Funded projects

Self-similar Markov processes and applications

• Alex Watson (Alex.Watson@manchester.ac.uk)

Positive self-similar Markov processes (ssMps) have an important status as scaling limits of Markov chains, and since Lamperti's pioneering work have been strongly connected with Levy processes. They have enjoyed a resurgence in recent years on account of the newly discovered connection with Markov additive processes enjoyed by real and R^d-valued ssMps. There are a number of interesting open questions which are under investigation at the moment, in particular the study of conditionings of ssMps and the behaviour as the starting point approaches the origin. These form the focus of this PhD project, and they have further applications, for instance in the field of fragmentation.

References:

A. E. Kyprianou (2014) Fluctuations of Lévy processes with applications. Chapter 13. https://doi.org/10.1007/978-3-642-37632-0

Growth-fragmentation phenomena

• Alex Watson (alex.watson@manchester.ac.uk)

Growth-fragmentation processes describe the evolution of a collection of cells which grow gradually and split apart (fragment) suddenly. They are motivated by applications to division of biological cells and of polymerisation phenomena, but they form a fascinating class of stochastic models in their own right. Pure fragmentation processes have been studied since Kolmogorov, and a framework for their understanding was introduced by Bertoin (2006). A much more recent development is the study of stochastic models incorporating growth. These were introduced in quite general form, again by Bertoin (2017), and there are many open questions, particularly concerning their behaviour at large times. A fascinating application of these processes is to models of 'random planar maps' which are motivated by statistical physics.

References:

J. Bertoin (2006) Random fragmentation and coagulation processes. https://doi.org/10.1017/CBO9780511617768

J. Bertoin (2017)Markovian growth-fragmentation processes. Bernoulli 23, no. 2, 1082–1101. https://doi.org/10.3150/15-BEJ770

Fast and Reliable Algorithms for High Peformance Numerical Linear Algebra

• Nick Higham (Nick.Higham@manchester.ac.uk)

The project focuses on developing a new generation of numerical linear algebra algorithms that exploit current and future computers. The algorithms need to be fast and to be accompanied by rigorous error analysis to guarantee their reliability, even for the largest and most difficult problems. The target problems will be drawn from linear equations, linear least squares problems, eigenvalue problems, the singular value decomposition, and matrix function evaluation. These are the innermost kernels in many scientific and engineering applications---in particular, in data science and in machine learning---so it is essential that they are fast, accurate, and reliable.

A key aspect of this work is the exploitation of variable precision arithmetic. Low precision arithmetic is now available in hardware and is increasingly being used in machine learning and scientific computing more generally because of its speed, but its limited precision and narrow range require careful treatment. High precision (quadruple precision and above) is available in software and may be used in small amounts to speed up or stabilize an algorithm.

A strong background in numerical linear algebra and programming skills in MATLAB or a high level language are essential.

This project provides the opportunity to be part of a large and vibrant numerical linear algebra group that has several strongly committed industrial partners (see https://nla-group.org/).  

Funding Notes

Open to all. Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a fees-only bursary for overseas students.

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. 

Instabilities of buoyancy driven flows in a confined environment

• Dr Julien Landel (julien.landel@manchester.ac.uk)

Buoyancy driven flows are important in many geophysical and environmental applications: from natural ventilation in buildings to the evolution of cloud particles after a volcanic eruption or an explosion. Much work has been done on these flows in both turbulent and laminar flow regimes. However, the impact of lateral confinement on the flow has not received as much attention. One of the distinctive features is the development of an instability due to lateral shear. This effect can have important consequences for the large scale dynamics as well as the small scale mixing and dispersion properties of these flows when carrying particles or other tracers.

In this project, we will explore experimentally and theoretically the origin of the instability for various fundamental buoyancy-driven flows such as fountains and thermals. New experimental techniques, based on novel experimental design and imaging techniques, recently developed in our laboratory have allowed to probe further into the complex dynamics of these flows. The goal of this project is to study the phenomenology of the flow in order to determine the source of the instability. We will also analyse how the instability affects mixing and dispersion of active or passive tracers.

The project is suitable for an enthusiastic and creative candidate who has some experience in laboratory experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary.

Funding Notes

Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a fees-only bursary for overseas students. Competitive bursaries are also available for overseas students to fully cover both fees and maintenance at RCUK level.

Anticipated start date: September 2019. 

