You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.
Title  LOT Groups 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Martin Edjvet 
Description  A labelled oriented graph is a connected, finite, directed graph in which each edge is labelled by a vertex. A labelled oriented graph gives rise to a group presentation whose generating set is the vertex set and whose defining relations say that the initial vertex of an edge is conjugated to its terminal vertex by its label. A group G is a LOG group if it has such a presentation. A LOT group is a LOG group for which the graph is a tree. Every classical knot group is a LOT group. In fact LOT groups are characterised as the fundamental groups of ribbon ndiscs in Dn+2.The most significant outstanding question on the topology of ribbon discs is: are they aspherical? The expected answer is yes. Howie has shown that it is sufficient to prove that LOT groups are locally indicable. An interesting research project would be to study the following two questions: is every LOT group locally indicable; is every LOT group an HNN extension of a finitely presented group? 
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Title  Equations over groups 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Martin Edjvet 
Description  Let G be a group. An expression of the form g_{1} t … g_{k} t=1 where each g_{i} is an element of G and the unknown t is distinct from G is called an equation over G. The equation is said to have a solution if G embeds in a group H containing an element h for which the equation holds. There are two unsettled conjectures here. The first states that if G is torsionfree then any equation over G has a solution. The second due to Kervaire and Laudenbach states that if the sum of the exponents of t is nonzero then the equation has a solution. There have been many papers published in this area. The methods are geometric making use of diagrams over groups and curvature. This subject is related to questions of asphericity of groups which could also be studied. 
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Title  Quadratic forms and forms of higher degree, nonassociative algebras 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Susanne Pumpluen 
Description  Dr. Pumplün currently studies forms of higher degree over fields, i.e. homogeneous polynomials of degree d greater than two (mostly over fields of characteristic zero or greater than d). The theory of these forms is much more complex than the theory of homogeneous polynomials of degree two (also called quadratic forms). Partly this can be explained by the fact that not every form of degree greater than two can be “diagonalized”, as it is the case for quadratic forms over fields of characteristic not two. (Every quadratic form over a field of characteristic not two can be represented by a matrix which only has nonzero entries on its diagonal, i.e. is diagonal.) A modern uniform theory for these forms like it exists for quadratic and symmetric bilinear forms (cf. the standard reference books by Scharlau or Lam) seems to be missing, or only exists to some extent. Many questions which have been settled for quadratic forms quite some time ago are still open as as soon as one looks at forms of higher degree. It would be desirable to obtain a better understanding of the behaviour of these forms. First results have been obtained. Another related problem would be if one can describe forms of higher degree over algebraic varieties, for instance over curves of genus zero or one. Dr. Pumplün is also studying nonassociative algebras over rings, fields, or algebraic varieties. Over rings, as modules these algebras are finitely generated over the base ring. Their algebra structure, i.e. the multiplication, is given by any bilinear map, such that the distributive laws are satisfied. In other words, the multiplication is not required to be associative any more, as it is usually the case when one talks about algebras. Her techniques for investigating certain classes of nonassociative algebras (e.g. octonion algebras) include elementary algebraic geometry. One of her next projects will be the investigation of octonion algebras and of exceptional simple Jordan algebras (also called Albert algebras) over curves of genus zero or one. Results on these algebras would also imply new insights on certain algebraic groups related to them. Another interesting area is the study of quadratic or bilinear forms over algebraic varieties. There are only few varieties of dimension greater than one where the Witt ring is known. One wellknown result is due to Arason (1980). It says that the Witt ring of projective space is always isomorphic to the Witt ring of the base field. If you want to investigate algebras or forms over algebraic varieties, this will always involve the study of vector bundles of that variety. However, even for algebraically closed base fields it is usually very rare to have an explicit classification of the vector bundles. Hence, most known results on quadratic (or symmetric bilinear) forms are about the Witt ring of quadratic forms, e.g. the Witt ring of affine space, the projective space, of elliptic or hyperelliptic curves. An explicit classification of symmetric bilinear spaces is in general impossible because it would involve an explicit classification of the corresponding vector bundles (which admit a form). There are still lots of interesting open problems in this area, both easier and very difficult ones. 
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Title  Cohomology Theories for Algebraic Varieties 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Alexander Vishik 
Description  After the groundbreaking works of V. Voevodsky, it became possible to work with algebraic varieties by completely topological methods. An important role in this context is played by the socalled Generalized Cohomology Theories. This includes classical algebraic Ktheory, but also a rather modern (and more universal) Algebraic Cobordism theory. The study of such theories and cohomological operations on them is a fascinating subject. It has many applications to the classical questions from algebraic geometry, quadratic form theory, and other areas. One can mention, for example: the Rost degree formula, the problem of smoothing algebraic cycles, and uinvariants of fields. This is a new and rapidly developing area that offers many promising directions of research. 
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Title  Quadratic Forms: Interaction of Algebra, Geometry and Topology 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Alexander Vishik 
Description  From the beginning of the 20th century it was observed that quadratic forms over a given field carry a lot of information about that field. This led to the creation of rich and beautiful Algebraic Theory of Quadratic Forms that gave rise to many interesting problems. But it became apparent that quite a few of these problems can hardly be approached by means of the theory itself. In many cases, solutions were obtained by invoking arguments of a geometric nature. It was observed that one of the central questions on which quadratic form theory depends is the socalled "Milnor Conjecture". This conjecture, as we now understand it, relates quadratic forms over a field to the socalled motivic cohomology of this field. Once proven, this would provide a lot of information about quadratic forms and about motives (algebrogeometric analogues of topological objects) as well. The Milnor Conjecture was finally settled affirmatively by V. Voevodsky in 1996 by means of creating a completely new world, where one can work with algebraic varieties with the same flexibility as with topological spaces. Later, this was enhanced by F. Morel, and now we know that quadratic forms compute not just the cohomology of a point in the "algebro geometric homotopic world", but also the socalled stable homotopy groups of spheres as well. It is thus no wonder that these objects indeed have nice properties. Therefore, by studying quadratic forms, one actually studies the stable homotopy groups of spheres, which should shed light on the classical problem of computing such groups (one of the central questions in mathematics as a whole). So it is fair to say that the modern theory of quadratic forms relies heavily on the application of motivic topological methods. On the other hand, the Algebraic Theory of Quadratic Forms provides a possibility to view and approach the motivic world from a rather elementary point of view, and to test the new techniques developed there. This makes quadratic form theory an invaluable and easy access point to the forefront of modern mathematics. 
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Title  Regularity conditions for Banach function algebras 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Joel Feinstein 
Description  Banach function algebras are complete normed algebras of bounded, continuous, complexvalued functions defined on topological spaces. There are very many different examples with a huge variety of properties. Two contrasting examples are the algebra of all continuous complexvalued functions on the closed unit disc, and the subalgebra of this algebra consisting of those functions which are continuous on the closed disc and analytic on the interior of the disc. In the second of these algebras, any function which is zero throughout some nonempty open set must be constantly zero. This is very much not the case in the bigger algebra: indeed Urysohn’s lemma shows that for any two disjoint closed subsets of the closed disc, there is a continuous, complexvalued function defined on the disc which is constantly 0 on one closed set and constantly 1 on the other (algebras of this type are called regular algebras). Most Banach function algebras have some features in common with one or the other of these two algebras. The aim of this project is to investigate a variety of conditions, especially regularity conditions, for Banach function algebras, and to relate these conditions to each other, and to other important conditions that Banach function algebras may satisfy. Regularity conditions have important applications in several areas of functional analysis, including automatic continuity theory and the theory of Wedderburn decompositions. There is also a close connection between regularity and the theory of decomposable operators on Banach spaces. 
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Title  Properties of Banach function algebras and their extensions 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Joel Feinstein 
Description  Banach function algebras are complete normed algebras of bounded continuous, complexvalued functions defined on topological spaces. There are very many different examples with a huge variety of properties. Two contrasting examples are the algebra of all continuous complexvalued functions on the closed unit disc, and the subalgebra of this algebra consisting of those functions which are continuous on the closed disc and analytic on the interior of the disc. In the second of these algebras, any function which is zero throughout some nonempty open set must be constantly zero. This is very much not the case in the bigger algebra: indeed Urysohn’s lemma shows that for any two disjoint closed subsets of the closed disc, there is a continuous, complexvalued function defined on the disc which is constantly 0 on one closed set and constantly 1 on the other (algebras of this type are called regular algebras). Most Banach function algebras have some features in common with one or the other of these two algebras. The aim of this project is to investigate a variety of conditions (including regularity conditions) for Banach function algebras, to relate these conditions to each other, and to other important conditions that Banach function algebras may satisfy, and to investigate the preservation or introduction of these conditions when you form various types of extension of the algebras (especially ‘algebraic’ extensions such as ArensHoffman or Cole extensions). 
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Title  Meromorphic Function Theory 

Group(s)  Algebra and Analysis 
Proposer(s)  Prof James Langley 
Description  A meromorphic function is basically one convergent power series divided by another: such functions arise in many branches of pure and applied mathematics. Professor Langley has supervised ten PhD students to date, and specific areas covered by his research include the following.

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Title  Compensated convex transforms and their applications 

Group(s)  Algebra and Analysis 
Proposer(s)  Prof Kewei Zhang 
Description  This aim of the project is to further develop the theory and numerical methods for compensated convex transforms introduced by the proposer and to apply these tools to approximations, interpolations, reconstructions, image processing and singularity extraction problems arising from applied sciences and engineering. 
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Title  Endomorphisms of Banach algebras 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Joel Feinstein 
Description  Compact endomorphisms of commutative, semisimple Banach algebras have been extensively studied since the seminal work of Kamowitz dating back to 1978. More recently the theory has expanded to include power compact, Riesz and quasicompact endomorphisms of commutative, semiprime Banach algebras. This project concerns the classification of the various types of endomorphism for specific algebras, with the aid of the general theory. The algebras studied will include algebras of differentiable functions on compact plane sets, and related algebras such as Lipschitz algebras. 
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Title  Iteration of quasiregular mappings 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Daniel Nicks 
Description  Complex dynamics is the study of iteration of analytic functions on the complex plane. A rich mathematical structure is seen to emerge amidst the chaotic behaviour. Its appeal is enhanced by the intricate nature of the Julia sets that arise, and fascinating images of these fractal sets are widely admired. 
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Title  Hydrodynamic limit of GinzburgLandau vortices 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Matthias Kurzke 
Description  Many quantum physical systems (for example superconductors, superfluids, BoseEinstein condensates) exhibit vortex states that can be described by GinzburgLandau type functionals. For various equations of motion for the physical systems, the dynamical behaviour of finite numbers of vortices has been rigorously established. We are interested in studying systems with many vortices (this is the typical situation in a superconductor). In the hydrodynamic limit, one obtains an evolution equation for the vortex density. Typically, these equations are relatives of the Euler equations of incompressible fluids: for the GrossPitaevskii equation (a nonlinear Schrödinger equation), one obtains Euler, for the timedependent GinzburgLandau equation (a nonlinear parabolic equation), one obtains a dissipative variant of the Euler equations. There are at least two interesting directions to pursue here. One is to extend recent analytical progress for the Euler equation to the dissipative case. Another one is to obtain hydrodynamic limits for other motion laws (for example, mixed or wavetype motions). This is a project mostly about analysis of PDEs, possibly with some numerical simulation involved. 
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Title  Dynamics of boundary singularities 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Matthias Kurzke 
Description  Some physical problems can be modelled by a function or vector field with a near discontinuity at a point. Specific examples include boundary vortices in thin magnetic films, and some types of dislocations in crystals. Typical static configurations can be found by minimizing certain energy functionals. As the core size of the singularity tends to zero, these energy functionals are usually well described by a limiting functional defined on point singularities. This project investigates how to obtain dynamical laws for singularities (typically in the form of ordinary differential equations) from the partial differential equations that describe the evolution of the vector field. For some such problems, results for interior singularities are known, but their boundary counterparts are still lacking. This project requires some background in the calculus of variations and the theory of partial differential equations. 
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Title  Where graphs and partial differential equations meet 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Yves van Gennip 
Description  Many problems in image analysis and data analysis can be represented mathematically as a network based This has lead to an interesting mix of theoretical questions (what is the dynamics on the network induced This project will investigate graph curvature and related quantities and make links to established 
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Title  Graph limits for faster computations 

Group(s)  Algebra and Analysis 
Proposer(s)  Dr Yves van Gennip 
Description  Many problems in image analysis and data analysis, such as image segmentation or data clustering, require Recent developments in the theory of (dynamics on) graph limits offer the hope that this subset can be This project will investigate this possibility and can be taken in a theoretical and/or application 
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Title  Vectorial Calculus of Variations, Material Microstructure, ForwardBackward Diffusion Equations and Coercivity Problems 

Group(s)  Algebra and Analysis, Algebra and Analysis 
Proposer(s)  Prof Kewei Zhang 
Description  This aim of the project is to solve problems in vectorial calculus of variations, forwardbackward diffusion equations, partial differential inclusions and coercivity problems for elliptic systems. These problems are motivated from the variational models for material microstructure, image processing and elasticity theory. Methods involve quasiconvex functions, quasiconvex envelope, quasiconvex hull, Young measure, weak convergence in Sobolev spaces, elliptic and parabolic partial differential equations, and other analytic and geometric tools. 
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Title  Mathematical image analysis of 4D Xray microtomography data for crack propagation in aluminium wire bonds in power electronics 

