volcano Eyjafjallajökul. This project will also explore the effect on mixing processes, which are very important for instance in chemical reactors where the efficiency of the reaction depends strongly on the efficiency of the mixing. These dense particleladen jets are still poorly understood due to the considerable challenges faced analytically and numerically. Technical difficulties have also prevented progress on the experimental side for a long time. New experimental techniques, based on novel experimental design and imaging techniques, recently developed in the laboratory have allowed to probe much further into the complex dynamics of these dense particle laden jet. The main goal of this project is to pursue the development of these techniques in order to address the questions on entrainment and mixing described above. The project is suitable for an enthusiastic and creative candidate who has some experience in experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary. The motivation and readiness of the candidate to learn new techniques and develop them to explore fundamental scientific questions will be key to the success of this project. 
Title 
Scheduling and Parallel Computing 

Group  Probability and Stochastic Analysis 
Supervisor  
Description 
Consider a large computational task, for instance, solving a large system of linear equations. This task can be split into many smaller jobs which are then scheduled and queued at a large number of different heterogeneous computing resources and are executed in parallel. The processing requirements of the different resources (for instance CPUs and GPUs) are different, and they may well have different communication costs. In this project we will be interested in understanding the stochastic effects before designing (and implementing) novel distributed scheduling algorithms. 
Title 
Scheduling and Parallel Computing 

Group  Numerical Analysis and Scientific Computing 
Supervisor  
Description 
Consider a large computational task, for instance, solving a large system of linear equations. This task can be split into many smaller jobs which are then scheduled and queued at a large number of different heterogeneous computing resources and are executed in parallel. The processing requirements of the different resources (for instance CPUs and GPUs) are different, and they may well have different communication costs. In this project we will be interested in understanding the stochastic effects before designing (and implementing) novel distributed scheduling algorithms. 
Title 
Representations of algebras and interpretations 

Group  Mathematical Logic 
Supervisor  
Description 
This is an indication of the area in which my current work is focussed, hence the area in which I would expect to supervise a student. 
Title 
Representations of algebras and interpretations 

Group  Algebra 
Supervisor  
Description 
This is an indication of the area in which my current work is focussed, hence the area in which I would expect to supervise a student. 
Title 
Optimal Experimental Designs 

Group  Statistics and its Applications 
Supervisor  
Description 
The success of many experimental studies in Biology, Chemistry, Engineering, Experimental Physics, Material Science, Medicine, etc., depends on the experimental designs that are used to collect the data. The aim of this project is to develop novel statistical methods for constructing designs that have desirable statistical properties. 
Title 
Development of group theory in the language of internal set theory 

Group  Mathematical Logic 
Supervisor  
Description 
The internal set theory, as proposed by Edward Nelson in 1977, blurs the line between finite and infinite sets in a very simple, effective and controlled way. This PhD project is aimed at a systematic development of the theory of finite and pseudofinite groups in the language of the internal set theory. This is motivated by problems in a branch of computational group theory, the socalled black box recognition of finite groups. Its typical object is a group generated by several matrices of large size, say, 100 by 100, over a finite field. Individual elements of such a group can be easily manipulated by a computer; however, the size of the whole group is astronomical, and arguments leading to identification of the structure of the group are being de facto carried out in an infinite object. The internal set theory provides tools that allow us to deal with finite objects and numbers that are, in effect, infinite. This is an exciting, unusual, but accessible topic for study. Read more at http://www.maths.manchester.ac.uk/~avb/pdf/PhD_Topic_Internal_Set_Theory.pdf. Prerequisites for the project: university level courses in algebra. Some knowledge of mathematical logic is desirable. 
Title 
Axiomatic approaches to the Hrushovski Programme 

Group  Mathematical Logic 
Supervisor  
Description 
The celebrated Hrushovski Programme is aimed at proving that the group of fixed points of a generic automorphism of a simple group of finite Morley rank behaves as a pseudofinite group and, with some luck, is pseudofinite indeed. The aim of the project is to analyse a few configurations where the assumptions of the Hrushovski Conjecture are strengthened. For example, an interesting case is where the fixed points sets of the automorphism in question have "size" with values in a linearly ordered ring which behaves in a strict analogy with cardinality of finite sets; will in that case the group of fixed points be pseudofinite? This question may perhaps involve some nontrivial model theory of the ring of "sizes" and some abstract versions of the LangWeil inequality linking the Morley rank of an invariant definable set and the "size" of the set of its fixed points. 
Title 
Axiomatic approaches to the Hrushovski Programme 