Marangoni driven film dynamics for diffusive mixtures

• Dr Julien Landel (julien.landel@manchester.ac.uk)

Marangoni-driven flows develop due to local surface tension gradients at the interface between two liquids. These flows have many implications for biochemical systems, microfluidic systems or chemical reactors. When a drop of a liquid substance such as ethanol is deposited at the surface of a liquid film layer, e.g. water, the surface tension gradient expands the drop rapidly to form a very thin film. Competition between the Marangoni-driven film flow and the diffusion of some or all of the film components into the other phase can lead to complex mixing dynamics as well as interfacial instabilities.

In this project, we will explore experimentally and theoretically the competition between convection and diffusion mechanisms and their impact on the dynamics and stability of these rapidly expanding thin films. Experimental techniques such as high-speed particle image velocimetry and dye attenuation will be used to understand the various flow phenomena and the mixing properties. Pure or mixed substances for the drop will be tested varying the miscibility and surface tension properties with the liquid film layer onto which it is deposited. We will also investigate the impact of the results of these studies for practical applications in biochemical systems and microfluidic systems.

The project is suitable for an enthusiastic and creative candidate who has some experience in laboratory experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary.

Funding Notes

Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a fees-only bursary for overseas students. Competitive bursaries are also available for overseas students to fully cover both fees and maintenance at RCUK level.

Anticipated start date: September 2019.

Instabilities of buoyancy driven flows in a confined environment

• Dr Julien Landel (julien.landel@manchester.ac.uk)

Buoyancy driven flows are important in many geophysical and environmental applications: from natural ventilation in buildings to the evolution of cloud particles after a volcanic eruption or an explosion. Much work has been done on these flows in both turbulent and laminar flow regimes. However, the impact of lateral confinement on the flow has not received as much attention. One of the distinctive features is the development of an instability due to lateral shear. This effect can have important consequences for the large scale dynamics as well as the small scale mixing and dispersion properties of these flows when carrying particles or other tracers.

In this project, we will explore experimentally and theoretically the origin of the instability for various fundamental buoyancy-driven flows such as fountains and thermals. New experimental techniques, based on novel experimental design and imaging techniques, recently developed in our laboratory have allowed to probe further into the complex dynamics of these flows. The goal of this project is to study the phenomenology of the flow in order to determine the source of the instability. We will also analyse how the instability affects mixing and dispersion of active or passive tracers.

The project is suitable for an enthusiastic and creative candidate who has some experience in laboratory experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary.

Funding Notes

Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a fees-only bursary for overseas students. Competitive bursaries are also available for overseas students to fully cover both fees and maintenance at RCUK level.

Anticipated start date: September 2019.

Marangoni driven film dynamics for diffusive mixtures

• Julien Landel (Julien.landel@manchester.ac.uk)

Marangoni-driven flows develop due to local surface tension gradients at the interface between two liquids. These flows have many implications for biochemical systems, microfluidic systems or chemical reactors. When a drop of a liquid substance such as ethanol is deposited at the surface of a liquid film layer, e.g. water, the surface tension gradient expands the drop rapidly to form a very thin film. Competition between the Marangoni-driven film flow and the diffusion of some or all of the film components into the other phase can lead to complex mixing dynamics as well as interfacial instabilities.

In this project, we will explore experimentally and theoretically the competition between convection and diffusion mechanisms and their impact on the dynamics and stability of these rapidly expanding thin films. Experimental techniques such as high-speed particle image velocimetry and dye attenuation will be used to understand the various flow phenomena and the mixing properties. Pure or mixed substances for the drop will be tested varying the miscibility and surface tension properties with the liquid film layer onto which it is deposited. We will also investigate the impact of the results of these studies for practical applications in biochemical systems and microfluidic systems.

The project is suitable for an enthusiastic and creative candidate who has some experience in laboratory experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary.

Funding Notes

Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a fees-only bursary for overseas students. Competitive bursaries are also available for overseas students to fully cover both fees and maintenance at RCUK level.

Anticipated start date: September 2019.

Thermo-visco-acoustic metamaterials for underwater applications

• Prof Will Parnell (William.J.Parnell@manchester.ac.uk)

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.

Thermo-visco-acoustic metamaterials for underwater applications

• Prof Will Parnell (william.j.parnell@manchester.ac.uk)

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.