Group(s)  Algebra and Analysis, Industrial and Applied Mathematics 
Proposer(s)  Dr Yves van Gennip 
Description  Supervised by Dr Yves van Gennip and Dr Pearl Agyakwa
In this project we will develop mathematical image processing methods for the analysis of 4D (3d+time) Xray microtomography data of wire bonds [1].
Wire bonds are an essential but lifelimiting component of most power electronic modules, which are critical for energy conversion in applications like renewable energy generation and transport.
The key issue we will examine is how we can use mathematical image processing and image analysis techniques to study how defects in wire bonds arise and evolve under operating conditions; this will facilitate more accurate lifetime prediction. 
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Other information  3D Xray microtomography provides nondestructive observations of defect growth, which allows the same wire bond to be evaluated over its lifetime, affording invaluable new insights into a still insufficiently understood process.
Analysis and full exploitation of the useful information contained within these large datasets is nontrivial and requires advanced mathematical techniques [2,3]. A major challenge is the ability to subsample compressible information without losing informative data features, to improve temporal accuracy.
Our goal is to construct new mathematical imaging and data analysis methods for evaluating 2D and 3D tomography data for the same wire bond specimen at various stages of wearout during its lifetime, to better understand the degradation mechanism(s). This will include methods for denoising of data, detection and segmentation of the wires and their defects in the tomography images, and image registration to quantify the wire deformations that occur over time. 
Title  Crystallisation in polymers 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Richard Graham 
Description  Polymers are very long chain molecules and many of their unique properties depend upon their long chain nature. Like simple fluids many polymer fluids crystallise when cooled. However, the crystallisation process is complicated by the way the constituent chains are connected, leading to many curious and unexplained phenomena. Furthermore, if a polymer fluid is placed under flow, this strongly affects both the ease with which the polymer crystallises and the arrangement of the polymer chains within the resulting crystal. This project will develop and solve models for polymer dynamics and phase transitions using a range of analytical, numerical and stochastic techniques, with the ultimate aim of improving our understanding of polymer crystallisation. The project offers the opportunity to collaborate with a wide range of scientists working in the field, including several worldleading experimental groups. 
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Title  Dynamics of entangled polymers 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Richard Graham 
Description  Polymers are extraordinarily long molecules, made out of chains of simpler molecules. They occur everywhere in our everyday lives, including in the DNA chains that make up our genetics, in many hightech consumer products and in the simple plastic bag. Often these applications depend crucially on the way that the polymer chains move. This is especially true in concentrated polymer liquids, where the chain dynamics are controlled by how the chains become entangled with each other. A powerful mathematical framework for describing these entangled systems has been under development for some time now, but the ideas have yet to be fully developed, tested and exploited in practical applications. Working on this PhD project will give the opportunity to train in a wide range of mathematical techniques including analytical work, numerical computations and stochastic simulation and to apply these to problems of real practical impact. This lively research field involves mathematicians, scientists and engineers and a keenness to learn from and cooperate with researchers from a range of backgrounds would be a real asset in this project. 
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Title  Instabilities of fronts 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Cox, Dr Paul Matthews 
Description  Chemical reactions often start at a point and spread through a reactant, much as a fire spreads through combustible material. The advancing zone in which the reactions take place is called a reaction front. In the simplest cases, the reaction front is smooth (flat, cylindrical or spherical), but it may develop irregularities due to instability. Sometimes the instability is so strong that it destroys the front itself; in other cases, it just results in a slight modulation to the front shape. This project involves studying a partial differential equation, the Nikolaevskiy equation, that describes the nonlinear development of the instability of a front. Numerical simulations of the Nikolaevskiy equation show highly complicated, chaotic solutions. This project will involve a mix of numerical simulations and analytical work to understand the behaviour of the Nikolaevskiy equation and of front instabilities in general. 
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Title  Power converters 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Cox, Dr Stephen Creagh 
Description  In a wide range of applications, it is necessary to convert one electrical power supply to another, of different voltage or frequency. Power converters are devices which achieve this, but they often suffer highly undesirable instabilities, which significantly compromise their operation. The goal of this project is to develop mathematical models for existing power converter technologies and to use these to provide a detailed description of their operation and a thorough understanding of the instability. Through mathematical modelling, it may prove possible to improve existing power converter designs to reduce or eliminate the stability problems! This project will be theoretical in nature, relying largely on analytical and numerical techniques for differential equations, and will involve significant interaction with the Power Electronics Group in the Department of Electrical and Electronic Engineering. 
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Title  ClassD audio amplifiers 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Cox, Dr Stephen Creagh 
Description  The holy grail for an audiophile is distortionfree reproduction of sound by amplifier and loudspeaker. This project concerns the mathematical modelling and analysis of classD audio amplifiers, which are highly efficient and capable of very low distortion. Designs for such amplifiers have been known for over 50 years, but only much more recently have electronic components been up to the job, making classD amplifiers a reality. (ClassD amplifiers rely on very high frequency – around 1MHz – sampling of the input signal, and so test their components to the limit.) Unfortunately, while the standard classD design offers zero distortion, it has poor noise characteristics; when the design is modified by adding negative feedback to reduce the noise, the amplifier distorts. By a further modification to the design it is possible to eliminate (most of) the distortion. This project involves modelling various classD designs and determining their distortion characteristics, with the aim of reducing the distortion. The project will be largely analytical, applying asymptotic methods and computer algebra to solve the mathematical models. Simulations in matlab or maple will be used to test the predictions of the mathematical models. 
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Title  Mathematical modelling and analysis of composite materials and structures 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Konstantinos Soldatos 
Description  Nottingham has established and maintained, for more than half a century, worldwide research leadership in developing the Continuum Theory of fibrereinforced materials and structures. Namely, a theoretical mechanics research subject with traditional interests to engineering and, more recently, to biological material applications. The subject covers extensive research areas of mathematical modelling and analysis which are of indissoluble adherence to basic understanding and prediction of the elastic, plastic, viscoelastic or even viscous (fluidtype) behaviour observed during either manufacturing or real life performance of anisotropic, composite materials and structural components. Typical research projects available in this as well as in other relevant research subjects are related with the following interconnected areas:
The large variety of topics and relevant problems emerging in these subjects of Theoretical Mechanics and Applied Mathematics allow considerable flexibility in the formation of PhD projects. A particular PhD project may accordingly be formed/designed around the strong subjects of knowledge of a potential postgraduate student. The candidate’s relevant cooperation is accordingly desirable and, as such, will be appreciated at the initial, but also at later stages of tentative research collaboration. 
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Title  Dynamics of coupled nonlinear oscillators 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Paul Matthews 
Description  Coupled oscillators arise in many branches of science and technology and also have applications to biological systems. One spectacular example is swarms of fireflies that flash in synchrony. This research field is an expanding area in applied mathematics because of the many applications within physics and biology and because of the variety of behaviour which such systems can exhibit. Recent work on coupled oscillators has revealed some interesting novel results: nonlinear oscillators can synchronise to a common oscillation frequency even if they have different natural frequencies, provided the coupling is above some threshold; the breakdown of synchronisation as the coupling strength decreases involves periodic behaviour and chaos. The project involves extending and improving this work in two ways. First, the oscillator model used in earlier work was simple and idealised; the model will be refined to make it more realistic. Second, most earlier work used a simple linear global coupling so that each oscillator is equally coupled to all of the others. In most practical examples this is not the case and a coupling law over a two or threedimensional lattice would be more appropriate, with stronger coupling between nearer pairs of oscillators. The research will be carried out using a combination of numerical and analytical techniques. 
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Title  Dynamo action in convection 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Paul Matthews 
Description  The magnetic fields of the Earth and Sun are maintained by dynamo action. Fluid motions are generated by thermal convection. The kinetic energy of these fluid motions is then converted to magnetic energy, in a manner similar to that of a bicycle dynamo. Dynamo theory studies how this conversion takes place. It is known that in order for a dynamo to work efficiently, the fluid flow must exhibit chaos. This project will investigate dynamo action in convection, using 3D numerical simulation of the equations for the fluid motion and the magnetic field. An existing computer program will be used to study the dynamo problem. A sequence of numerical simulations will be carried out to determine
The project is also suitable for analytical work, either based on an asymptotic analysis of the equations, or in investigating or proving 'antidynamo' theorems. 
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Title  Nonlinear penetrative convection 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Paul Matthews 
Description  The phenomenon of convection, in which heat is transferred by fluid motion, occurs very commonly in nature. Examples include in the Earth's atmosphere, the interior of the Sun, the Earth's liquid outer core, lakes and oceans. The most commonly used mathematical model for convection assumes that a layer of fluid is bounded above and below by boundaries that are maintained at a fixed temperature. This is not a good model for most of the environmental applications, where typically part of the fluid layer is thermally unstable and part is stable. Convection in the unstable layer overshoots and penetrates into the stable layer. This phenomenon, known as 'penetrative convection', has received relatively little investigation. The research project will study penetrative convection in the nonlinear regime. An existing computer program will be adapted to investigate penetrative convection numerically, and analytical work will be carried out using asymptotic methods and methods of bifurcation theory. In particular, the extent of penetration into the stable layer and the possibility of instability to a mean flow will be explored. 
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Title  Coupling between optical components 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Creagh 
Description  Evanescent coupling between different optical components is a very important process in optical communications. In this effect, light travelling along an optical fibre effectively spills out a little bit into the region of space immediately surrounding the fibre itself and can then leak into and become captured by other, nearby optical components. Among other uses, this mechanism forms a basis for optical switches, which transfer light from one fibre to another, and for wavelength filters, which selectively transmit or redirect light in only certain frequency ranges. This project will investigate the coupling between cylindrical and spherical optical components in two and three dimensions using the geometry of the underlying ray solutions. The aim will be to exploit and generalise approximations which have been developed in the context of quantum waves but which should be equally applicable to the optics problem. 
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Title  Is Periodic Behaviour an Emergent Phenomenon? 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Keith Hopcraft 
Description  Periodic behaviours can be described with great power and economy using the rather simple mathematical machinery associated with wave phenomena. However periodic effects can also be ‘observed’ in collections of discrete objects, be they individuals sending emails, fireflies signaling to attract mates, synapses firing in the brain or photons emerging from a cavity. The manifestation of periodicity requires both a dynamical process and a ‘medium’ in which it operates and the project will seek to identify the essential properties of the dynamics and the structure of the medium required to do this without invoking the ideas of the continuum, determinism or reversibility. A very simple but surprisingly rich model has been constructed, that involves purely random dynamics acting on a graph, which nevertheless exhibits amorphous, coherent and collapsed states as a single control parameter is changed. The coherent states indeed do exhibit periodic behaviours, and the criteria for this emergence to occur have been identified. Periodicity requires a minimum of three nodes in the graph, for there to be a bias in the direction for flow of information around the network and for the control parameter to exceed a threshold. It also requires the concept of ‘action at a distance’, which is familiar to any field theory. The project will investigate some of the other emergent properties this model possesses, before seeing whether the assumption of a field can be relaxed by considering the selfinteraction of a node’s dynamics. 
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Title  Modeling acoustic emission energy propagation 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Creagh, Prof Gregor Tanner 
Description  This project will develop modelling techniques to predict acoustic emissions radiated from a water jet hitting a target object and subsequently reverberating within the structure. The approach taken will be to exploit dynamical systems and raypropagation approaches developed in the context of wave and quantum chaos. The source is complex and statistically characterized in this scenario, so raytracing techniques must be adapted to predict features such as average intensities and correlations rather than the field amplitude itself. The motivation for this project is provided by manufacturing processes in which such water jets are used to create deformation or attrition on the target object, and will be undertaken in collaboration with Dr Amir Rabani of the School of Mechanical, Materials and Manufacturing Engineering. In the experiments, highenergy fluid jet milling where a multi phase media is used as the source of attrition is an example of such processes. These deformations or attritions are primarily due to the mechanical energy applied to the target object. This mechanical energy propagates within the target object in the shape of high frequency elastic and plastic waves that can be picked up using acoustic emission sensors [1]. Modeling the propagation of the mechanical energy of the propagating waves and their attenuation can provide valuable information about the applied energy to the target object by the source. This information can potentially provide means to monitor/control the deformation or attrition process of the target object. The applications of the acoustic emission energy propagation model can go beyond the manufacturing arena and can be used in condition monitoring to diagnose the causes of deformations and attritions. This provides measures to take fail preventative actions that other methods such as nondestructive evaluation (NDE) methods fail to provide.
References: [1] Rabani, A., Marinescu, I., & Axinte, D. (2012). Acoustic emission energy transfer rate: a method for monitoring abrasive waterjet milling. International journal of machine tools and Manufacture, 61, 8089. 
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Title  Solitons in higher dimensions 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Prof Jonathan Wattis 
Description  The localisation of energy and its transport is of great physical interest in many applications. The mechanisms by which this occurs have been widely studied in onedimensional systems; however, in two and threedimensional systems a greater variety of waves and wave phenomena can be observed; for example, waves can be localised in one or both directions. This project will start with an analysis of the nonlinear Schrodinger equation (NLS) in higher space dimensions, and with more general nonlinearities (that is, not just $\gamma=1$). Current interest in the BoseEinstein Condensates which are being investigated in the School of Physics and Astronomy at Nottingham makes this topic particularly timely and relevant. The NLS equation also arises in the study of astrophysical gas clouds, and in the reduction of other nonlinear wave equations using small amplitude asymptotic expansions. For example, the reduction of the equations of motion for atoms in a crystal lattice; this application is particularly intriguing since the lattice structure defines special directions, which numerical simulations show are favoured by travelling waves. Also the motion of a wave through a hexagonal arrangement of atoms will differ from that through a square array of atoms. The project will involve a combination of theoretical and numerical techniques to the study such systems. 
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Title  Modelling the vibroacoustic response of complex structures 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Prof Gregor Tanner 
Description  The vibroacoustic response of mechanical structures (cars, airplanes, ...) can in general be well approximated in terms of linear wave equations. Standard numerical solution methods comprise the finite or boundary element method (FEM, BEM) in the low frequency regime and socalled Statistical Energy Analysis (SEA) in the highfrequency limit. Major computational challenges are posed by socalled midfrequency problems  that is, composite structures where the local wave length may vary by orders of magnitude across the components. The PhD project wil deal with extending these techniques towards a DEA/FEM hybrid method as well as considering FEM formulations of the method. The work will comprise a mix of analytic and numerical skills and will be conducted in close collaboration with our industrial partners inuTech GmbH, Nurenberg, Germany and Jaguar/landrover, Gaydon, UK. 
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Title  Ruin, Disaster, Shame! 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Keith Hopcraft 
Description  Naturally occurring disasters, such as a freak wave that inundates a ship, a bear market that plunges an economy into recession, or those caused by extremes in weather resulting from ‘global warming’, cannot be avoided. But they can be planned for so that their devastating effects can be ameliorated. This project will study the mathematical properties extremal events that are caused by a stochastic process exceeding a threshold. It forms part of a larger programme that will generate data from an optical analogue of extremal events – the generation of caustics, and from analyses of financial and climate data. The project will investigate the extrema produced by a nongaussian stochastic process that is represented mathematically by the nonlinear filtering of a signal, and will determine such useful quantities as the fluctuations in number of extremal events, and the time of occurrence to the next event. The project will involve modelling of stochastic processes, asymptotic analysis, simulation and data processing. Direct involvement with the experimental programme will also be encouraged. A Case Award supplement may be available for a suitably qualified candidate. 
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Title  Projects in the mechanics of crystals 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Gareth Parry 
Description  The aim is to understand different aspects of plastic behaviour in complex defective crystals. It is not surprising that methods of traditional continuum mechanics play a role in this area of materials science, but it is perhaps unexpected that classical ideas of differential geometry are central to an appreciation of the issues involved. A student of traditional background in either pure or applied mathematics will be guided, first of all, in reading and in other preparatory exercises, in such a way as to strengthen his or her knowledge appropriately. Possible research projects in this area are the following.

