# Funded projects

All projects listed on the PhD projects page 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 below. Please contact the supervisor/grant holder for further information.

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 non-linear 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 non-linear 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 Ramsey-theoretic criterion of Rado to systems of degree greater than one?

Higher order Fourier analysis of non-linear 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 non-linear 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 non-linear 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 Ramsey-theoretic criterion of Rado to systems of degree greater than one?

Higher order Fourier analysis of non-linear 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 differential-algebraic 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 differential-algebraic 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

#### Thermo-visco-acoustic 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 “in-air” 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 semi-infinite 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

#### Thermo-visco-acoustic 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 “in-air” 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 semi-infinite 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

Group Continuum Mechanics
Supervisors
Description

Turbulent particle-laden 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 two-phase 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 particle-laden 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

#### Convective mass transfer for cleaning and decontamination

Group Continuum Mechanics
Supervisors
Description

Cleaning and decontamination processes can rely on different mechanisms to remove a patch of alien substance attached to a substrate. A shear flow covering the substrate can remove the substance through mechanical forces, potentially combined with chemical surfactant agent decreasing the adhesion of the substance onto the surface. However, this project is concerned with a second type of mechanism which is based on the dissolution of the substance into the cleaning fluid flow covering the substance.

This second type of cleaning process establishes a convective mass transfer between the alien phase and the cleaning phase. Several applications rely on this process, particularly when the dispersion of the substance is unwanted, such as in the decontamination process of toxic chemical spills. In our daily life, the cleaning mechanism more and more favoured in dishwashers relies also on a convective mass transfer as it has been shown empirically to reduce energy and water consumption.

This project will focus on the case of a film flow covering a single droplet containing several substances. Many fundamental questions are still unresolved in this multiphase convective mass transfer problem. In particular, we will study how advection processes inside the drop can influence the convective mass transfer. Effect of solubility and surface tension on the overall mass transfer can also be analysed. The project will explore these questions using a combination of experimentation, numerical simulations and theoretical analysis.

The project is suitable for an enthusiastic and creative candidate who has good knowledge in fluid mechanics and some experience in experimentation and numerical simulations.

Reference: J. R. Landel, A. L. Thomas, H. McEvoy and S. B. Dalziel. Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 2016.

Title

Group Industrial and Applied Mathematics
Supervisors
Description

Turbulent particle-laden 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 two-phase 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 particle-laden 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

#### Convective mass transfer for cleaning and decontamination

Group Industrial and Applied Mathematics
Supervisors
Description

Cleaning and decontamination processes can rely on different mechanisms to remove a patch of alien substance attached to a substrate. A shear flow covering the substrate can remove the substance through mechanical forces, potentially combined with chemical surfactant agent decreasing the adhesion of the substance onto the surface. However, this project is concerned with a second type of mechanism which is based on the dissolution of the substance into the cleaning fluid flow covering the substance.

This second type of cleaning process establishes a convective mass transfer between the alien phase and the cleaning phase. Several applications rely on this process, particularly when the dispersion of the substance is unwanted, such as in the decontamination process of toxic chemical spills. In our daily life, the cleaning mechanism more and more favoured in dishwashers relies also on a convective mass transfer as it has been shown empirically to reduce energy and water consumption.

This project will focus on the case of a film flow covering a single droplet containing several substances. Many fundamental questions are still unresolved in this multiphase convective mass transfer problem. In particular, we will study how advection processes inside the drop can influence the convective mass transfer. Effect of solubility and surface tension on the overall mass transfer can also be analysed. The project will explore these questions using a combination of experimentation, numerical simulations and theoretical analysis.

The project is suitable for an enthusiastic and creative candidate who has good knowledge in fluid mechanics and some experience in experimentation and numerical simulations.

Reference: J. R. Landel, A. L. Thomas, H. McEvoy and S. B. Dalziel. Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 2016.

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.

We seek a student with skills in probability, optimization and mathematical modeling. A student with good programming skills (particularly in C/C++) would be preferred.

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.