Fluid-structure interaction effects in the sedimentation of thin elastic sheets

• Professor Matthias Heil (School of Mathematics) (Matthias.Heil@manchester.ac.uk)
• Professor Anne Juel (School of Physics and Astronomy) (Anne.Juel@manchester.ac.uk)
• Dr Draga Pihler-Puzovic (School of Physics and Astronomy) (draga.pihler-puzovic@manchester.ac.uk)

There is much current interest in so-called two-dimensional materials 
because of their unusual and attractive mechanical and electrical properties. 
Much of their processing is performed in a fluid environment, e.g., during
the size selection of dispersed graphene flakes by centrifugation, or
their deposition by ink-jet printing. The flakes' large aspect ratio 
implies that despite their impressive in-plane stiffness they have a
very small bending stiffness and are therefore easily deformed by the 
traction that the surrounding fluid exerts on them. The resulting
strong fluid-structure interaction affects not only the dynamics of 
individual flakes but also their collective behaviour.

The aim of this project is to perform a systematic study of 
the behaviour of thin elastic sheets in a viscous fluid. 
Specifically, we wish to establish how the flow-induced 
deformation affects the sedimentation of such sheets, paying
particular attention to

-- the effect of the sheets' aspect ratio; long narrow sheets are
likely to behave in a manner similar to elastic rods: at which point
does their finite aspect ratio become significant?

-- the effect of wrinkling instabilities and the development of
symmetry-breaking frustrated patterns: how do they arise in sheets of 
canonical shapes (circular, rectangular, polygonal,...) and how do
they affect the sheets' sedimentation?

The focus of this specific project is on computational/semi-analytical
approaches and would suit a student with a good background in Applied
Mathematics (especially fluid and solid mechanics) and Scientific
Computing. There is an opportunity for hands-on involvement
in an associated experimental study in the School of Physics
and Astronomy.

MAIN SUPERVISOR: Professor Matthias Heil (School of Mathematics)

CO-SUPERVISORS: Professor Anne Juel (School of Physics and Astronomy)
Dr Draga Pihler-Puzovic (School of Physics and
Astronomy)

START DATE: September 2018 (or as soon as possible thereafter)

OTHER ASSOCIATED PROJECT AREAS: Physics

FUNDING: Funding is available and would provide fees and maintenance
at RCUK level for home/EU students, or a fees-only bursary
for overseas students.

DEADLINE: Applications are accepted at any time until the position
is filled.

Fluid-structure interaction effects in the sedimentation of thin elastic sheets

• Professor Matthias Heil (School of Mathematics) (Matthias.Heil@manchester.ac.uk)
• Professor Anne Juel (School of Physics and Astronomy) (Anne.Juel@manchester.ac.uk)
• Dr Draga Pihler-Puzovic (School of Physics and Astronomy) (draga.pihler-puzovic@manchester.ac.uk)

There is much current interest in so-called two-dimensional materials 
because of their unusual and attractive mechanical and electrical properties. 
Much of their processing is performed in a fluid environment, e.g., during
the size selection of dispersed graphene flakes by centrifugation, or
their deposition by ink-jet printing. The flakes' large aspect ratio 
implies that despite their impressive in-plane stiffness they have a
very small bending stiffness and are therefore easily deformed by the 
traction that the surrounding fluid exerts on them. The resulting
strong fluid-structure interaction affects not only the dynamics of 
individual flakes but also their collective behaviour.

The aim of this project is to perform a systematic study of 
the behaviour of thin elastic sheets in a viscous fluid. 
Specifically, we wish to establish how the flow-induced 
deformation affects the sedimentation of such sheets, paying
particular attention to

-- the effect of the sheets' aspect ratio; long narrow sheets are
likely to behave in a manner similar to elastic rods: at which point
does their finite aspect ratio become significant?

-- the effect of wrinkling instabilities and the development of
symmetry-breaking frustrated patterns: how do they arise in sheets of 
canonical shapes (circular, rectangular, polygonal,...) and how do
they affect the sheets' sedimentation?

The focus of this specific project is on computational/semi-analytical
approaches and would suit a student with a good background in Applied
Mathematics (especially fluid and solid mechanics) and Scientific
Computing. There is an opportunity for hands-on involvement
in an associated experimental study in the School of Physics
and Astronomy.

OTHER ASSOCIATED PROJECT AREAS: Physics

FUNDING: Funding is available and would provide fees and maintenance
at RCUK level for home/EU students, or a fees-only bursary
for overseas students.