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Title  The frequency of catastrophes 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Keith Hopcraft 
Description  We have recently developed analytical stochastic models that are capable of describing the frequency of discrete events that have (essentially) an arbitrary distribution, including such extreme cases as when the mean does not exist. Such models can be used to investigate the frequency of rare or extremal events, and can be used to quantify the size of fluctuations that are generated by systems that are close to a critical point, where correlations have a dominating role. The current interest on global climate change provides an interesting and important area with which to apply these models. Climate records provide a detailed source of data from which one can deduce extremal events, such as the number of times the temperature or precipitation exceeds the mean during a period and the models then provide the capacity to estimate the future frequency of such occurrences. The work will involve timeseries analysis of climate records, stochastic model building and solution of those models using analytical and numerical techniques. 
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Title  Caustics: optical paradigms of complex systems 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Keith Hopcraft 
Description  A complex system is multicomponent and heterogeneous in character, the interactions between its component parts leading to collective, correlated and selforganising behaviours. Manifestations of these behaviours are diverse and can range from descriptions of matter near a critical point, through turbulence, to the organising structures that emerge in societies. The interactions which generate these behaviours are always nonlinear and often triggered by the system crossing a threshold, the frequency of crossing this barrier provides an important characteristic of the system under consideration. The pattern of caustics observed on the bottom of a swimming pool is one commonly experienced manifestation of such a threshold phenomenon, the caustics being caused by the stationary points of the water's surface. This illustrates how a continuous fluctuation i.e. the water's surface, leads to the occurrence of a discrete the number of events — the caustics. The project will investigate the how the number of caustics depends on the properties of the surface and propagation distance (i.e. the depth of the swimming pool). The work will be mainly analytical in nature, involving elements of stochastic model building and their solution, with some simulation. There is a possibility of comparing models with experimental data of light propagation through 'model swimmingpools' and entrained fluids. 
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Title  The discrete random phasor 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Keith Hopcraft 
Description  In 1965 Richard Feynman wrote ‘I think I can safely say that nobody understands quantum mechanics’, and that situation has not changed in the intervening years despite its continued predictive capacity. One of the many paradoxes that the theory presents is waveparticle duality – for example an electric field behaves as a continuous wave disturbance according to Maxwell’s theory, but also presents phenomenology associated with discrete photons at microscopic scalesizes. In the first instance this project will investigate how a very simple representation of electric field behaviour, a phasor of constant amplitude but random phase, has real and imaginary parts that can be represented by a population of classically interacting particles (photons). The project will proceed by seeking a generalization to this population model with characteristic that can be interpreted as being the addition of two random phasors, each of constant amplitude but independent phase. Such a model leads to interference effects. No prior knowledge of quantum mechanics is required. 
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Title  Machine learning for firstprinciples calculation of physical properties. 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Richard Graham 
Description  The physical properties of all substances are determined by the interactions between the molecules that make up the substance. The energy surface corresponding to these interactions can be calculated from firstprinciples, in theory allowing physical properties to be derived abinitio from a molecular simulation; that is by theory alone and without the need for any experiments. Recently we have focussed on applying these techniques to model carbon dioxide properties, such as density and phase separation, for applications in Carbon Capture and Storage. However, there is enormous potential to exploit this approach in a huge range of applications. A significant barrier is the computational cost of calculating the energy surface quickly and repeatedly, as a simulation requires. In collaboration with the School of Chemistry we have recently developed a machinelearning technique that, by using a small number of precomputed abinitio calculations as training data, can efficiently calculate the entire energy surface. This project will involve extending the approach to more complicated molecules and testing its ability to predict macroscopic physical properties. This project will be jointly supervised by Dr Richard Wheatley in the School of Chemistry. 
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Title  Modelling Thermal Effects within ThinFilm Flows 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Hibberd 
Description  A number of technologies in aerospace gas turbine transmission systems must maintain appropriate cooling of component surfaces and mitigate contact by the use of thin fluid films. In many cases the operating requirements for these components include high rotation speeds, high pressures and high temperatures. Modern design processes for aeroengine components depend increasingly on high quality modelling tools to guide the creation of new products to obtain a comprehensive understanding of the underlying flow characteristics. This project aims to develop detailed understanding of heat transfer in highly sheared thinfilm flows through the creation of sophisticated modelling approaches and numerical tools. Using this capability there will be an opportunity to perform detailed analysis of several enginerelevant configurations.
The classical theory of thinfilm flow is associated with solutions typically for low fluid speeds (Reynolds equation). For higher speed flow an important physical process, often neglected from many current thinfilm flow models, is the generation, transfer and effect of heat within the film and from the surrounding structures. Advanced modelling requires the careful development of fully representative equations and the specification of appropriate boundary conditions. A new model to incorporate nonisothermal effects relevant to a bearing chamber context is provided by This multidisciplinary project will be undertaken by a graduate student in mathematics, engineering or related degree with a strong applied mathematics background and with an interest in fluid mechanics, mathematical and numerical methods. 
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Other information  The supervision team will include include Prof H Power, Faculty of Engineering and a project partner at RollsRoyce plc.
This project is eligible as an EPSRC Industrial CASE award supported by Rolls –Royce plc that includes an additional stipend and a period of experience working locally at RollsRoyce plc. 
Title  On mathematical models for high speed nonisothermal air bearings 

Group(s)  Industrial and Applied Mathematics 
Proposer(s)  Dr Stephen Hibberd 
Description  There are a number of technologies that must maintain appropriate cooling of component surfaces and may include resisting contact by the use of thin fluid films. Modern design processes increasingly on high quality numerical modelling tools to guide the creation of new products and identify operating requirements that include high rotation speeds, high pressures and high temperatures. These demanding operating conditions make a comprehensive understanding of the underlying flow characteristics essential.
Advanced modelling requires the careful development of representative equations for all thermal effects, coupling with an appropriate (film) Reynolds equation and the specification of appropriate boundary conditions. Further, implementation of appropriate numerical methods and analysis is required in these demanding model systems.
High speed gas film bearing bearings (and seals), as proposed for future aeroengines, are designed to work with no contact and very small gaps and applicable to a wide range of industrial applications. Airriding bearings have inherent dynamic advantages in making use of local structural features to maintain sufficient gap between the rotating parts but these may lead to significant instabilities as a result of the dynamic behaviour of the gas film and potential thermal and mechanical distortions.
Building on recent PhD studies this project aims to develop detailed understanding of heat transfer in highly sheared thinfilm flows through the creation of sophisticated and numerical and modelling approaches.

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Other information  The supervision team will include Prof H Power, Faculty of Engineering 
Title  Classical and quantum Chaos in 3body Coulomb problems 

Group(s)  Industrial and Applied Mathematics, Mathematical Physics 
Proposer(s)  Prof Gregor Tanner 
Description  The realisation that the dynamics of 2 particles interacting via central forces is fundamentally different from the dynamics of three particles can be seen as the birth of modern dynamical system theory. The motion of two particles (for example the earthmoon problem neglecting the sun and other planets) is regular and thus easy to predict. This is not the case for three or more particles (especially if the forces between all these particles are of comparable size) and the resulting dynamics is in general chaotic, a fact first spelt out be Poincaré at the end of the 19th century. An important source for chaos in the threebody problem is the possibility of triple collisions, that is, events where all three particles collide simultaneously. Triple collisions form essential singularities in the equation of motions, that is, trajectories can not be smoothly continued through triple collision events. This is related to the fact, that the dynamics at the triple collision point itself takes place on a collision manifold of nontrivial topology. During the project, the student will be introduced to scaling techniques which allow to study the dynamics at the triple collision point. We will in particular consider threebody Coulomb problems, such as twoelectron atoms, and study the influence of the triplecollision on the total dynamics of the problem. As a long term goal, we will try to uncover the origin of approximate invariants of the dynamics whose existence is predicted by experimental and numerical quantum spectra of twoelectron atoms such as the helium atom. 
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Title  Electromagnetic compatibility in complex environments: predicting the propagation of electromagnetic waves using wavechaos theory 

Group(s)  Industrial and Applied Mathematics, Mathematical Physics 
Proposer(s)  Dr Stephen Creagh, Prof Gregor Tanner 
Description  The focus of this project is the development of a mathematical framework to understand the propagation of electromagnetic fields within complicated environments – a challenging task especially in the high frequency limit. Modern technology is typically stuffed with electronic componentry. Devices ranging from a mobile phone to a pc to an Airbus A380 will have many internal electronic components operating at high frequencies and therefore radiating electromagnetic waves. If the waves radiated from one component are strong enough, they can interfere with the functioning of another component somewhere else in the unit. The field of Electromagnetic Compatibility (EMC) aims to mitigate these effects by better understanding the emitted radiation. The outcome of the research will help to design electronic devices, which are protected from interference from other EM sources within buildings, pc enclosures or even planes. The innovative idea in the proposed approach rests on combining EMfield propagation with ideas of chaos theory and nonlinear dynamics. In particular, the representation of waves emitted from a complex source is described in terms of their raydynamics in phase space using the socalled Wigner distribution function (WDF) formalism. It allows us to replace the wave propagation problem with one of propagating classical densities within phase space.

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Title  Wave propagation in complex builtup structures – tackling quasiperiodicity and inhomogeneity 

Group(s)  Industrial and Applied Mathematics, Mathematical Physics 
Proposer(s)  Prof Gregor Tanner, Dr Stephen Creagh 
Description  Computing the dynamic response of modern aerospace, automotive and civil structures can be a computationally challenging task. Characterising the structural dynamics in terms of waves in a uniform or periodic medium is often an important first step in understanding the principal propagating wave modes. Real mechanical structures are rarely fully periodic or homogeneous – variations in shape or thickness, boundaries and intersections as well as curvature destroy the perfect symmetry. The aim of the project is to extend periodic structure theory to wave propagation in quasiperiodic and inhomogeneous media such as stiffened structures. The modelling of waves can then be recast in terms of Bloch theory, which will be modified by using appropriate energy or flux conservation assumptions. The information about the propagating modes will then be implemented into modern highfrequency wave methods – such as the socalled Dynamical Energy Analysis developed in Nottingham  making it possible to compute the vibrational response of structures with arbitrary complexity at large frequencies.

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Title  Network performance subject to agentbased dynamical processes 

Group(s)  Industrial and Applied Mathematics, Statistics and Probability 
Proposer(s)  Dr Keith Hopcraft, Dr Simon Preston 
Description  Networks – systems of interconnected elements – form structures through which information or matter is conveyed from one part of an entity to another, and between autonomous units. The form, function and evolution of such systems are affected by interactions between their constituent parts, and perturbations from an external environment. The challenge in all application areas is to model effectively these interactions which occur on different spatial and timescales, and to discover how i) the microdynamics of the components influence the evolutionary structure of the network, and ii) the network is affected by the external environment(s) in which it is embedded. Activity in nonevolving networks is well characterized as having diffusive properties if the network is isolated from the outside world, or ballistic qualities if influenced by the external environment. However, the robustness of these characteristics in evolving networks is not as well understood. The projects will investigate the circumstances in which memory can affect the structural evolution of a network and its consequent ability to function. Agents in a network will be assigned an adaptive profile of goal and costrelated criteria that govern their response to ambitions and stimuli. An agent then has a memory of its past behaviour and can thereby form a strategy for future actions and reactions. This presents an ability to generate ‘lumpiness’ or granularity in a network’s spatial structure and ‘burstiness’ in its time evolution, and these will affect its ability to react effectively to external shocks to the system. The ability of externally introduced activists to change a network’s structure and function  or agonists to test its resilience to attack  will be investigated using the models. The project will use data of real agent’s behaviour. 
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Title  Fluctuation Driven Network Evolution 