Group  Algebra 
Supervisor  
Description 
The celebrated Hrushovski Programme is aimed at proving that the group of fixed points of a generic automorphism of a simple group of finite Morley rank behaves as a pseudofinite group and, with some luck, is pseudofinite indeed. The aim of the project is to analyse a few configurations where the assumptions of the Hrushovski Conjecture are strengthened. For example, an interesting case is where the fixed points sets of the automorphism in question have "size" with values in a linearly ordered ring which behaves in a strict analogy with cardinality of finite sets; will in that case the group of fixed points be pseudofinite? This question may perhaps involve some nontrivial model theory of the ring of "sizes" and some abstract versions of the LangWeil inequality linking the Morley rank of an invariant definable set and the "size" of the set of its fixed points. 
Title 
Interdefinability of abelian functions 

Group  Mathematical Logic 
Supervisor  
Description 
Recently there has been a great deal of interaction between model theorists and number theorists on topics around `unlikely intersections', see for example [3]. One outcome of this is that there are now various functional transcendence results known for certain covering maps. The original example of this is Ax's functional version [1] of Schanuel's conjecture. This result and its more recent descendants have been used to study interdefinability of Weierstrass elliptic functions [2] and the initial aim of this project is to extend this to abelian functions. This would involve a mixture of model theory, differential algebra and number theory, although these are not all required to get started. It should also lead naturally to further interesting questions in these areas. 
Title 
Lie algebra actions on noncommutative rings 

Group  Algebra 
Supervisor  
Description 
My interests in Lie theory are focused on Lie algebra actions on noncommutative rings. An example is the representation of a Lie group, and its Lie algebra, on exterior powers of a finitedimensional module, or on a Clifford algebra. Many methods of the ChevalleyKostant theory still apply, but often need to be combined with tools of noncommutative algebra. 
Title 
Reflection groups in noncommutative algebra 

Group  Algebra 
Supervisor  
Description 
Finite linear groups generated by reflections arise in many areas of algebra, Lie theory being a prominent example. In the work of Chevalley, Shephard, Todd and Serre, reflection groups are seen to be the groups which have ""good"" rings of invariants when acting on a ring of polynomials. 
Title 
Morita equivalences of finite groups 

Group  Algebra 
Supervisor  
Description 
Most of my current research is focused on the problem of identifying Morita equivalence classes of blocks of finite groups. This is part of the study of the representation theory of finite groups with respect to fields of prime characteristic. Briefly, Morita equivalence is an equivalence of module categories, preserving the structure of modules for an algebra. 
Title 
Complex deformations of biological soft tissues 

Group  Continuum Mechanics 
Supervisors  
Description 
The answers to many open questions in medicine depend on understanding the mechanical behaviour of biological soft tissues. For example, which tendon is most appropriate to replace the anterior cruciate ligament in reconstruction surgery? what causes the onset of aneurysms in the aorta? and how does the mechanics of the bladder wall affect afferent nerve firing? Current work at The University of Manchester seeks to understand how the microstructure of a biological soft tissue affects its macroscale mechanical properties. Most of the work to date has focused on simple deformations (e.g. longitudinal extension under tension) for which analytical solutions can be found. However, the geometry and deformation of many soft tissues in vivo is sufficiently complex to prohibit analytical solutions. 
Title 
Efficient Uncertainty Quantification for PDEs with Random Data 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
Uncertainty Quantification (UQ) is the science of accounting for uncertainty in mathematical models. Research in this area has undergone rapid growth in the last few years and is currently considered a 'hot topic'. This growth has been driven by the need for scientists in today's world to provide decision makers with ever more accurate and reliable predictions that are based on results obtained from mathematical models. Projects on this topic would suit students who have taken undergraduate courses in numerical analysis and applied mathematics who have a keen interest in computational mathematics and developing practical algorithms. Some prior programming experience is essential. 
Title 
Mathematical theory of diffraction 

Group  Continuum Mechanics 
Supervisor  
Description 
There is a long history of mathematicians working on canonical diffraction (or scattering) problems. The mathematical theory of diffraction probably started with the work of Sommerfeld at the end of the 19th century and his famous solution to the diffraction of acoustic waves by a solid halfplane. Since, some very ingenious mathematical methods have been developed to tackle such problems. One of the most famous being the WienerHopf technique. 
Title 
Combustion instabilities 