DEADLINE: Applications are accepted at any time until the position
is filled.

Model Theory of Fields with Operators

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

Model theory is a branch of Mathematical logic that has had several remarkable applications with other areas of mathematics, such as Combinatorics, Algebraic Geometry, Number Theory, Arithmetic Geometry, Complex and Real Analysis, Functional Analysis, and Algebra (to name a few). Some of these applications have come from the study of model-theoretic properties of fields equipped with a family of operators. For instance, this includes differential/difference fields. In this project, we will look at the model theory of fields equipped with a general class of operators (that unifies other known approaches) and also within certain natural classes of fields (such as real closed fields). Several foundational questions remain open around what is called "model-companion", "elimination of imaginaries", and the "trichotomy", this is a small sample of the problems that will be tackled.

Quantitative Aspects of Number Theory

• Dr Christopher Frei (christopher.frei@manchester.ac.uk)

The aim of this project is to investigate problems in number theory that have a quantitative component. Typically, the problems would involve some algebraic objects, such as as algebraic number fields or rational points on algebraic varieties. The questions that we ask are, however, of analytic nature. For example, how many number fields are there of a given type and with given properties, such that their discriminant is bounded by a (large) number B? The answer that we are seeking would then be an asymptotic formula or, if this is too hard, bounds for this number as B tends to infinity.

Quantitative results of this kind are of interest by themselves, but moreover they are useful in other proofs, often to establish the existence of objects with certain properties. This holds, for example, for many applications of the Hardy-Littlewood circle method from analytic number theory.

A candidate for this project would have some background in analytic or algebraic number theory, as well as an interest in learning the other field. An interest in algebraic geometry could be useful, but is by no means required. The concrete problems to work on will be chosen to match the student's interests.

Fluid Mechanics of Cleaning and Decontamination

• Dr Julien Landel (julien.landel@manchester.ac.uk)

Cleaning and decontamination processes are important in many applications: from the daily chores of doing the dishes (with or without a dishwasher), to ensuring clean hygiene in hospitals, the food industry, or pharmaceutical companies. Although a lot of research has been done in chemistry and chemical engineering to improve detergents and cleaning devices, much less work has been done on the modelling of the underlying physical and chemical processes. In some cleaning applications, such as the neutralisation of toxic chemicals after a spill, it is crucial to avoid using strong mechanical forces in order to prevent the dispersion of the toxic material in the environment. Instead, a localised dissolution process, aided by chemical reactions neutralising the material, is used. This PhD project will investigate the advection, diffusion and reaction processes involved in this scenario.

Through a combination of experiments and modelling work the student will study the influence of flow properties: such as the Reynolds number and the Péclet number; geometry: whether the material is attached to a permeable or impermeable surface; and chemical properties such as solubility, reactivity and diffusivity.

This project is directly motivated by industrial applications and will suit candidates interested in using mathematical approaches to solve real challenges. Suitable candidates should have experience in the lab or a keen interest to support theoretical work in fluid dynamics by experimental evidence.

Reference: Landel, Thomas, McEvoy & Dalziel (2016). Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 789: 630.

Fluid Mechanics of Cleaning and Decontamination

• Dr Julien Landel (julien.landel@manchester.ac.uk)

Cleaning and decontamination processes are important in many applications: from the daily chores of doing the dishes (with or without a dishwasher), to ensuring clean hygiene in hospitals, the food industry, or pharmaceutical companies. Although a lot of research has been done in chemistry and chemical engineering to improve detergents and cleaning devices, much less work has been done on the modelling of the underlying physical and chemical processes. In some cleaning applications, such as the neutralisation of toxic chemicals after a spill, it is crucial to avoid using strong mechanical forces in order to prevent the dispersion of the toxic material in the environment. Instead, a localised dissolution process, aided by chemical reactions neutralising the material, is used. This PhD project will investigate the advection, diffusion and reaction processes involved in this scenario.

Through a combination of experiments and modelling work the student will study the influence of flow properties: such as the Reynolds number and the Péclet number; geometry: whether the material is attached to a permeable or impermeable surface; and chemical properties such as solubility, reactivity and diffusivity.

This project is directly motivated by industrial applications and will suit candidates interested in using mathematical approaches to solve real challenges. Suitable candidates should have experience in the lab or a keen interest to support theoretical work in fluid dynamics by experimental evidence.