Group(s)  Industrial and Applied Mathematics, Statistics and Probability 
Proposer(s)  Dr Keith Hopcraft, Dr Simon Preston 
Description  A network’s growth and reorganisation affects its functioning and is contingent upon the relative timescales of the dynamics that occur on it. Dynamical timescales that are short compared with those characterizing the network’s evolution enable collectives to form since each element remains connected with others in spite of external or internally generated ‘shocks’ or fluctuations. This can lead to manifestations such as synchronicity or epidemics. When the network topology and dynamics evolve on similar timescales, a ‘plastic’ state can emerge where form and function become entwined. The interplay between fluctuation, form and function will be investigated with an aim to disentangle the effects of structural change from other dynamics and identify robust characteristics. 
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Title  Excitability in biology  the role of noisy thresholds 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Ruediger Thul, Prof Stephen Coombes 
Description  Excitability is ubiquitous in biology. Two important examples are the membrane potential of neurons or the dynamics of the intracellular calcium concentration. What characterises excitable systems is the presence of a threshold. For instance, neurons only fire when the membrane potential crosses a critical value. Importantly, the dynamics of excitable systems is often driven by fluctuations such as the opening of ion channels or the binding of hormones to a receptor. A mathematically and computationally appealing approach is to represent this biological noise by a random excitability threshold. This concept has already provided great insights into the dynamics of neurons that process sounds [1]. In this project, we will investigate the role of correlations of the noisy threshold in shaping cellular responses. Our applications will come from neuroscience in the form of single cell and neural field models as well as from cell signalling when we investigate travelling calcium waves. This will help us to understand the emergence of unusual firing patterns in the brain as well as of the wide variety of travelling calcium waves observed in numerous cell types. 
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Title  Spatiotemporal patterns with piecewiselinear regulatory networks 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Etienne Farcot 
Description  A number of fascinating and important biological processes involve 
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Title  Spine morphogenesis and plasticity 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes, Dr Ruediger Thul 
Description  Mathematical Neuroscience is increasingly being recognised as a powerful tool to complement neurobiology to understand aspects of the human central nervous system. The research activity in our group is concerned with developing a sound mathematical description of subcellular processes in synapses and dendritic trees. In particular we are interested in models of dendritic spines [1], which are typically the synaptic contact point for excitatory synapses. Previous work in our group has focused on voltage dynamics of spineheads [2]. We are now keen to broaden the scope of this work to include developmental models for spine growth and maintenance, as well as models for synaptic plasticity [3]. Aberrations in spine morphology and density are well known to underly certain brain disorders, including Fragile X syndrome (which can lead to attention deficit and developmental delay) and depression [4]. Computational modelling is an ideal method to do insilico studies of drug treatments for brain disorders, by modelling their action on spine development and plasticity. This is an important complementary tool for drug discovery in an area which is struggling to make headway with classical experimental pharmaceutical tools. The mathematical tools relevant for this project will be drawn from dynamical systems theory, biophysical modelling, statistical physics, and scientific computation. 
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Title  Rare event modelling for the progression of cancer 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Richard Graham, Prof Markus Owen 
Description  Purpose This project will apply cuttingedge mathematical modelling techniques to solve computational and modelling issues in predicting the evolution of cancerous tumours. The project will combine rare event modelling from the physical sciences and cellularlevel models from mathematical biology. The aim is to produce new cancer models with improved biological detail that can be solved on clinically relevant timescales, which can be decades.
Background A widespread problem in treating cancer is to distinguish indolent (benign) tumours from metastaticcapable primary tumours (tumours that can spread to other parts of the body). Although therapies for metastatic disease exist, metastatic disease is a significant cause of death in cancer patients. This problem can lead to misdiagnosis, unnecessary treatment and a lack of clarity on which treatments are most effective.
A predictive mathematical model of cancer development could assist with the above issues. However, as the progression of cancer to metastasis is a rare event, in a direct simulation, virtually all of the computational time is consumed in simulating the quasistable behaviour of the indolent tumour, revealing no information about progression. This generic problem of rare events is common in the physical sciences, where modern techniques have enabled rare events to be simulated and understood. This project will extend these techniques to cancer modelling. The project will build on a stateoftheart spatiotemporal cancer model, which models individual cancer cells in a host tissue, vascular networks and angiogenesis. In this model cells can divide, migrate or die, in response to their microenvironment of cell crowding and cell signalling. To this framework the project will add transitions between cell types, driven by random mutation events and intravasation events.
The project will use a rare event algorithm, forward flux sampling (FFS), to create a statistical map of the transition from indolent cancer to metastatic cancer. In a typical rare event transition the system spends the overwhelming majority of the time close to the start. Consequently, the sampling of the trajectory space is very uneven. Thus, despite a very long simulation the statistical resolution of the mechanism and crossing rate are very poor. FFS solves this problem by dividing the phase space into a series of interfaces that represent sequential advancement towards the rare event. The algorithm logs forward crossings of these interfaces and a series of trajectories are begun at these crossing points. This produces a far more even sampling of the trajectory space and so better statistics of the whole mechanism from a shorter simulation.

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Title  Stochastic Neural Network Modelling 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes, Dr Ruediger Thul 
Description  Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrateandfire model, which is described by a nonsmooth dynamical system with a threshold [1]. It has recently been shown [2] that one way to model the variability of neuronal firing is to introduce noise at the threshold level. This project will develop the analysis of networks of synaptically coupled noisy neurons. Importantly it will go beyond standard phase oscillator approaches to treat strong coupling and nonGaussian noise. One of the main mathematical challenges will be to extend the MasterStability framework for networks of deterministic limit cycle oscillators to the noisy nonsmooth case that is relevant to neural modelling. This work will determine the effect of network dynamics and topology on synchronisation, with potential application to psychiatric and neurological disorders. These are increasingly being understood as disruptions of optimal integration of mental processes subserved by distributed brain networks [3]. 
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Title  Cell signalling 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof John King 
Description  Cell signalling effects have crucial roles to play in a vast range of biological processes, such as in controlling the virulence of bacterial infections or in determining the efficacy of treatments of many diseases. Moreover, they operate over a wide range of scales, from subcellular (e.g. in determining how a particular drug affects a specific type of cell) to organ or population (such as through the quorum sensing systems by which many bacteria determine whether or not to become virulent). There is therefore an urgent need to gain greater quantitative understanding of these highly complex systems, which are wellsuited to mathematical study. Experience with the study of nonlinear dynamical systems would provide helpful background for such a project. 
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Title  Modelling DNA Chain Dynamics 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Jonathan Wattis 
Description  Whilst the dynamics of the DNA double helix are extremely complicated, a number of welldefined modes of vibration, such as twisting and bending, have been identified. At present the only accurate models of DNA dynamics involve largescale simulations of molecular dynamics. Such approaches suffer two major drawbacks: they are only able to simulate short strands of DNA and only for extremely short periods (nanoseconds). the aim of this project is to develop simpler models that describe vibrations of the DNA double helix. The resulting systems of equations will be used to simulate the dynamics of longer chains of DNA over long timescales and, hence, allow largerscale dynamics, such as the unzipping of the double helix, to be studied. 
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Title  Multiscale modelling of vascularised tissue 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Markus Owen 
Description  Most human tissues are perfused by an evolving network of blood vessels which supply nutrients to (and remove waste products from) the cells. The growth of this network (via vasculogenesis and angiogenesis) is crucial for normal embryonic and postnatal development, and its maintenance is essential throughout our lives (e.g. wound healing requires the repair of damaged vessels). However, abnormal remodelling of the vasculature is associated with several pathological conditions including diabetic retinopathy, rheumatoid arthritis and tumour growth. The phenomena underlying tissue vascularisation operate over a wide range of time and length scales. These features include blood flow in the existing vascular network, transport within the tissue of bloodborne nutrients, cell division and death, and the expression by cells of growth factors such as VEGF, a potent angiogenic factor. We have developed a multiscale model framework for studying such systems, based on a hybrid cellular automaton which couples cellular and subcellular dynamics with tissuelevel features such as blood flow and the transport of growth factors. This project will extend and specialise our existing model to focus on particular applications in one of the following areas: wound healing, retinal angiogenesis, placental development, and corpus luteum growth. This work would require a significant element of modelling, numerical simulation and computer programming. 
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Title  Selfsimilarity in a nanoscale islandgrowth 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Jonathan Wattis 
Description  Molecular Beam Epitaxy is a process by which single atoms are slowly deposited on a surface. These atoms diffuse around the surface until they collide with a cluster or another atom and become part of a cluster. Clusters remain stationary. The distribution of cluster sizes can be measured, and is observed to exhibit selfsimilarity. Various systems of equations have been proposed to explain the scaling behaviour observed. The purpose of this project is to analyse the systems of differential equations to verify the scalings laws observed and predict the shape of the sizedistribution. The relationship of equations with other models of deposition, such as reactions on catalytic surfaces and polymer adsorption onto DNA, will also be explored. 
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Title  Sequential adsorption processes 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Jonathan Wattis 
Description  The random deposition of particles onto a surface is a process which arises in many subject areas, and determining its efficiency in terms of the coverage attained is a difficult problem. In onedimension the problem can be viewed as how many cars can be parked along a road of a certain length; this problem is similar to a problem in administering gene therapy in which polymers need to be designed to package and deliver DNA into cells. Here one wishes to know the coverage obtained when one uses a variety of polymer lengths to bind to strands of DNA. The project will involve the solution of recurrence relations, and differential equations, by a mixture of asymptotic techniques and stochastic simulations. 
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Title  Robustness of biochemical network dynamics with respect to mathematical representation 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Etienne Farcot 
Description  In the recent years, a lot of multidisciplinary efforts have been 
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Title  Neurocomputational models of hippocampusdependent place learning and navigation 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  This project will be based at the University of Nottingham in the School of Mathematical Sciences and the School of Psychology. 
Relevant Publications 

Other information  Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics (or other highly mathematical field such as Physics or Chemistry), preferably at MMath/MSc level, or an equivalent overseas degree (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). Apply: This studentship is available to start from September 2017 and remain open until it is filled. To apply please visit the University Of Nottingham application page: http://www.nottingham.ac.uk/pgstudy/apply/applyonline.aspx Funding NotesSummary: UK/EU students  Tuition Fees paid, and full Stipend at the RCUK rate, which is £14,296 per annum for 2016/17. There will also be some support available for you to claim for limited conference attendance. The scholarship length will be 3 or 3.5, depending on the qualifications and training needs of the successful applicant. 
Title  Spirals and autosoliton scattering: interface analysis in a neural field model 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes, Dr Daniele Avitabile 
Description  Neural field models describe the coarse grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in 2D, where they are well known to generate rich patterns of spatiotemporal activity. Typical patterns include localised solutions in the form of travelling spots as well as spiral waves [1]. These patterns are naturally defined by the interface between low and high states of neural activity. This project will derive the dimensionally reduced equations of motion for such interfaces from the full nonlinear integrodifferential equation defining the neural field. Numerical codes for the evolution of the interface will be developed, and embedded in a continuation framework for performing a systematic bifurcation analysis. Weakly nonlinear theory will be developed to understand the scattering of multiple spots that behave as autosolitons, whilst strong scattering solutions will be investigated using the scattor theory that has previously been developed for multicomponent reaction diffusion systems [2]. 
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Other information  S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9 
Title  Modelling signal processing and sexual recognition in mosquitoes: neural computations in insect hearing systems 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Daniele Avitabile, Prof Stephen Coombes 
Description  Insects have evolved diverse and delicate morphological structures in order to When a sound wave reaches the head of a mosquito, the antenna oscillates under the Recent studies have shown that mosquitoes of either sex use both their antenna and Even though some models of mosquitoes hearing systems have been proposed in the past, 
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Title  Nonsmooth dynamical systems: from nodes to networks 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes, Dr Ruediger Thul 
Description  There is a growing appreciation in the applied mathematics community that many real world systems can be described by nonsmooth dynamical systems. This is especially true of impacting mechanical systems or systems with switches [1]. The latter are ubiquitous in fields ranging from electrical engineering to biology. In a neuroscience context nonsmooth models now pervade the field, with exemplars being low dimensional piecewise linear models of excitable tissue, integrateandfire neurons, and the Heaviside nonlinearity invoked in neural mass models of cortical populations. Despite the relevance and preponderance of such models their mathematical analysis lags behind that of their smooth counterparts. This PhD project will redress this balance, by translating recent advances from nonsmooth dynamical systems to neuroscience as well as developing new approaches. The initial phase of the project will consider the periodic forcing of a nonsmooth node, as a precursor to exploring recurrent network dynamics. The Arnol'd tongue structure will be explored for modelocked states of oscillatory systems, as well as bifurcation diagrams for excitable systems. This will rely heavily on the construction of socalled saltation operators, to ensure the proper propagation of perturbations. Similarly, chaos will be studied using a suitable generalisation of the Liapunov exponent. The subsequent work will address emergent network dynamics, particularly in neural systems with chemical and electrical connections. Explicit analysis at the network level will build upon results at the single node level, with a focus on understanding patterns of synchrony, clustering, and more exotic chimera states [2]. This aspect of the project will first pursue the extension of the Master Stability framework for assessing stability of the synchronous state to treat nonsmooth systems with nonsmooth interactions [3]. The next stage will develop more general techniques, tapping into tools from computational group theory [4], to provide a more complete understanding of the spatiotemporal states that can be generated in realistic neural networks. 
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Title  Pattern formation in biological neural networks with rebound currents 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  Waves and patterns in the brain are well known to subserve natural computation. In the case of spatial navigation the geometric firing fields of grid cells is a classic example. Grid cells fire at the nodes of a hexagonal lattice tiling the environment. As an animal approaches the centre of a grid cell firing field, their spiking output increases in frequency. Interestingly the spacing of the hexagonal lattice can range from centimetres to metres and is thought to underly the brain's internal positioning system. The mechanism for controlling this global spatial scale is linked to a local property of neurons within an inhibitory coupled population, namely rebound firing. This arises through the activation of hyperpolarisationactivated channels. For the case of grid cells in the medial enthorinal cortex this gives rise to a socalled I_h current. Many other cells types also utilise rebound currents for firing, and in particular thalamocortical relay cells do so via slow Ttype calcium channels (the I_T current). This gives rise to saltatory lurching waves in thalamic slices. Both of these examples show that rebound currents can contribute significantly to important spatiotemporal brain dynamics. This project will investigate such important phenomenon from a mathematical perspective. One of the most successful approaches to modelling a spiking neuron involves using an integrateandfire process. This couples an ODE model with a reset rule for generating firing events. Almost by definition this precludes analysis using traditional approaches from the theory of smooth dynamical systems. This mathematical challenge is compounded at the network level when recognising that synaptic currents that mediate interactions between neurons are event driven rather than directly state dependent. Fortunately there is a growing appreciation that these mathematical biology challenges can benefit from a crossfertilisation of ideas with those being developed in the engineering community for impact oscillators and piecewise linear systems. This PhD will translate and develop mathematical methodologies from nonsmooth dynamical systems and apply them to two important neurobiological problems. The first being to analytically determine grid cell firing fields in a two dimensional spiking neural field model with an I_h rebound current, and the second to determine lurching wave speed and stability in a firing rate neural field model with an I_T rebound current. As well as mathematical techniques from nonsmooth dynamics, the project will involve large scale simulations of spiking networks, Evans functions for determining wave stability, and require an enthusiasm for learning about neuroscience. 
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Title  Mechanistic models of airway smooth muscle cells  application to asthma 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Bindi Brook 
Description  Lung inflammation and airway hyperresponsiveness (AHR) are hallmarks of asthma, but their interrelationship is unclear. Excessive shortening of airway smooth muscle (ASM) in response to bronchoconstrictors is likely an important determinant of AHR. Hypercontractility of ASM could stem from a change in the intrinsic properties of the muscle, or it could be due to extrinsic factors such as chronic exposure of the muscle to inflammatory mediators in the airways with the latter being a possible link between lung inflammation and AHR. The aim of this project will be to investigate the influence of chronic exposure to a contractile agonist on the forcegenerating capacity of ASM via a celllevel model of an ASM cell. Previous experimental studies have suggested that the muscle adapts to basal tone in response to application of agonist and is able to regain its contractile ability in response to a second stimulus over time. This is thought to be due to a transformation in the cytoskeletal components of the cell enabling it to bear force, thus freeing up subcellular contractile machinery to generate more force. Force adaptation in ASM as a consequence of prolonged exposure to the many spasmogens found in asthmatic airways could be a mechanism contributing to AHR seen in asthma. We will develop and use a cell model in an attempt to either confirm this hypothesis or determine other mechanisms that may give rise to the observed phenomenon of force adaptation. 
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Title  Synchronisation and propagation in human cortical networks 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Reuben O'Dea 
Description  Around 25% of the 50million epilepsy sufferers worldwide are not responsive to antiepileptic medication; improved understanding of this disorder has the potential to improve diagnosis, treatment and patient outcomes. The idea of modelling the brain as a complex network is now well established. However, the emergence of pathological brain states via the interaction of large interconnected neuronal populations remains poorly understood. Current theoretical study of epileptic seizures is flawed by dynamical simulation on inadequate network models, and by the absence of customised network measures that capture pathological connectivity patterns. 
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Title  Multiscale modelling of cell signalling and mechanics in tissue development and cancer 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof John King, Dr Reuben O'Dea 
Description  Cells respond to their physical environment through mechanotransduction, the translation of mechanical forces into biochemical signals; evoked cell phenotypic changes can lead to an altered cell microenvironment, creating a developmental feedback. Interplay between such mechanosentive pathways and other inter and intracellular signalling mechanisms determines cell differentiation and, ultimately, tissue development. Such developmental mechanisms have key relevance to the initiation and development of cancer, a disease of such inherent complexity (involving the interaction of a variety of processes across disparate spatiotemporal scales, from intracellular signalling cascades to tissuelevel mechanics) that, despite a wealth of theoretical and experimental studies, it remains a leading cause of mortality and morbidity: in the UK, more than one in three people will develop some form of cancer. There is therefore an urgent need to gain greater quantitative understanding of these highly complex systems, which are wellsuited to mathematical study. 
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Title  Patterns of synchrony in discrete models of gene networks 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Etienne Farcot 
Description  One of the greatest challenges of biology is to decipher the relation between genotype and 
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Title  Cell cycle desynchronization in growing tissues 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Etienne Farcot 
Description  A very general phenomenon is the fact that coupled oscillators tend to naturally 
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Title  Bottomup development of multiscale models of airway remodelling in asthma: from cell to tissue. 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Bindi Brook, Dr Reuben O'Dea 
Description  Airway remodelling in asthma has until recently been associated almost exclusively with inflammation over long timescales. Current experimental evidence suggests that bronchoconstriction (as a result of airway smooth muscle contraction) itself triggers activation of proremodelling growth factors that causes airway smooth muscle growth over much shorter timescales. This project will involve the coupling of subcellular mechanotransduction signalling pathways to biomechanical models of airway smooth muscle cells and extracellular matrix proteins with the aim of developing a tissuelevel biomechanical description of the resultant growth in airway smooth muscle. The mechanotransduction pathways and biomechanics of airway smooth muscle contraction are extremely complex. The cytoskeleton and contractile machinery within the cell and ECM proteins surrounding it are thought to rearrange dynamically (order of seconds). The cell is thought to adapt its length (over 10s of seconds). To account for all these processes from the bottomup and generate a tissue level description of biological growth will require the combination of agentbased models to biomechanical models governed by PDEs. The challenge will be to come up with suitably reduced models with elegant mathematical descriptions that are still able to reproduce observed experimental data on cell and tissue scales, as well as the different timescales present. While this study will be aimed specifically at airway remodelling, the methodology developed will have application in multiscale models of vascular remodelling and tissue growth in artificially engineered tissues. Initially models will be informed by data from ongoing experiments in Dr Amanda Tatler's lab in Respiratory Medicine but there will also be the opportunity to design new experiments based on model results. 
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Title  From molecular dynamics to intracellular calcium waves 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Ruediger Thul, Prof Stephen Coombes 
Description  Intracellular calcium waves are at the centre of a multitude of cellular processes. Examples include the generation of a heartbeat or the beginning of life when egg cells are fertilised. A key driver of intracellular calcium waves are ion channels, which are large molecules that control the passage of calcium ions across a cell. Importantly, these ion channels display stochastic behaviour such as random opening and closing. A key challenge in mathematical physiology and computational biology is to link this molecular stochasticity to travelling calcium waves. In this project, we will use a firediffusefire (FDF) model of intracellular calcium waves and couple it to Markov chains of ion channels. Traditionally, simulating large numbers of Markov chains is computationally expensive. Our goal is to derive an effective description for the stochastic ion channel dynamics. This will allow us to incorporate the molecular fluctuations from the ion channels into the FDF model without having to evolve Markov chains. This will put us in an ideal position to answer current questions in cardiac dynamics (How does an irregular heart beat emerge, leading to a potentially lifethreatening condition?) as well as to elucidate fundamental concepts in cell signalling.