We seek a student with skills in probability, optimization and mathematical modeling. A student with good programming skills (particularly in C/C++) would be preferred.

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.

First, it's algebra with input from model theory and category theory.
The area of algebra is module (= representation) theory, especially representations of algebras.

In extremely general terms, the aim is to understand the structure of the category of modules.  This might mean getting a description of some of the most interesting modules and the maps between them or it might mean finding some structure (topological, geometric, algebraic, ...) on a set of these, and investigating that 'larger-scale' structure on (part of) the category of modules.

The input of model theory (part of mathematical logic) in the specific context of the representation theory of finite-dimensional algebras, where interest is typically focussed on finite-dimensional representations, leads us to extend our interest to at least some of the infinite-dimensional representations, even if our eventual applications are back in the context of the finite-dimensional ones.  The same general pattern, of looking at (somewhat) 'large' representations, can be seen over algebras which are not finite-dimensional.

Another input of model theory is the concept of interpretation which, in this context, can be seen as a certain kind of functor between categories of modules.  Understanding how these link categories of modules is another rather general aim.

My website  (www.maths.manchester.ac.uk/~mprest/publications.html)  gives more (too much) information but some flavour of the area can be got by browsing around there.

Any offer of a place will include a description of a broad research problem but a specific project will be determined taking account of a variety of factors, in particular, the current state of knowledge and activity in the area and the interests and development of the student.  It can also be that the direction of the project changes as it develops, in the light of what is discovered.

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.

First, it's algebra with input from model theory and category theory.
The area of algebra is module (= representation) theory, especially representations of algebras.

In extremely general terms, the aim is to understand the structure of the category of modules.  This might mean getting a description of some of the most interesting modules and the maps between them or it might mean finding some structure (topological, geometric, algebraic, ...) on a set of these, and investigating that 'larger-scale' structure on (part of) the category of modules.

The input of model theory (part of mathematical logic) in the specific context of the representation theory of finite-dimensional algebras, where interest is typically focussed on finite-dimensional representations, leads us to extend our interest to at least some of the infinite-dimensional representations, even if our eventual applications are back in the context of the finite-dimensional ones.  The same general pattern, of looking at (somewhat) 'large' representations, can be seen over algebras which are not finite-dimensional.

Another input of model theory is the concept of interpretation which, in this context, can be seen as a certain kind of functor between categories of modules.  Understanding how these link categories of modules is another rather general aim.

My website  (www.maths.manchester.ac.uk/~mprest/publications.html)  gives more (too much) information but some flavour of the area can be got by browsing around there.

Any offer of a place will include a description of a broad research problem but a specific project will be determined taking account of a variety of factors, in particular, the current state of knowledge and activity in the area and the interests and development of the student.  It can also be that the direction of the project changes as it develops, in the light of what is discovered.

Title

#### Segmentation and mathematical modelling of cerebrospinal fluid in the vertebral column

Group Industrial and Applied Mathematics
Supervisor
Description

In the human body, the spinal cord, transmitting bi-directional information between brain and body, is found within the bony vertebral column. It is contained inside a membranous sac, the dura, and bathed in cerebrospinal fluid (CSF), which circulates throughout the central nervous system in the subarachnoid space between dura and cord. Blood vessels and nerves enter and exit through the walls of the dural sac along its length. During each cardiac cycle a cyclic flow and pressure change has been observed within the CSF having the same period as the cardiac cycle. However, little is known about the precise nature or mechanism of this “CSF circulation”, including how the geometry of the subarachnoid space may affect the pressure of the CSF and its flow pattern. This knowledge would further our understanding of several pathological conditions of the central nervous system.