Reference: Landel, Thomas, McEvoy & Dalziel (2016). Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 789: 630.

Bubble dynamics in confined geometries

• Dr Alice Thompson (alice.thompson@manchester.ac.uk)
• Prof Andrew Hazel (andrew.hazel@manchester.ac.uk)

The motion of deformable bubbles, drops, capsules and cells surrounded by viscous fluid and confined in a narrow channel has applications in micro-fluidic devices and lab-on-a-chip design. When viewed from above, the motion often appears two dimensional, suggesting that we might be able to use two dimensional, depth-averaged models to predict, design and control system behaviour.

Depth-averaged models are known to be accurate in some situations, such as when the bubble is very flattened and far from any walls, and where thin films are relatively passive. However, the typical assumptions break down if two bubbles are very close to each other (e.g. coalescence or breakup), or to walls, constrictions or obstacles, when it highly likely that three-dimensional effects come into play. Furthermore, the typical depth-averaged model is of lower order and dimension than the Stokes equations and hence neglects certain free boundary effects.

The aim of this project is  to establish the limits of 2D modelling, and to determine how we can extend our models to be valid in these extreme scenarios. This project will involve asymptotic analysis, model development, finite element computations of two or three dimensional systems in oomph-lib,  and likely also collaborations with experimentalists in the Manchester Centre for Nonlinear Dynamics.

Thermodynamic Quantum Chaos and large networks

• Dr Tuomas Sahlsten (tuomas.sahlsten@manchester.ac.uk)

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

Additive combinatorics in dynamics and spectral theory

• Dr Tuomas Sahlsten (tuomas.sahlsten@manchester.ac.uk)

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

Scenery flow and fine structure of fractals

• Dr Tuomas Sahlsten (tuomas.sahlsten@manchester.ac.uk)

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

Open Maps

• Dr Nikita Sidorov (nikita.a.sidorov@manchester.ac.uk)

The study of open maps is an exciting relatively new area of the theory of dynamical systems. The characterisation of the holes involves geometry (‘shape’) and measure theory (‘size’), so their study involves a complex interplay between dynamics, geometry and analysis.

The standard dynamical approach is to assume that the survivor set (= the points whose orbits do not fall into a hole) is “sufficiently large” and investigate ergodic and geometric properties of the induced map. The novelty of the proposed project is to take a step back and give sufficient conditions that the survivor set is indeed "sufficiently large" (uncountable, say) for the induced map to be meaningful. This involves a detailed analysis of the class of holes under investigation.

The classes of maps under investigation include - but are not limited to - expanding maps of the interval, algebraic toral automorphisms and subshifts.

Analytic Number Theory and mean values of L-functions

• Dr Hung Bui (hung.bui@manchester.ac.uk)

The Riemann zeta-function and other L-functions play a central role in analytic number theory and in mathematics in general. For example, the Riemann zeta-function satisfies an Euler product, which underlines a connection between the natural numbers and the prime numbers. The problem of determining the properties of prime numbers has a long history, from the ancient theorem of Euclid that there are infinitely many primes, to the celebrated eight page paper of Riemann on the zeta-function in the mid-nineteenth century. Since that time, several important problems in analytic number theory have been solved, and Riemann's ideas have been the inspiration behind much of this progress.

Investigating the properties of the Riemann zeta-function and L-functions in various contexts leads to many other interesting problems, which now represent major challenges in modern mathematics. In fact both the Riemann Hypothesis, which asserts that all the non-trivial zeros of the Riemann zeta-function lie on a particular line, and the Birch and Swinnerton-Dyer Conjecture, which concerns some properties of the L-functions associated to elliptic curves, have been included in the seven Millennium Prize Problems.

The aim of the project is to study various questions related to the moments of the Riemann zeta-function and L-functions, which are the mean values over certain families of these functions. These questions have applications to the distribution of zeros of the Riemann zeta-function (partial answers to the Riemann Hypothesis), the order of magnitude of L-functions (partial answers to the Lindelof Hypothesis), order of vanishing of L-functions at the central point (analytic progress towards the Birch and Swinnerton-Dyer Conjecture), and many others. There is a remarkable connection between the subject and Random Matrix Theory, an area of Mathematical Physics used to describe complex quantum systems.