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Title  Waves on a folded brain 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Daniele Avitabile, Prof Stephen Coombes 
Description  The human brain has a wonderfully folded cortex with regions of both negative and positive curvature at gyri and sulci respectively. As the state of the brain changes waves of electrical activity spread and scatter through this complicated surface geometry. This project will focus on the mathematical modelling of realistic cortical tissue and the analysis of wave propagation and scattering using techniques from dynamical systems theory and scientific computation. 
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Title  Modelling macrophage extravasation and phenotype selection 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Prof Markus Owen 
Description  Macrophages are a type of white blood cell, a vital component of the immune system, and play a complex role in tumour growth and other diseases. Macrophage precursors, called monocytes, are produced in the bone marrow and enter the blood, before leaving the bloodstream (extravasating). Monocyte extravasation requires adhesion to, and active movement through, the blood vessel wall, both of which are highly regulated processes. Once in the tissue, monocytes begin to differentiate into macrophages, and it has become clear that the tissue microenvironment is a crucial determinant of macrophage function [1]. A spectrum of phenotypes have been identified: at one end, macrophages produce a variety of signals that are beneficial to a tumour, including those that promote the formation of new blood vessels and suppress inflammation. At the other end of the scale, inflammation is promoted and appropriately stimulated macrophages can kill tumour cells. 
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Title  Next generation neural field models on spherical domains 

Group(s)  Mathematical Medicine and Biology 
Proposer(s)  Dr Rachel Nicks 
Description  The number of neurons in the brain is immense (of the order of 100 billion). A popular approach to modelling such cortical systems is to use neural field models which are mathematically tractable and which capture the large scale dynamics of neural tissue without the need for detailed modelling of individual neurons. Neural field models have been used to interpret EEG and brain imaging data as well as to investigate phenomena such as hallucinogenic patterns, shortterm (working) memory and binocular rivalry. A typical formulation of a neural field equation is an integrodifferential equation for the evolution of the activity of populations of neurons within a given domain. Neural field models are nonlinear spatially extended pattern forming systems. That is, they can display dynamic behaviour including spatially and temporally periodic patterns beyond a Turing instability in addition to localised patterns of activity. The majority of research on neural field models has been restricted to the line or planar domains, however the cortical white matter system is topologically close to a sphere. It is relevant to study neural field models as pattern forming systems on spherical domains, particularly as the periodic boundary conditions allow for natural generation (via interference) of the standing waves observed in EEG signals. This project will build on recent developments in neural field theory, focusing in particular on extending to spherical geometry the neural field equations arising from “Next generation neural mass models” (which incorporate a description of the evolution of synchrony within the system). Techniques from dynamical systems theory, including linear stability analysis, weakly nonlinear analysis, symmetric bifurcation theory and numerical simulation will be used to consider the global and local patterns of activity that can arise in these models. 
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Title  Exploiting network symmetries for analysis of dynamics on neural networks 

Group(s)  Mathematical Medicine and Biology, Industrial and Applied Mathematics 
Proposer(s)  Dr Rachel Nicks, Prof Stephen Coombes, Dr Paul Matthews 
Description  Networks of interacting dynamical systems occur in a huge variety of applications including gene regulation networks, food webs, power networks and neural networks where the interacting units can be individual neurons or brain centres. The challenge is to understand how emergent network dynamics results from the interplay between local dynamics (the behaviour of each unit on its own), and the nature and structure of the interactions between the units. Recent work has revealed that real complex networks can exhibit a large number of symmetries. Network symmetries can be used to catalogue the possible patterns of synchrony which could be present in the network dynamics, however which of these exist and are stable depends on the local dynamics and the nature of the interactions between units. Additionally, the more symmetry a network has the more possible patterns of synchrony it may possess. Computational group theory can be used to automate the process of identifying the spatial symmetries of synchrony patterns resulting in a catalogue of possible network cluster states. This project will extend current methods for analysing dynamics on networks of (neural) oscillators through automating the process of determining possible phase relations between oscillators in large networks in addition to spatial symmetries. This will be used to investigate dynamics on coupled networks of simplified (phaseamplitude reduced or piecewiselinear) neuron and neural population models. We will also consider the effect on the network dynamics of introducing delays in the coupling between oscillators which will give a more realistic representation of interactions in real world networks. 
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Title  Analysing and interpreting neuroimaging data using mathematical frameworks for network dynamics 

Group(s)  Mathematical Medicine and Biology, Mathematical Medicine and Biology 
Proposer(s)  Prof Stephen Coombes 
Description  Modern noninvasive probes of human brain activity, such as magnetoencephalography (MEG), give high temporal resolution and increasingly improved spatial resolution. With such a detailed picture of the workings of the brain, it becomes possible to use mathematical modelling to establish increasingly complete mechanistic theories of spatiotemporal neuroimaging signals. There is an ever expanding toolkit of mathematical techniques for addressing the dynamics of oscillatory neural networks allowing for the analysis of the interplay between local population dynamics and structural network connectivity in shaping emergent spatial functional connectivity patterns. This project will be primarily mathematical in nature, making use of notions from nonlinear dynamical systems and network theory, such as coupledoscillator theory and phaseamplitude network dynamics. Using experimental data and data from the output of dynamical systems on networks with appropriate connectivities, we will obtain insights on structural connectivity (the underlying network) versus functional connectivity (constructed from similarity of real time series or from timeseries output of oscillator models on networks). The project will focus in particular on developing techniques for the analysis of dynamics on “multilayer networks” to better understand functional connectivity within and between frequency bands of neural oscillations. This project will be in collaboration with Dr Matt Brookes from the Nottingham MEG group. 
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Title  Mathematical modelling of macromolecular capillary permeability 

Group(s)  Mathematical Medicine and Biology, Scientific Computation 
Proposer(s)  Dr Reuben O'Dea, Dr Matthew Hubbard 
Description  The primary function of blood vessels is to transport molecules to tissues. In diseases such as cancer and diabetes this transport, particularly of large molecules such as albumin, can be an order of magnitude higher than normal. The project is to model transient flow of macromolecules across the vascular wall in physiology and pathology. With additional supervision from Dr Kenton Arkill and Professor David Bates (Medicine), the doctoral student will join a team that includes medical researchers, biophysicists and mathematicians acquiring structural and functional data. Detailed microscale models of vascular wall hydrodynamics and transport properties will be employed; in addition, powerful multiscale homogenisation techniques will be exploited that enable permeability and convection parameters on the nanoscale to be linked through the microscale into translatable information on the tissue scale. Computational simulations will be used to investigate and understand the model behaviour, including, for example, stochastic and multiphysics effects in the complex diffusionconvection nanoscale environment. The project will afford a great opportunity to form an information triangle where modelling outcomes will determine physiological experiments to feedback to the model. Furthermore, the primary results will inform medical researchers on potential molecular therapeutic targets. 
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Title  Optimising experiments for developing ion channel models 

Group(s)  Mathematical Medicine and Biology, Statistics and Probability 
Proposer(s)  Dr Gary Mirams, Dr Simon Preston 
Description  Background: in biological systems ion channel proteins sit in cell membranes and selectively allow the passage of particular types of ions, creating currents. Ion currents are important for many biological processes, for instance: regulating ionic concentrations within cells; passing signals (such as nerve impulses); or coordinating contraction of muscle (skeletal muscle and also the heart, diaphragm, gut, uterus etc.). Mathematical ion channel electrophysiology models have been used for thousands of studies since their development by Hodgkin & Huxley in 1952 [1], and are the basis for whole research fields, such as cardiac modelling and brain modelling [2]. It has been suggested that there are problems in identifying which set of equations is most appropriate as an ion channel model. Often it appears different structures and/or parameter values could fit the training data equally well, but they may make different predictions in new situations [3]. Eligibility/Entry Requirements: this PhD will suit a graduate with a 1st class degree in Mathematics (or other highly mathematical field such as Physics), ideally at the MMath/MSc level, or an equivalent overseas degree. Prior knowledge of biology is not essential. 
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Other information  Please see Gary Mirams' research homepage for more information. 
Title  Critical random matrix ensembles 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  In Random Matrix Theory (RMT) one deals with matrices whose entries are given by random variables. RMT has a great number of applications in physics, mathematics, engineering, finance etc. In this project, a particular class of random matrix ensembles  critical random matrix models will be studied. These models describe statistical properties of disordered systems at a point of the quantum phase transition. Using RMT one can compute various critical exponents, correlation functions and other physically important quantities. 
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Title  Models of Quantum Geometry 