Group  Continuum Mechanics 
Supervisor  
Description 
Combustion is essential to energy generation and transport needs, and will remain so for the foreseeable future. Mitigating its impact on the climate and human health, by reducing its associated emissions, is thus a priority. One suggested strategy to reduce NOx is to operate combustors at lean conditions. Unfortunately, combustion instability is more likely to occur in the lean regime, and may have catastrophic consequences on the components of combustion chambers, such as vibrations and structural fatigue. Ramjet engines, rocket engines and in general any type of gas turbine engines may be subject to this detrimental instability. The ability to predict and control the instability is crucial for implementing the lean burn strategy. Combustion instability involves an intricate interplay of several key physical processes, which take place in regions of different length scales. Due to this multiscale, multiphysics nature of the problem, direct numerical simulations of realistic combustors are extremely challenging. For this reason, simplified mathematical models capturing qualitatively and quantitatively the main characteristics of combustion instability are essential. In particular, by exploiting the scale disparity, systematic asymptotic analyses may be carried out to derive relevant models on first principles, and to provide guidance for developing reliable and efficient numerical algorithms. 
Title 
Microstructural models of the constitutive behaviour of soft tissue 

Group  Continuum Mechanics 
Supervisors  
Description 
Soft tissue such as tendon, ligament, skin, and the brain possess complex nonlinear viscoelastic constitutive behaviour which arises due to the intricate microstructures inherent in such materials. The majority of existing models for the constitutive behaviour of soft tissue are phenomenological so that the parameters involved in the model are not derivable from experiments. In this project the objective is to build models that are based on the microstructure and we will liaise with experimentalists, particularly those in imaging science, in order to ensure that the parameters involved can be directly measured. This project would suit those with a strong background in continuum mechanics and modelling and although not essential some background knowledge in nonlinear elasticity would be useful. 
Title 
Fractional differential equations and anomalous transport 

Group  Continuum Mechanics 
Supervisor  
Description 
This project is concerned with anomalous transport, which cannot be described by standard calculus. Instead it requires the use of fractional differential equations involving fractional derivatives of non integer order. This is a new, exciting area of research because anomalous transport is a widespread natural phenomenon. Examples include flight of albatross, stock prices, human migration, social networks, transport on fractal geometries, proteins on cell membranes, bacterial motion, and signalling molecules in the brain. 
Title 
Selfaffine sets: geometry, topology and arithmetic 

Group  Analysis and Dynamical Systems 
Supervisor  
Description 
Iterated function systems (IFS) are commonly used to produce fractals. While selfsimilar IFS are well studied, selfaffine IFS are still relatively new. 
Title 
Integral Geometry of Cones 

Group  Numerical Analysis and Scientific Computing 
Supervisor  
Description 
Science and technology depends increasingly on the efficient acquisition, storage and processing of vast amounts of data. 
Title 
Stability and separation in R>>1 flows 

Group  Continuum Mechanics 
Supervisor  
Description 
I have several projects available in the area of high Reynolds number flows, including the study of laminar separation and stability of thin films, cavity flows, breakup of separation bubbles, crossflow instability. The work can be theoretical, numerical or a mixture of both. 
Title 
Environmental fluid mechanics 

Group  Continuum Mechanics 
Supervisor  
Description 
Many problems of environmental significance require the effective prediction of particulate (contaminant) transport in a fluid system (which constitutes a `twophase' fluid/particle problem). The primary focus of this project is a suspension of solid particles (dust/ash) in a viscous incompressible fluid. Most practical cases of interest have particles that are typically fractions of a millimetre in size, but still occupy a nonsmall fraction of the total mixture mass and exist in large numbers. The simultaneous treatment of all individual particles (and the correspondingly complicated fluid domain) is computationally impractical, a state of affairs that will remain for the foreseeable future. Furthermore, the behaviour of a single particle cannot be solved in isolation of the other particles, owing to particleparticle interactions through the motion of the interstitial fluid, or by direct particle collisions at high concentration levels. In such cases, both phases of the mixture exchange momentum with the other, so that the fluid motion and the particle motion remain coupled together. Furthermore, the presence of bounding surfaces for the fluid mixture can have crucial consequences for the structural and temporal development of the flow and the distribution of suspended material. This project aims to continue the development of existing macro scale models, in which both phases are treated as coexisting (coupled) continua, through a combination of analytical and computational methods. 
Title 
Flow and transport in the placenta 

Group  Continuum Mechanics 
Supervisor  
Description 
The placenta provides an interface beween fetal and maternal blood, supplying essential nutrients to the growing fetus. Within the placenta, fetal blood is confined to a treelike network of blood vessels that are bathed in a pool of maternal blood. The placenta's effectiveness as a transporter of oxygen, glucose, and other molecules is critically determined by its complex geometric structure; this may be compromised in disease, with adverse consequences for fetal growth and development. This project will build on recent studies of the maternal circulation [13], developing analogies with models flow through porous media and exploring new multiscale approximation techniques. The project offers opportunities for analysis, computation and interaction with experimentalists. References:

Title 
Plant tissue mechanics 

Group  Continuum Mechanics 
Supervisor  
Description 
Plant growth arises through the coordinated expansion of individual cells, allowing a plant to adapt to its environment to harness light, water and essential nutrients. Growth is driven by the high internal turgor pressure of cells and is regulated by physical and biochemical modifications of plant cell walls. Many features of this immensely complex process remain poorly understood, despite its profound societal and environmental importance. Mathematical models describing the mechanical properties of a growing plant tissue integrate features ranging from molecular interactions within an individual cell wall to the expansion, bending or twisting of a multicellular root or stem. Building on current biological understanding, this project will address the development and analysis of new multiscale models for plant tissues, exploiting a variety of computational and asymptotic techniques. Background references:

Title 
Optimal prediction problems driven by Lévy processes 

Group  Probability and Stochastic Analysis 
Supervisor  
Description 
An optimal prediction problem is a type of optimal control problem very similar to classic optimal stopping problems, however with the crucial difference that the gains process is not adapted to the filtration generated by the driving process. Consider for instance the problem of stopping a Brownian motion at a time closest (in some suitable metric) to the time the Brownian motion attains its ultimate supremum. This is an optimal prediction problem: the gains are determined by the difference between the chosen stopping time and the time at which the Brownian motion actually does attain its ultimate supremum, and as the latter quantity is not known until the whole path of the process is revealed the gains is indeed not adapted. 
Title 
Revenue Management 

Group  Mathematical Finance and Actuarial Science 
Supervisor  
Description 
By Revenue Management (RM), we mean the process of understanding, forecasting and influencing consumer behaviour in order to maximise a firm's revenues. Put simply, RM is all about selling the right product to the right customer at the right time for the right price. RM originated in the airline industry under the term Yield Management. Today, RM is widely employed in many other major industries, such as hotels, restaurants, car rentals and carparks. A prospective research student in this area should expect to gain knowledge on Optimisation, Markov Processes, Dynamic programming and HJB equations. 
Title 
Numerical Analysis and Computational Methods for Solving PDEs with Uncertainty 

Group  Numerical Analysis and Scientific Computing 
Supervisor  
Description 
Uncertainty Quantification (UQ) is the science of accounting for uncertainty in mathematical models. Research in this area has undergone rapid growth in the last few years and is currently considered a 'hot topic'. This growth has been driven by the need for scientists in today's world to provide decision makers with ever more accurate and reliable predictions that are based on results obtained from mathematical models. Many physical processes such as fluid flows are governed by partial differential equations (PDEs). In practical applications in the real world, it is unlikely that all the inputs (boundary conditions, geometry, coefficients) for the chosen PDE model will be known. One possibility is to model the quantities that we don't know as random variables. Solving these problems is not always hard in theory but solving them efficiently in practice is a massive challenge. I am interested in working with students who want to develop numerical analysis and numerical methods for solving partial differential equations with uncertain inputs. I welcome any enquiries to work in this area. Specific projects could be theoretical or computational, according to the strengths of the student. Projects on this topic would suit students who have taken undergraduate courses in numerical analysis and applied mathematics who have a keen interest in computational mathematics and developing practical algorithms. Some prior programming experience is essential. Background reference: An Introduction to Computational Stochastic PDEs (Cambridge Texts in Applied Mathematics), G. J. Lord, C.E. Powell and T. Shardlow, 2014. 
Title 
Interactions between rocks and ice 

Group  Continuum Mechanics 
Supervisor  
Description 
Many glaciers are covered by a debris layer whose presence has multiple, competing effects on the glacier's melt rate. The debris layer shields the ice from incoming solar radiation and thus reduces its melt rate. However, since the albedo of the debris layer is much smaller than that of the ice, the debris layer is heated up very rapidly by the solar radiation, an effect that is likely to increase the melt rate. The project aims to develop theoretical/computational models to study how solid objects (rocks) which are placed on (or embedded in) an ice layer affect the ice's melt rate. The work will employ (and contribute to) the objectoriented multiphysics finiteelement library oomphlib, developed by M. Heil and A.L. Hazel and their collaborators, and available as open source software at http://www.oomphlib.org. The project would suit students with an interest in mathematical modelling, continuum mechanics and scientific computing and will be performed in close collaborations with Glaciologists at the University of Sheffield and the Bavarian Academy of Science. 