With this in mind, the objective of this project is to understand the dynamics of the CSF, contained within the subarachnoid space of the spinal column, throughout a cardiac cycle. To achieve this goal the student will use 4D phase-contrast MR images, which are capable of detecting the velocity field of the CSF. In the beginning stages the focus will be on developing segmentation methods capable of creating a three-dimensional model of the subarachnoid space allowing visualization of the changes throughout a cycle. This segmentation is difficult due to the complicated geometry of the subarachnoid space including vasculature and exiting nerves. Once this first stage is complete the student will use the resulting segmentation to produce a computational model for the flow of CSF throughout the cardiac cycle. The goal of this modeling will be to measure local relative pressure changes of the CSF, as well as to understand the interaction between the CSF pressure and the geometry of the subarachnoid space.

Though based in the school of mathematics, this project will also involve close collaboration with researchers at the Royal Preston Hospital’s Neurosurgery and Neuroradiology departments who will be providing the MR images, produced as part of an ongoing study into CSF circulation in health and disease.

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.

The applicants are expected to have deep knowledge in Statistics and Mathematics, as well as good computational skills. The focus of the project will be decided to suit the background and the strengths of the student.

Reference:
Atkinson, A.C., Donev, A.N. and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press.

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 so-called 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.

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 non-trivial model theory of the ring of "sizes" and some abstract versions of the Lang-Weil 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 non-trivial model theory of the ring of "sizes" and some abstract versions of the Lang-Weil 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.

[1] Ax, James On Schanuel's conjectures. Ann. of Math. (2) 93 1971 252–268. See  http://www.jstor.org/stable/1970774

[2] Jones, G., Kirby, J. and Servi, T., Local interdefinability of Weierstrass elliptic functions. Journal of the Institute of Mathematics of Jussieu, To appear. See http://dx.doi.org/10.1017/S1474748014000425

[3] Zannier, Umberto, Some problems of unlikely intersections in arithmetic and geometry. Annals of Mathematics Studies, 181. Princeton University Press, Princeton, NJ, 2012

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 finite-dimensional module, or on a Clifford algebra. Many methods of the Chevalley-Kostant theory still apply, but often need to be combined with tools of noncommutative algebra.

A project in this area may suit a student with background in Lie algebras and/or representation theory.

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.

More recently, reflection groups have been studied in connection with noncommutative rings that are obtained from commutative rings via deformation (or quantisation) construction inspired by quantum mechanics. In particular, this has led to the rich theory of Cherednik algebras.

I am interested in reflection groups, and their quantum analogues, acting on noncommutative rings arising from quantum algebra. Projects might focus on open conjectures in this area, and should be suitable for students with background, and interest, in representation theory and/or quantum groups.

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.

This problem is fundamental to the area, and ties in with another of my areas of interest, global-local relationships in finite groups.

Problems range from Donovan's conjecture, which is a finiteness conjecture concerning the number of Morita equivalence classes, to classification of Morita equivalence classes in specific cases.

A tool I use frequently is the classification of finite simple groups, but there is scope for a variety of projects suited to different interests. The precise nature of the project would be open for discussion with the prospective student.

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.

In this project, we will use our “in house” finite element software oomph-lib  to investigate complex deformations of biological soft tissues. The work will require development and implementation of novel strain energy functions as well as formulation of non-standard problems in solid mechanics. The project is likely to appeal to students with an interest in continuum mechanics, computational mathematics and interdisciplinary science. The work will be carried out in close collaboration with the School of Materials Science, and the student will have the opportunity to perform experiments at the state-of-the-art Henry Moseley X-ray Imaging Facilities.

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.

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 (e.g. solvers, error estimators) for solving partial differential equations with uncertain inputs (stochastic PDEs). 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

#### 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 half-plane. Since, some very ingenious mathematical methods have been developed to tackle such problems. One of the most famous being the Wiener-Hopf technique.

However, despite tremendous efforts in this field, some canonical problems remain open, in the sense that no clear analytical solution is available for them.

One of this problem is the quarter-plane problem, the problem of diffraction of acoustic waves by a solid quarter-plane. Thus far, it has not been possible to apply classical methods such as the Wiener-Hopf method successfully in that case, and hence some new mathematical methods need to be developed in order to tackle this problem. This makes it very interesting as it implies that many different types of mathematics can be used and it makes the subject intrinsically multidisciplinary.