Group(s)  Mathematical Physics 
Proposer(s)  Prof John Barrett 
Description  Noncommutative geometry is a generalisation of differential geometry where the "functions" on the space are not required to commute when multiplied together. This study is based on the approach to noncommutative geometry pioneered by Alain Connes. It has a number of applications, the most spectacular being the discovery that the fields in the standard model of particle physics have the structure of a noncommutative geometry. This noncommutativity relates to the "internal space" i.e. a geometric structure at every point of spacetime, and reveals itself in the nonabelian gauge groups, the Higgs and their couplings to fermion fields. The new idea is to use the noncommutative geometry also for spacetime itself, which one hopes will eventually give a coherent explanation of the structure of spacetime at the Planck scale. There are a number of projects investigating aspects of these quantum geometry models and related mathematics. It also uses techniques from topology, algebra, category theory and geometry, as well as numerical computations. The motivation is to study models that include gravity, working towards solving the problem of quantum gravity, and to study implications for particle physics. For the latest information on this research, please see my homepage https://johnwbarrett.wordpress.com/ 
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Title  Hydrodynamic simulations of rotating black holes 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Silke Weinfurtner 
Description  We are currently carrying out an experiment to study the effects occurring around effective horizons in an analogue gravity system. In particular, the scientific goals are to explore superradiant scattering and the black hole evaporation process. To address this issue experimentally, we utilize the analogy between waves on the surface of a stationary draining fluid/superfluid flows and the behavior of classical and quantum field excitations in the vicinity of rotating black. This project will be based at the University of Nottingham at the School of Mathematical Sciences. The two external collaborators are Prof. Josef Niemela (ICTP, Trieste in Italy) and Prof. Stefano Liberati (SISSA, Trieste in Italy). The external consultant for the experiment is Prof. Bill Unruh, who will be a regular visitor. The PhD student will be involved in all aspects of the experiments theoretical as well experimental. We require an enthusiastic graduate with a 1st class degree in Mathematics/Physics/Engineering (in exceptional circumstances a 2(i) class degree can be considered), preferably of the MMath/MSc level. Candidates would need to be keen to work in an interdisciplinary environment and interested in learning about quantum field theory in curved spacetimes, fluid dynamics, analogue gravity, and experimental techniques such as flow visualisation (i.g. Particle Imaging or Laser Doppler Velocimetry) and surface measurements (i.g. profilometry methods). 
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Title  Gravity as a theory of connections 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Kirill Krasnov 
Description  General Relativity is normally described as a dynamical theory of spacetime metrics. However, GR is a rather complicated theory  think about the rather nontrivial exercise of deriving Schwarzschild solution, with its computation of Christoffel symbols, then the curvature tensor, then Ricci tensor. At the same time, it has been appreciated for a long time that one can simplify the GR formalism by using differential forms. Indeed, the exercise leading to Schwarzschild solution does become simpler if one uses tetrads and the spin connection instead of the metric and the affine connection. Also in 3 spacetime dimensions General Relativity is best thought of as a theory of flat Poincare connections, with the action describing the dynamics being that of ChernSimons theory. A point of view on 3D gravity as a theory of connections has been extremely successful both classically (in describing the space of all possible solutions of 3D GR) and quantum mechanically (in quantising the space of solutions and obtaining an explicit description of the arising Hilbert space). This PhD project will concern itself with developing a similar language for 4D GR. Thus, it turns out to be possible to describe 4D GR as a dynamical theory of connections rather than metrics. The metric appears as a derived notion, and is constructed in a certain way from the curvature of the connection. There are many possible projects within this general area of development. One can either explore how some concrete solutions of GR are obtained in this way, or study the quantum mechanics of gravity (i.e. perturbative quantum gravity) in this language. The language of connections is simpler in many aspects than the usual metric formalism for GR, and the hope is that this simplicity will lead to qualitatively new understanding of what gravity really is. 
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Title  Acceleration, black holes and thermality in quantum field theory 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Jorma Louko 
Description  Hawking's 1974 prediction of black hole radiation continues to inspire the search for novel quantum phenomena associated with global properties of spacetime and with motion of observers in spacetime, as well as the search for laboratory systems that exhibit similar phenomena. At a fundamental level, a study of these phenomena provides guidance for developing theories of the quantum mechanical structure of spacetime, including the puzzle of the microphysical origin of black hole entropy. At a more practical level, a theoretical control of the phenomena may have applications in quantum information processing in situations where gravity and relative motion are significant, such as quantum communication via satellites. Specific areas for a PhD project could include: Model particle detectors as a tool for probing nonstationary quantum phenomena in spacetime, such as the onset of Hawking radiation during gravitational collapse. See arXiv:1406.2574 and references therein. Black hole structure behind the horizons as revealed by quantum field observations outside the horizons. See arXiv:1001.0124 and references therein. Quantum fields in accelerated cavities. See arXiv:1210.6772 and arXiv:1411.2948 and references therein. 
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Title  Scattering in disordered systems with absorption: beyond the universality 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  The study of wave scattering in quantum systems with disorder or underlying classical chaotic dynamics is essential for an understanding of many different physical systems. These include, for example, light propagation in random media, transport of electrons in quantum dots, transmission of microwaves in waveguides and cavities, and many others. An important feature of any real experiment on scattering is the presence of absorption. As the result, not all the incoming flux is either reflected or transmitted through system, but part of it is irreversibly lost in the environment. In recent years, considerable progress has been made in the study of scattering in disordered or chaotic quantum systems in the presence of absorption, see e.g Fyodorov, Savin & Sommers, (2005). However almost all results known so far are restricted by the so called "universal limit" described by the conventional Random Matrix Theory. The idea of the suggested project is to go beyond the "universal limit" and to investigate properties of the scattering matrix in lossy systems for the case of a quasionedimensional disordered waveguide. This model describes for example electron dynamics in a thick disordered wire or propagation of light or microwave radiation in a slab geometry. There are two recent advances making an analytical treatment of this problem feasible. The first one is a discovery of a kind of fluctuation dissipation relation between the properties of an open system in the presence of absorption and a certain correlation function of its closed counterpart. This can be exploited, for example, to relate statistics of scattering characteristics to eigenfunction fluctuations in closed systems (Ossipov & Fyodorov, 2005). The second one is a new analytical insight into properties of quasionedimensional disordered conductors, see Skvortsov & Ostrovsky, (2006). 
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Title  Quantum learning for large dimensional quantum systems 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Madalin Guta 
Description  This project stems from the ongoing collaboration with Theo Kypraios and Ian Dryden (Statistics group, Nottingham), Cristina Butucea (Univesite Paris Est) and Thomas Monz and Philipp Schindler (Rainer Blatt trapped ions experimental group, University of Innsbruck). The aim is to explore and investigate new methods for learning quantum states of large dimensional quantum systems. The efficient statistical reconstruction of such states is a crucial enabling tool for current quantum engineering experiments in which multiple qubits can be controlled and prepared in exotic entangled states. However, standard estimation methods such as maximum likelihood become practically unfeasible for systems of merely 10 qubits, due to the exponential growth of the Hilbert space with the number of qubits. Therefore new methods are needed which are able to "learn" the structure of the quantum state by making use of prior information encoded in physically relevant low dimensioanal models. 
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Other information  Click here to find more information on this topic and some illustrations of different types of estimators. For more about my reasearch interests you can visit my homepage. 
Title  Feedback control of quantum dynamical systems and applications in metrology 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Madalin Guta 
Description 
The ability to manipulate, control and measure quantum systems is a central issue in Quantum Technology applications such as quantum computation, cryptography, and high precision metrology [1]. Most realistic systems interact with an environment and it is important to understand how this affects the performance of quantum protocols and how it can be used to improve it. The inputoutput theory of quantum open systems [2] offers a clear conceptual understanding of quantum dynamical systems and continuoustime measurements, and has been used extensively at interpreting experimental data in quantum optics. Mathematically, we deal with an extension of the classical filtering theory used in control engineering at estimating an unobservable signal of interest from some available noisy data [3]. This projects aims at investigating the identification and control of quantum dynamical systems in the framework of the inputoutput formalism. As an example, consider a quantum system (atom) interacting with an incoming "quantum noise" (electromagnetic field); the output fields (emitted photons) emerging from the interaction can be measured, in order to learn about the system's dynamical parameters (e.g. its hamiltonian). The goal is to find optimal system identification strategies which may involve input state preparation, output measurement design, and quantum feedback control. An interesting related question is to understand the informationdisturbance tradeoff which in the context of quantum dynamical systems becomes identificationcontrol tradeoff. The first steps in this direction were made in [4] which introduce the concept of asymptotic quantum Fisher information for "nonlinear" quantum Markov processes, and [5] which investigates system identification for linear quantum systems, using transfer functions techniques from control theory. A furhter goal is to develop genearal Central Limit theory for quantum output processes as a probablistic underpinning of the asymptotic estimation theory. Another direction is the recently found connection between dynamical phase transitions in manybody open systems and high precision metrology for dynamical parameters (see arXiv:1411.3914).

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Other information  Click here to find more information on this topic and some illustrations of different types of estimators. For more about my reasearch interests you can visit my homepage. 
Title  Quantum correlations in manybody systems 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Gerardo Adesso 
Description  The behaviour of physical systems at the microscopic scale obeys the laws of quantum mechanics. Quantum systems can share a form of quantum correlations known as entanglement, which is nowadays acknowledged as a resource for enhanced information processing. However, there are more general types of quantum correlations, beyond entanglement, that can be present in separable quantum states. This project deals with the characterisation of the nonclassicality of correlations in multipartite quantum systems. Interesting aspects of this project are the elucidation of the relationship between these more general forms of quantum correlations, as quantified e.g. by the "quantum discord", and entanglement in mixed multipartite quantum states. Another theme will be the identification of experimentally friendly schemes to engineer quantum correlations, and detect them in practical demonstrations, as well as rigorously assessing the usefulness of quantum correlations beyond entanglement as resources for nextgeneration quantum information protocols. 
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Title  Quantum aspects of frustration in spin lattices 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Gerardo Adesso 
Description  Recently, a number of tools developed in the framework of quantum information theory have proven useful to tackle founding open questions in condensed matter physics, such as the characterization of quantum phase transitions and the scaling of correlations at critical points. Our contribution to the field dealt with a method, based on quantum informational concepts, to identify analytically factorized (unentangled) ground states in manybody spin models, which constitute an exact solution to generally nonexactly solvable models for specific values of the Hamiltonian parameters. In presence of frustration, ground state factorization is suppressed. Therefore the factorizability provides a qualitative handle on the degree of quantum frustration. This project will build on these premises and will seek for genuine signatures of quantum versus classical frustration in spin systems, a topic of great relevance for condensed matter. Frustrated quantum models may play a key role for hightemperature superconductivity and for certain biological processes. The relationship between frustration, disorder and entanglement is yet largely unexplored. 
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Title  Quantum information with nonGaussian states 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Gerardo Adesso 
Description  Quantum information with continuous variable systems is a burgeoning area of research which has recorded astonishing theoretical and experimental successes, mainly thanks to the manipulation and exploitation of Gaussian states of light and matter. However, quite recently a number of tasks have been individuated which can not be perfectly implemented by using Gaussian states and operations only, and another set of processes is being explored where some nonGaussianity has been recognised as an advantageous ingredient to sharply improve performances of quantum communication. In this project the student will investigate the limitations of the Gaussian scenario in different contexts such as quantum communication, computation and estimation and, more generally, quantum technology. This is paralleled by recent progresses in the experimental generation of nonGaussian states, which further motivate their application in quantum information science. Special emphasis will be put on devising efficient methods to quantify the entanglement in selected classes of nonGaussian states, using techniques whose complexity is not exceedingly large compared to the usual tools (quadrature measurements, homodyne detection) which are effective for Gaussian states. 
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Title  Developing new relativistic quantum technologies 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Ivette Fuentes 
Description  Relativistic quantum information is an emerging field which studies how to process information using quantum systems taking into account the relativistic nature of spacetime. The main aim of this PhD project is to find ways to exploit relativity to improve quantum information tasks such as teleportation and to develop new relativistic quantum technologies. Moving cavities and UnruhDewitt type detectors promise to be suitable systems for quantum information processing [1,2]. Interestingly, motion and gravity have observable effects on the quantum properties of these systems [2,3]. In this project we will find ways to implement quantum information protocols using localized systems such as cavities and detectors. We will focus on understanding how the protocols are affected by taking into account the nontrivial structure of spacetime. We will look for new protocols which exploit not only quantum but also relativistic resources for example, the nonlocal quantum correlations present in relativistic quantum fields. 
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Title  Homotopical algebra and quantum gauge theories 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Schenkel 
Description  A problem which frequently arises in mathematics is that one would like to treat certain classes of maps as if they were isomorphisms, even though they are not in the strict sense. Examples are homotopy equivalences between topological spaces  remember the famous doughnut and coffee mug  or quasiisomorphisms between chain complexes of modules. Homotopical algebra was introduced by Quillen in the late 1960s as an abstract framework to address these and related problems. Since then it has found many important applications in algebra, topology, geometry and also in mathematical physics. In quantum field theory, homotopical algebra turns out to be essential as soon as one deals with models involving gauge symmetries. Recent results showed that quantum gauge theories do not satisfy the standard axioms of algebraic quantum field theory, hence they are not quantum field theories in this strict sense. To solve these problems, we initiated the development of a novel and promising approach called “homotopical algebraic quantum field theory”, which combines the basic concepts of algebraic quantum field theory with homotopical algebra and which is expected to be a suitable mathematical framework for quantum gauge theories. Specific areas for a PhD project could include: 1.) Examples of homotopical algebraic quantum field theory. This project is about investigating the symplectic geometry of solution "spaces" of gauge theories, which are generalised spaces called stacks, and developing new techniques for their quantisation. An important part will be to analyse localtoglobal properties (called descent) of the resulting quantum gauge theories. 2.) Operadic structure of homotopical algebraic quantum field theory. The algebraic operations in homotopical algebraic quantum field theory are expected to be captured in an abstract structure called a coloured operad. This project is about constructing this coloured operad and using it to obtain modelindependent results in homotopical algebraic quantum field theory. 
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Title  Manybody localization in quantum spin chains and Anderson localization 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  Properties of wave functions in manybody systems is very active topic of research in modern condensed matter theory. Quantum spin chains are very useful models for studying quantum manybody physics. They are known to exhibit complex physical behaviour such as quantum phase transitions. Recently, they have been studied intensively in the context of manybody localization. 
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Title  Entanglement of noninteracting fermions at criticality 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  Entanglement of the ground state of manyparticle systems has recently attracted a lot of attention. For noninteracting fermions, the ground state entanglement can be calculated from the eigenvalues of the correlation matrix of the single particle wavefunctions. For this reason, the nature of the single particle wavefunctions is crucially important for understanding of the entanglement properties of a manybody system. 
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Title  Gravity at all scales 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Thomas Sotiriou 
Description  Various projects are available on the interplay between any of the following areas: quantum gravity, alternative theories of gravity, strong gravity and black holes. 
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Title  Topological Resonances on Graphs 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  If a light wave in a resonator between two almost perfect mirrors shows resonance if the wavelength is commensurate with the distance between the two mirrors. If this condition is satisfied it will decay much slower than at other wavelengths which are not commensurate. This is one of the simplest mechanisms for a resonace in a wave system. There are other weill known mechanisms that rely on complexity and disorder. It has recently been observed that a netork of wire may have a further mechanism that leads to resonances. This mechanism relies on cycles in the network and leads to various signatures which cannot be explained using other wellknown mechanisms for resonances. In this project these signatures will be analysed in detail. 
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Title  Quantum Chaos in Combinatorial Graphs 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Graphs consist of V vertices connected by B bonds (or edges). They are used in many branches of science as simple models for complex structures. In mathematics and physics one is strongly interested in the eigenvalues of the V x V connectivity matrix C of a graph. The matrix element C_ij of the latter is defined to be the number of bonds that connect the i'th vertex to the j'th vertex. In this PhD project the statistical properties of the connectivity spectra in (generally large) graph structures will be analysed using methods known from quantum chaos. These methods have only recently been extended to combinatorial graphs (Smilansky, 2007) and allow to represent the density of states and similar spectral functions of a graph as a sum over periodic orbits. The same methods have been applied successfully to metric graphs and quantum systems in the semiclassical regime for more than two decades. 
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Title  Supersymmetric field theories on quantum graphs and their application to quantum chaos 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Quantum graphs are a paradigm model for quantum chaos. They consist of a system of wires along which waves can propagate. Many properties of the excitation spectrum and the spatial distribution of standing waves can be mapped exactly onto a supersymmetric field theory on the network. In a meanfield approximation one may derive various universal properties for large quantum graphs. In this project we will focus on deviations from universal behaviour for finite quantum graphs with the fieldtheoretic approach. 
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Title  Pseudoorbit expansions in quantum graphs and their application 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Quantum graphs are a paradigm model to understand and analyse the effect of complexity on wave propagation and excitations in a network of wires. They have also been used as a paradigm model to understand topics in quantum and wave chaos where the complexity has a different origin while the mathematical framework is to a large extent analogous. Many properties of the waves that propagate through such a network can be described in terms of trajectories of a point particle that propagates through the network. The ideas is to write a property of interest as a sum over amplitudes (complex numbers) connected to all possible trajectories of the point particle. These sums remain challenging objects for explicit evaluations. Recently a numer of advanced methods for their summation have been introduced. The latter are built on socalled pseudoorbits. In this project these methods will be develloped further and applied to questions related to quantum chaos and randommatrix theory. 
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Title  The tenfold way of symmetries in quantum mechanics. An approach using coupled spin operators. 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  About 50 years ago Wigner and Dyson proposed a threefold symmetry classification for quantum mechanical systems  these symmetry classes consisted of timereversal invariant systems with integer spin which can be described by real symmetric matrices, timereversal invariant systems with halfinteger spin which can be described by real quaternion matrices, and systems without any timereversal symmetry which are described by complex hermitian matrices. These three symmetry classes had their immediate application in the three classical Gaussian ensembles of randommatrix theory: the Gaussian orthogonal ensemble GOE, the Gaussian symplectic ensemble GSE, and the Gaussian unitary ensemble GUE. In the 1990's this classification was extended by adding charge conjugation symmetries  symmetries which relate the positive and negative part of a spectrum and which are described by anticommutators. 
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Title  Nonlinear waves in waveguide networks 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Many wave guides (such as optical fibres) show a Kerrtype effect that leads to nonlinear wave propagation. If th wave guides are coupled at junctions then there is an additional element of complexity due to the nontrivial connectivity of wave guides. In this project the impact of the structure and topology of the network on wave propagation will be studied starting from simple geometries such as a Yjunctions (three waveguides coupled at one junction), a star (many waveguides at one junction), or a lasso (a waveguide that forms a loop and is connected at one point to a second waveguide). 
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Title  The statistics of nodal sets in wavefunctions 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  If a membrane vibrates at one of its resonance frequencies there are certain parts of the membrane that remain still. These are called nodal points and the collection of nodal points forms the nodal set. Building on earlier work this project will look at the statistical properties of the nodal set  e.g. for 3dimensional waves the nodal set consists of a coillection of surfaces and one may ask questions about how the area of the nodal set is distributed for an ensemble of membranes or for an ensemble of different resonances of the same membrane. This project will involve a strong numerical component as wavefunctions of irregular membranes need to be found and analysed on the computer. Effective algorithms to find the area of the nodal set, or the number of domain in which the sign does not change (nodal domains) will need to be developed andimplemented. 
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Title  Coherent states, nonhermitian Quantum Mechanics and PTsymmetry 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Heisenberg's uncertainty principle states that momentum and position cannot be sharp at the same time because there is a lower bound for the product of the uncertaincies. Coherent states can be defined as the states that minimize the uncertainty  in this sense they are as close as quantum mechanics allows to describe a classical point particle. When a quantum system starts in a coherent states it's expectation values follow the classical equations of motion while the shape of the wave function often changes only very slowly. Coherent states are an important tool to understand the corresp[ondence between quantum and classical dynamics. In this project this correspondence will be analysed for a generalized quantum dynamics where the Hamilton operator is not required to be Hermitian. Such dynamics can arise in practice as an effective description for an open quantum system with eitehr decay or gain. Accordingly the energy eigenvalues may have an imaginary part that describes the loss or gain. Recently there have also be suggestions that nonhermitian Hamilton operators could play a fundamental role in quantum mechanics if the Hamilton operator remains symmetric with respect to a combined operatyion of parity P and time reversal T. Such PTsymmetric dynamics have a balance between gain and loss which can lead to real energy eigenvalues. Classical to quantum correspondence for such systems remains an open research topic and this project will aim at getting a clear understanding of the underlying classical dynamics using coherent states as the main tool. 
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Title  Geometry and analysis of Schubert varieties 

