Funded projects

Additive combinatorics and Diophantine problems

• Dr Sean Prendiville (sean.prendiville@manchester.ac.uk)

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?

Additive combinatorics and Diophantine problems

• Dr Sean Prendiville (sean.prendiville@manchester.ac.uk)

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?

Algebraic differential equations and model theory

• Dr Omar Leon Sanchez (omar.sanchez@manchester.ac.uk)

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.

Algebraic differential equations and model theory

• Dr Omar Leon Sanchez (omar.sanchez@manchester.ac.uk)

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.

Rational points on algebraic varieties

• Dr Daniel Loughran (Daniel.loughran@manchester.ac.uk)

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).

Rational points on algebraic varieties

• Dr Daniel Loughran (Turing Fellow) (daniel.loughran@manchester.ac.uk)

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).

Rational points on algebraic varieties

• Dr Daniel Loughran (Turing Fellow) (daniel.loughran@manchester.ac.uk)

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).

Turbulent particle-laden jets

• Dr Julien Landel (julien.landel@manchester.ac.uk )
• Dr Rich Hewitt (richard.hewitt@manchester.ac.uk)

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.

Convective mass transfer for cleaning and decontamination

• Dr Julien Landel (julien.landel@manchester.ac.uk)
• Prof Matthias Heil (matthias.heil@manchester.ac.uk)

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.

Turbulent particle-laden jets

• Julien Landel (julien.landel@manchester.ac.uk)
• Dr Rich Hewitt (richard.hewitt@manchester.ac.uk)

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.

Convective mass transfer for cleaning and decontamination

• Dr Julien Landel (julien.landel@manchester.ac.uk)
• Prof Matthias Heil (matthias.heil@manchester.ac.uk)

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.

Scheduling and Parallel Computing

• Dr Neil Walton (Neil.Walton@manchester.ac.uk)

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.

Scheduling and Parallel Computing

• Dr Neil Walton (Neil.Walton@manchester.ac.uk)

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.

Representations of algebras and interpretations

• Mike Prest (mprest@manchester.ac.uk)

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.

Representations of algebras and interpretations

• Mike Prest (mprest@manchester.ac.uk)

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.

Segmentation and mathematical modelling of cerebrospinal fluid in the vertebral column

• Dr Sean Holman (sean.holman@manchester.ac.uk)

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. 

Optimal Experimental Designs

• Alexander Donev (a.n.donev@manchester.ac.uk)

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.

Development of group theory in the language of internal set theory

• Alexandre Borovik (borovik@manchester.ac.uk)

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.

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.

Axiomatic approaches to the Hrushovski Programme

• Alexandre Borovik (borovik@manchester.ac.uk)

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.

Axiomatic approaches to the Hrushovski Programme

• Alexandre Borovik (borovik@manchester.ac.uk)

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.

Interdefinability of abelian functions

• Gareth Jones (gareth.jones-3@manchester.ac.uk)

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

Lie algebra actions on noncommutative rings

• Yuri Bazlov (yuri.bazlov@manchester.ac.uk)

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.

Reflection groups in noncommutative algebra

• Yuri Bazlov (yuri.bazlov@manchester.ac.uk)

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.

Morita equivalences of finite groups

• Charles Eaton (charles.eaton@manchester.ac.uk)

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.

Complex deformations of biological soft tissues

• Andrew Hazel (Andrew.Hazel@manchester.ac.uk)
• Tom Shearer (Tom.Shearer@manchester.ac.uk)

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.

Efficient Uncertainty Quantification for PDEs with Random Data

• Dr. Catherine Powell (c.powell@manchester.ac.uk)

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.

Mathematical theory of diffraction

• Raphael Assier (raphael.assier@manchester.ac.uk)

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.

Combustion instabilities

• Raphael Assier (raphael.assier@manchester.ac.uk)

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