Many industrial problems can be linked to the theory of diffraction, for example the noise generated by a jet engine (acoustic waves) or radar detection (electro-magnetic waves) and defect detection in materials (elastic waves).

PhD projects are available in this field.

References:
-- R. C. Assier and N. Peake. On the diffraction of acoustic waves by a quarter-plane. Wave Motion, 49(1):64-82, 2012
-- R. C. Assier and N. Peake. Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane. IMA J. Appl. Math., 77(5):605-625, 2012.

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 multi-scale, multi-physics 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.

Recent progress have been made in the mathematical modelling of such instabilities, using refining and implementing this model would make a good PhD project.

Reference:
R. C. Assier and X. Wu. Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field. J. Fluid Mech. (2014), vol. 758, pp. 180-220

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

#### Self-affine sets: geometry, topology and arithmetic

Group Analysis and Dynamical Systems
Supervisor
Description

Iterated function systems (IFS) are commonly used to produce fractals. While self-similar IFS are well studied, self-affine IFS are still relatively new.

In a recent paper Kevin Hare and I considered a simple family of two-dimensional self-affine sets ($=$ attractors of self-affine IFS) and proved several results on their connectedness, interior points, convex hull and corresponding simultaneous expansions. A great deal of natural questions (simple connectedness, set of uniqueness, dimensions, etc.) remain open - even for this most natural family.

The project is aimed at closing these gaps as well as generalising our results to other 2D families (which are completely classified) as well as higher dimensions.

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.
While the seemingly endless availability of data is a blessing from a statistical point of view, it poses enormous challenges from a computational perspective.

An exciting recent development has been the emergence of ideas for efficient information acquisition and identification that take advantage of simple underlying structure of the problems considered . This new genre of ideas, encompassing the burgeoning field of compressed sensing, has experienced tremendous growth in recent years.

Numerical optimization plays an important role in these developments, and its effectiveness depends crucially on deep geometric properties of the underlying problems. I am interested in studying such problems in convex geometry and geometric probability that help explain the reach and limitations of convex optimization. Besides background in numerical analysis, this project would also benefit from knowledge in computational complexity, differential geometry, and probability.

Literature:
{\bf Schneider, Weil}, Stochastic and Integral Geometry.
{\bf Foucart, Rauhut}, A Mathematical Introduction to Compressive Sensing.

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, break-up of separation bubbles, cross-flow 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 two-phase' 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 non-small 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 particle-particle 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 co-existing (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 tree-like 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 [1-3], 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:

1. Chernyavsky, IL, Jensen, OE et al. (2010) Placenta 31, 44
2. Chernyavsky, IL, Leach, L et al.  (2011) Phil Trans Roy Soc A 369, 4162
3. Chernyavsky, IL, Dryden, IL et al. (2012) IMA J Appl Math 77, 697
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:

1. Dyson, RJ & Jensen, OE (2010) J Fluid Mech 655, 472
2. Dyson, RJ, Band, L & Jensen, OE (2012) J Theor Biol 307, 125
3. Baskin, TI & Jensen, OE (2013) J Exp Bot 64, 4697
4. Dyson, RJ et al. (2014) New Phytologist 202, 1212
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.

Besides being mathematically very appealing, optimal prediction problems have many applications, for instance in mathematical finance, insurance, medicine and engineering, to name a few. They were relatively recently introduced, and the bulk of the work done so far is for diffusions only. In this project, several types of optimal prediction problems driven by Lévy processes rather than diffusions will be studied. Lévy processes play a role in mathematical finance as an extension of the classic Black & Scholes model and in insurance as an extension of the classic Cramer-Lundberg model for instance.

Note that this project requires a very strong background in probability theory, in particular stochastic processes, and mathematical analysis. Preferably the student should already be familiar with Lévy processes. Experience with computer programming and implementing numerical schemes is very helpful.

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 object-oriented multi-physics finite-element library oomph-lib, developed by M. Heil and A.L. Hazel and their collaborators, and available as open source software at http://www.oomph-lib.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.