Group(s)  Number Theory and Geometry 
Proposer(s)  Dr Sergey Oblezin 
Description  Schubert varieties is a basic tool of classical algebraic and enumerative geometry. In modern mathematics these geometric object arise widely in representation theory, theory of automorphic forms and in harmonic analysis. In particular, it appears that the classical geometric structures can be naturally extended to infinitedimensional setting (loop groups and, more generally, KacMoody groups), and such generalizations provide new constructions in infinitedimensional geometry. Moreover, many of the arising constructions are supported by (hidden) symmetries and dualities of quantum (inverse) scattering theory and quantum integrability. Possible PhD projects will be devoted to extensive development of harmonic analysis on Schubert varieties with further applications to automorphic forms, arithmetic goemetry and number theory. 
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Title  Orthogonal polynomials in probability, representation theory and number theory 

Group(s)  Number Theory and Geometry 
Proposer(s)  Dr Sergey Oblezin 
Description  Orthogonal ensembles play a crucial role in many areas including random matrix theory, probability and harmonic analysis. In the recent decades a new striking connections with the theory of automorphic forms and number theory appeared. However, there is a definite lack of general results and implementations at present. Possible PhD projects will be aiming at developing these recent interactions among the group theory, harmonic analysis and number theory. 
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Title  Number theory in a broad context 

Group(s)  Number Theory and Geometry 
Proposer(s)  Prof Ivan Fesenko 
Description  Ivan Fesenko studies zeta functions in number theory using zeta integrals. These integrals are better to operate with than the zeta functions, they translate various properties of zeta functions into properties of adelic objects. This is a very powerful tool to understand and prove fundamental properties of zeta functions in number theory. In the case of elliptic curves over global fields, associated zeta functions are those of regular models of the curve, i.e. the zeta function of a two dimensional object. Most of the classical work has studied arithmetic of elliptic curves over number fields treating them as one dimensional objects and working with with generally noncommutative Galois groups over the number field, such as the one generated by all torsion points of the curve. The zeta integral gadget works with adelic objects associated to the two dimensional field of functions of the curve over a global field and using commutative Galois groups. The latter has already been investigated in two dimensional abelian class field theory and it is this theory which supplies adelic objects on which the zeta integral lives. For example, Fourier duality on adelic spaces associated to the model of the curve explains the functional equation of the zeta function (and of the Lfunction of the curve). The theory uses many parts of mathematics: class field theory, higher local fields and several different adelic structures, translation invariant measure and integration on higher local fields (arithmetic loop spaces), functional analysis and harmonic analysis on such large spaces, groups endowed with sequential topologies, parts of algebraic Ktheory, algebraic geometry. This results in a beautiful conceptual theory. There are many associated research problems and directions at various levels of difficulty and opportunities to discover new objects, structures and laws. 
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Title  Computational methods for elliptic curves and modular forms 

Group(s)  Number Theory and Geometry 
Proposer(s)  Dr Christian Wuthrich 
Description  Computational Number Theory is a fairly recent part of pure mathematics even if computations in number theory are a very old subject. But over the last few decades this has changed dramatically with the modern, powerful and cheap computers. In the area of explicit computations on elliptic curves, there are two subjects that underwent a great development recently: elliptic curves over finite fields (which are used for cryptography) and 'descent' methods on elliptic curves over global fields, such as the field of rational numbers. It is a difficult question for a given elliptic curve over a number field to decide if there are infinitely many solutions over this field, and if so, to determine the rank of the MordellWeil group. Currently, there are only two algorithms implemented for finding this rank, one is the descent method that goes back to Mordell, Selmer, Cassels,... and the other is based on the work of Gross, Zagier, Kolyvagin... using the link of elliptic curves to modular forms. While the first approach works very well over number fields of small degree, it becomes almost impossible to determine the rank of elliptic curves over number fields of larger degree. The second method unfortunately is not always applicable, especially the field must be either the field of rational numbers or a quadratic extension thereof. There is another way of exploiting the relation between elliptic curves and modular forms by using the padic theory of modular forms and the socalled Iwasawa theory for elliptic curves. Results by Kato, Urban, Skinner give us a completely new algorithm for computing the rank and other invariants of the elliptic curve, but not much of this has actually been implemented. Possible PhD projects could concern the further development of these new methods and their implementation. 
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Title  Variants of automorphic forms and their Lfunctions 

Group(s)  Number Theory and Geometry 
Proposer(s)  Dr Nikolaos Diamantis 
Description  Classical automorphic forms are a powerful tool for handling difficult number theoretic problems. They provide links between analytic, algebraic and geometric aspects of the study of arithmetic problems and, as such, they are at the heart of the major research programmes in Number Theory, e.g. Langlands programme. Crucial for these links are certain functions associated to automorphic forms, called Lfunctions, which are the subject of some of the most important conjectures of Mathematics. 
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Title  Foundations of adaptive finite element methods for PDEs 

Group(s)  Scientific Computation, Algebra and Analysis 
Proposer(s)  Dr Kris van der Zee 
Description  Foundations of adaptive finite element methods for PDEs Adaptive finite element methods allow the computation of solutions to partial differential equations (PDEs) in the most optimal manner that is possible. In particular, these methods require the least amount of degreesoffreedom to obtain a solution up to a desired accuracy! In recent years a theory has emerged that explains this behaviour. It relies on classical a posteriori error estimation, Banach contraction, and nonlinear approximation theory. Unfortunately, the theory so far applies only to specific model problems. Challenges for students: Depending on the interest of the student, several of these issues (or others) can be addressed. 
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Title  Multiscale models for growing tissues – simulation and analysis. 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Donald Brown 
Description  A fundamental barrier to advancing the understanding of biological tissue growth lies in its inherently multiscale nature: interactions between processes acting at disparate scales can profoundly influence the emergent dynamics. A unified description of such phenomena requires reconciling insight obtained by theoretical or experimental study at one scale with observations at another. For example, diseases that manifest at the organ scale often arise through the interaction of microscopic events at the cellular scale; moreover, the resulting macroscale changes influence the microscopic dynamics. [1] Effective equations governing an active poroelastic medium. J Collis, DL Brown, ME Hubbard, RD O'Dea Proceedings of the Royal Society A 473 (2198) 
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Title  Partitioneddomain concurrent multiscale modelling 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Kris van der Zee 
Description  Partitioneddomain concurrent multiscale modelling Multiscale modeling is an active area of research in all scientific disciplines. The main aim is to address problems involving phenomena at disparate length and/or time scales that span several orders of magnitude! An important multiscalemodeling type is known as partitioneddomain concurrent modelling. This type addresses problems that require a finescale model in only a small part of the domain, while a coarse model is employed in the remainder of the domain. By doing this, significant computational savings are obtained compared to a full finescale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem. Challenges for students: Depending on the interest of the student, several of these issues (or others) can be addressed. 
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Title  Phasefield modelling of evolving interfaces 

Group(s)  Scientific Computation, Mathematical Medicine and Biology 
Proposer(s)  Dr Kris van der Zee 
Description  Phasefield modelling of evolving interfaces Evolving interfaces are ubiquitous in nature, think of the melting of the polar ice caps, the separation of oil and water, or the growth of cancerous tumours. Two mathematical descriptions exist to model evolving interfaces: those with sharpinterface descriptions, such as parametric and levelset methods, and those with diffuseinterface descriptions, commonly referred to as phasefield models. Challenges for students: Depending on the interest of the student, one of these issues (or others) can be addressed. 
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Title  Numerical methods for stochastic partial differential equations 

Group(s)  Scientific Computation, Statistics and Probability 
Proposer(s)  Prof Michael Tretyakov 
Description  Numerics for stochastic partial differential equations (SPDEs) is one of the central topics in modern numerical analysis. It is motivated both by applications and theoretical study. SPDEs essentially originated from the filtering theory and now they are also widely used in modelling spatially distributed systems from physics, chemistry, biology and finance acting in the presence of fluctuations. The primary objectives of this project include construction, analysis and testing of new numerical methods for SPDEs.