Distributional approximation by Stein's method

• Dr Robert Gaunt (robert.gaunt@manchester.ac.uk)

Stein's method is a powerful (and elegant) technique for deriving bounds on the distance between two probability distributions with respect to a probability metric.  Such bounds are of interest, for example, in statistical inference when samples sizes are small; indeed, obtaining bounds on the rate of convergence of the central limit theorem was one of the most important problems in probability theory in the first half of the 20th century.

The method is based on differential or difference equations that in a sense characterise the limit distribution and coupling techniques that allow one to derive approximations whilst retaining the probabilistic intuition.   There is an active area of research concerning the development of Stein's method as a probabilistic tool and its application in areas as diverse as random graph theory, statistical mechanics and queuing theory.

There is an excellent survey of Stein's method (see below) and, given a strong background in probability, the basic method can be learnt quite quickly, so it would be possible for the interested student to make progress on new problems relatively shortly into their PhD.  Possible directions for research (although not limited) include: extend Stein's method to new limit distributions; generalisations of the central limit theorem; investigate `faster than would be expected' convergence rates and establish necessary and sufficient conditions under which they occur; applications of Stein's method to problems from, for example, statistical inference.

Literature:

Ross, N. Fundamentals of Stein's method.  Probability Surveys 8 (2011), pp. 210-293.

Generalized Flame Balls and their Stability

• Dr Joel Daou (joel.daou@manchester.ac.uk)

Flame balls are balls of burnt gas in a reactive mixture, which constitute stationary solutions to non-linear Poisson's equations. These were first described by the famous Russian physicist Zeldovich (the father of Combustion Theory) about 70 years ago. The fact that these solutions are typically unstable provides a powerful fundamental criterion for successful ignition, i.e. determines the minimum energy (of the spark) required to generate propagating flames. Several projects are available to extend the study of these fascinating flames (mainly their existence and stability) to take into account realistic effects such as the presence of flow-field, non-uniformity of the reactive mixture, proximity of walls, etc. 

Methodology: The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs), depending  on the preference of the candidate.  

Laminar aspects of turbulent combustion/ Flame propagation in a multi-scale flow

• Dr Joel Daou (joel.daou@manchester.ac.uk)

The idea is to ask if the fundamental questions of turbulent combustion can be answered for simple laminar flows. Since the answer is often no, we shall formulate and study problems to answer these questions in simpler laminar-flow situations. An exciting topic!

Methodology: The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs), depending  on the preference of the candidate.  

Taylor dispersion and hydrodynamic lubrication theory in premixed combustion

• Dr Joel Daou (joel.daou@manchester.ac.uk)

In 1953, the British physicist G.I. Taylor published an influential paper describing the enhancement of diffusion processes by a (shear) flow, a phenomenon later termed Taylor dispersion. This has generated to date thousands of publications in various areas involving transport phenomena, none of which, surprisingly, in the field of combustion. In 1940, the German chemist G. Damköhler postulated two hypotheses which have largely shaped current views on the propagation of premixed flames in turbulent flow fields. The project consists of pioneering investigations linking Taylor dispersion and Damköhler’s hypotheses, and is expected to provide significant insight into turbulent combustion. The work will be carried out in the framework of lubrication theory, generalized to combustion situations, and will include interesting stability problem such as the Saffman-Taylor instability in a reactive mixture.

Methodology:  The approach will typically adopt a combination of analytical techniques (asymptotic methods) and/or numerical techniques (solution of ODEs or PDEs), depending  on the preference of the candidate.  

Mathematical Combustion and Flame Instabilities

• Dr Joel Daou (joel.daou@manchester.ac.uk)

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

Suggested sample projects:

• Ignition in a flow field (such as a Poiseuille flow) and in mixing-layers. The main aim is to determine the critical energy of the initial hot kernel (or spark) to ignite a flowing reactive mixture.

• Propagating Flames and their Stability:  This involves the investigation of the various instabilities of flames using analytical and/or numerical approaches. The   flames will be modelled as travelling wave solutions to reaction-diffusion-convection equations, which may, or may not, include full coupling with the hydrodynamics (the Navier-Stokes equation).

• Flame initiation and propagation in spatially non-uniform mixtures: This is a problem of considerable interest in combustion, whenever the reactants are spatially separated. The approach will be based on asymptotic and/or numerical methods.  The Combustion basics needed for the projects will be provided and explained to the candidate.

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.

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.

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.

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.

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.

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