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Title  Property Prediction of Composite Components Prior to Production 

Group(s)  Scientific Computation, Statistics and Probability 
Proposer(s)  Prof Michael Tretyakov, Prof Frank Ball 
Description  Property Prediction of Composite Components Prior to Production Supervisors: Dr Frank Gommer^{1*}, Prof Michael Tretyakov^{2*}, Prof Frank Ball^{2} , Dr Louise P. Brown^{1 } University of Nottingham, University Park, Nottingham NG7 2RD, UK ^{1} Polymer Composites Group, Faculty of Engineering ^{2} School of Mathematical Sciences ^{*} Contact: F.Gommer@nottingham.ac.uk or Michael.Tretyakov@nottingham.ac.uk
This is an exciting opportunity for a postgraduate student to join a vibrant interdisciplinary team and to work in the modern area of Uncertainty Quantification. Fibre reinforced composites are increasingly used in the transport industry to decrease the structural weight of a vehicle and thus increase its fuel efficiency. The importance of the UK composite sector is reflected in the current growth rate of 17% pa for high performance composite components and the expected gross value of £2 billion in 2015 [1]. However, due to the large number of production steps and the necessary saturation of the fibre preform with a resin matrix, a significant amount of waste is produced, which may range between 2% and 20% of the production volume [2]. A major cause of rejecting parts is variability in the reinforcement, such as varying yarn spacing and yarn path waviness, which can significantly influence subsequent properties. For example, these variabilities can affect resin flow and may cause dry spots or reduce mechanical properties. This PhD project will enable the successful candidate to work at the forefront of material science, combining engineering standards, applied mathematics and statistics, with a potential of making an impact on the way of manufacturing composite parts in the future. This proposed doctoral study aims to demonstrate that properties of lightweight fibre reinforced plastics can be predicted in real time before a part is actually manufactured. Data gained from images taken of each layer of a composite during the stacking process are used to determine local geometries and variabilities, within and inbetween individual layers [3]. For example, based on the measured textile geometries it will be possible to predict the resin flow within a preform during a liquid composite moulding (LCM) process considering individual variabilities before injection. These specific flow predictions will allow adjustments of the process parameters during the impregnation process to ensure full saturation of the entire preform with a liquid resin matrix. This will be especially useful when a number of inlet and outlet ports are present such as in the case of complex or large parts. The formation of dry spots will be avoided, which will reduce immediate wastage. For these predictions, faster solutions than currently available are necessary. To find such solutions, appropriate advanced statistical techniques and stochastic modelling for quantifying uncertainties in composites production will be developed in the course of the PhD project. In addition, the developed techniques will also allow virtual testing of a finished component with its specific inherent reinforcement variability. This will make it feasible to customise predictions for every fabricated component. In combination with continuous health monitoring of a structure, it may be possible to estimate the influence of loading conditions, load cycles and damage evaluation. This will also make it possible to predict an individual life expectancy of a part in service. These data can then be used to determine customised inspection intervals for each component. We require an enthusiastic graduate with a 1st class degree in Mathematics or Engineering, preferably of the MMath/MSc level, with good programming skills and willing to work as a part of an interdisciplinary team. A candidate with a solid background in statistics will have an advantage. References [1] CompositesUK. www.compositesuk.co.uk/Information/FAQs/UKMarketValues.aspx. [2] A. C. Long, Design and Manufacture of Textile Composites: Woodhead Publ, 2005. [3] F. Gommer, L. P. Brown, and R. Brooks, “Quantification of mesoscale variability and geometrical reconstruction of a textile”, submitted to Compos Part AAppl S, 2015. 
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Other information  This project is supported by EPSRC DTG Centre in Complex Systems and Processes, see elligibility and how to apply at http://www.nottingham.ac.uk/complexsystems/index.aspx 
Title  Index policies for stochastic optimal control 

Group(s)  Statistics and Probability 
Proposer(s)  Dr David Hodge 
Description  Since the discovery of Gittins indices in the 1970s for solving multiarmed bandit processes the pursuit of optimal policies for this very wide class of stochastic decision processes has been seen in a new light. Particular interest exists in the study of multiarmed bandits as problems of optimal allocation of resources (e.g. trucks, manpower, money) to be shared between competing projects. Another area of interest would be the theoretical analysis of computational methods (for example, approximative dynamic programming) which are coming to the fore with ever advancing computer power.

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Other information  Keywords: multiarmed bandits, dynamic programming, Markov decision processes 
Title  SemiParametric Time Series Modelling Using Latent Branching Trees 

Group(s)  Statistics and Probability 
Proposer(s)  Dr Theodore Kypraios 
Description  A class of semiparametric discrete time series models of infinite order where we are be able to specify the marginal distribution of the observations in advance and then build their dependence structure around them can be constructed via an artificial process, termed as Latent Branching Tree (LBT). Such a class of models can be very useful in cases where data are collected over long period and it might be relatively easy to indicate their marginal distribution but much harder to infer about their correlation structure. The project is concerned with the development of such models in continuoustime as well as developing efficient methods for making Bayesian inference for the latent structure as well as the model parameters. Moreover, the application of such models to real data would be also of great interest. 
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Title  Ion channel modelling 

Group(s)  Statistics and Probability 
Proposer(s)  Prof Frank Ball 
Description  The 1991 Nobel Prize for Medicine was awarded to Sakmann and Neher for developing a method of recording the current flowing across a single ion channel. Ion channels are protein molecules that span cell membranes. In certain conformations they form pores allowing current to pass across the membrane. They are a fundamental part of the nervous system. Mathematically, a single channel is usually modelled by a continuous time Markov chain. The complete process is unobservable but rather the state space is partitioned into two classes, corresponding to the receptor channel being open or closed, and it is only possible to observe which class of state the process is in. The aim of single channel analysis is to draw inferences about the underlying process from the observed aggregated process. Further complications include (a) the failure to detect brief events and (b) the presence of (possibly interacting) multiple channels. Possible projects include the development and implementation of Markov chain Monte Carlo methods for inferences for ion channel data, Laplace transform based inference for ion channel data and the development and analysis of models for interacting multiple channels. 
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Title  Optimal control in yield management 

Group(s)  Statistics and Probability 
Proposer(s)  Dr David Hodge 
Description  Serious mathematics studying the maximization of revenue from the control of price and availability of products has been a lucrative area in the airline industry since the 1960s. It is particularly visible nowadays in the seemingly incomprehensible price fluctuations of airline tickets. Many multinational companies selling perishable assets to mass markets now have large Operations Research departments inhouse for this very purpose. This project would be working studying possible innovations and existing practices in areas such as: customer acceptance control, dynamic pricing control and choicebased revenue management. Applications to social welfare maximization, away from pure monetary objectives, and the resulting game theoretic problems are also topical in home energy consumption and mass online interactions. 
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Title  Stochastic Processes on Manifolds 

Group(s)  Statistics and Probability 
Proposer(s)  Prof Huiling Le 
Description  As well as having a wide range of direct applications to physics, economics, etc, diffusion theory is a valuable tool for the study of the existence and characterisation of solutions of partial differential equations and for some major theoretical results in differential geometry, such as the 'Index Theorem', previously proved by totally different means. The problems which arise in all these subjects require the study of processes not only on flat spaces but also on curved spaces or manifolds. This project will investigate the interaction between the geometric structure of manifolds and the behaviour of stochastic processes, such as diffusions and martingales, upon them. 
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Title  Statistical Theory of Shape 

Group(s)  Statistics and Probability 
Proposer(s)  Prof Huiling Le 
Description  Devising a natural measure between any two fossil specimens of a particular genus, assessing the significance of observed 'collinearities' of standing stones and matching the observed systems of cosmic 'voids' with the cells of given tessellations of 3spaces are all questions about shape. It is not appropriate however to think of 'shapes' as points on a line or even in a euclidean space. They lie in their own particular spaces, most of which have not arisen before in any context. PhD projects in this area will study these spaces and related probabilistic issues and develop for them a revised version of multidimensional statistics which takes into account their peculiar properties. This is a multidisciplinary area of research which has only become very active recently. Nottingham is one of only a handful of departments at which it is active. 
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Title  Automated tracking and behaviour analysis 

Group(s)  Statistics and Probability 
Proposer(s)  Dr Christopher Brignell 
Description  In collaboration with the Schools of Computer and Veterinary Science we are developing an automated visual surveillance system capable of identifying, tracking and recording the exact movements of multiple animals or people. The resulting data can be analysed and used as an early warning system in order to detect illness or abnormal behaviour. The threedimensional targets are, however, viewed in a two dimensional image and statistical shape analysis techniques need to be adapted to improve the identification of an individual's location and orientation and to develop automatic tests for detecting specific events or individuals not following normal behaviour patterns. 
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Title  Asymptotic techniques in Statistics 

Group(s)  Statistics and Probability 
Proposer(s)  Prof Andrew Wood 
Description  Asymptotic approximations are very widely used in statistical practice. For example, the largesample likelihood ratio test is an asymptotic approximation based on the central limit theorem. In general, asymptotic techniques play two main roles in statistics: (i) to improve understanding of the practical performance of statistics procedures, and to provide insight into why some proceedures perform better than others; and (ii) to motive new and improved approximations. Some possible topics for a Ph.D. are

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Title  Statistical Inference for Ordinary Differential Equations 

Group(s)  Statistics and Probability 
Proposer(s)  Dr Theodore Kypraios, Dr Simon Preston, Prof Andrew Wood 
Description  Ordinary differential equations (ODE) models are widely used in a variety of scientific fields, such as physics, chemistry and biology. For ODE models, an important question is how best to estimate the model parameters given experimental data. The common (nonlinear least squares) approach is to search parameter space for parameter values that minimise the sum of squared differences between the model solution and the experimental data. However, this requires repeated numerical solution of the ODEs and thus is computationally expensive; furthermore, the optimisation's objective function is often highly multimodal making it difficult to find the global optimum. In this project we will develop computationally less demanding likelihoodbased methods, specifically by using spline regression techniques that will reduce (or eliminate entirely) the need to solve numerically the ODEs. 
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Title  Statistical shape analysis with applications in structural bioinformatics 

Group(s)  Statistics and Probability 
Proposer(s)  Dr Christopher Fallaize 
Description  In statistical shape analysis, objects are often represented by a configuration of landmarks, and in order to compare the shapes of objects, their configurations must first be aligned as closely as possible. When the landmarks are unlabelled (that is, the correspondence between landmarks on different objects is unknown) the problem becomes much more challenging, since both the correspondence and alignment parameters need to be inferred simultaneously. 
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Title  Highdimensional molecular shape analysis 

Group(s)  Statistics and Probability 
Proposer(s)  Prof Ian Dryden 
Description  In many application areas it is of interest to compare objects 
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Title  Statistical analysis of neuroimaging data 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Dr Christopher Brignell 
Description  The activity of neurons within the brain can be detected by function magnetic resonance imaging (fMRI) and magnetoencephalography (MEG). The techniques record observations up to 1000 times a second on a 3D grid of points separated by 110 millimetres. The data is therefore highdimensional and highly correlated in space and time. The challenge is to infer the location, direction and strength of significant underlying brain activity amongst confounding effects from movement and background noise levels. Further, we need to identify neural activity that are statistically significant across individuals which is problematic because the number of subjects tested in neuroimaging studies is typically quite small and the intersubject variability in anatomical and functional brain structures is quite large. 
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Title  Identifying fibrosis in lung images 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Dr Christopher Brignell 
Description  Many forms of lung disease are characterised by excess fibrous tissue developing in the lungs. Fibrosis is currently diagnosed by human inspection of CT scans of the affected lung regions. This project will develop statistical techniques for objectively assessing the presence and extent of lung fibrosis, with the aim of identifying key factors which determine longterm prognosis. The project will involve developing statistical models of lung shape, to perform object recognition, and lung texture, to classify healthy and abnormal tissue. Clinical support and data for this project will be provided by the School of Community Health Sciences. 
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Title  Modelling hospital superbugs 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Philip O'Neill, Dr Theodore Kypraios 
Description  The spread of socalled superbugs such as MRSA within healthcare settings provides one of the major challenges to patient welfare within the UK. However, many basic questions regarding the transmission and control of such pathogens remain unanswered. This project involves stochastic modelling and data analysis using highly detailed data sets from studies carried out in hospital, addressing issues such as the effectiveness of patient isolation, the impact of different antibiotics, the way in which different strains interact with each other, and the information contained in data on highresolution data (e.g. whole genome sequences). 
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Title  Modelling of Emerging Diseases 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Frank Ball 
Description  When new infections emerge in populations (e.g. SARS; new strains of influenza), no vaccine is available and other control measures must be adopted. This project is concerned with addressing questions of interest in this context, e.g. What are the most effective control measures? How can they be assessed? The project involves the development and analysis of new classes of stochastic models, including intervention models, appropriate for the early stages of an emerging disease. 
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Title  StructuredPopulation Epidemic Models 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Frank Ball 
Description  The structure of the underlying population usually has a considerable impact on the spread of the disease in question. In recent years the Nottingham group has given particular attention to this issue by developing, analysing and using various models appropriate for certain kinds of diseases. For example, considerable progress has been made in the understanding of epidemics that are propogated among populations made up of households, in which individuals are typcially more likely to pass on a disease to those in their household than those elsewhere. Other examples of structured populations include those with spatial features (e.g. farm animals placed in pens; school children in classrooms; trees planted in certain configurations), and those with random social structure (e.g. using random graphs to describe an individual's contacts). Projects in this area are concerned with novel advances in the area, including developing and analysing appropriate new models, and methods for statistical inference (e.g. using pseudolikelihood and Markov chain Monte Carlo methods). 
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Title  Bayesian Inference for Complex Epidemic Models 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Philip O'Neill, Dr Theodore Kypraios 
Description  Dataanalysis for reallife epidemics offers many challenges; one of the key issues is that infectious disease data are usually only partially observed. For example, although numbers of cases of a disease may be available, the actual pattern of spread between individuals is rarely known. This project is concerned with the development and application of methods for dealing with these problems, and involves using the latest methods in computational statistics (e.g. Markov Chain Monte Carlo (MCMC) methods, Approximate Bayesian Computation, Sequential Monte Carlo methods etc). 
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Title  Bayesian model choice assessment for epidemic models 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Philip O'Neill, Dr Theodore Kypraios 
Description  During the last decade there has been a significant progress in the area of parameter estimation for stochastic epidemic models. However, far less attention has been given to the issue of model adequacy and assessment, i.e. the question of how well a model fits the data. This project is concerned with the development of methods to assess the goodnessoffit of epidemic models to data, and methods for comparing different models. 
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Title  Epidemics on random networks 

Group(s)  Statistics and Probability, Mathematical Medicine and Biology 
Proposer(s)  Prof Frank Ball 
Description  There has been considerable interest recently in models for epidemics on networks describing social contacts. In these models one first constructs an undirected random graph, which gives the network of possible contacts, and then spreads a stochastic epidemic on that network. Topics of interest include: modelling clustering and degree correlation in the network and analysing their effect on disease dynamics; development and analysis of vaccination strategies, including contact tracing; and the effect of also allowing for casual contacts, i.e. between individuals unconnected in the network. Projects in this area will address some or all of these issues. 
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Title  Statistical analysis of fibre variability in composites manufacture 

Group(s)  Statistics and Probability, Scientific Computation 
Proposer(s)  Prof Frank Ball, Prof Michael Tretyakov 
Description  Multidisciplinary collaborations are a critical feature of material science research enabling integration of data collection with computational and/or mathematical modelling. This PhD study provides an exciting opportunity for an individual to participate in a project spanning research into composite manufacturing, stochastic modelling, statistical analysis and scientific computing. The project is integrated into the EPSRC Centre for Innovative Manufacturing in Composites, which isled by the University of Nottingham and delivers a coordinated programme of research in composites manufacturing. This project focuses on the development of a manufacturing route for composite materials capable of producing complex components in a single process chain based on advancements in the knowledge, measurement and prediction of uncertainty in processing. The outcome of this work will enable a step change in the capabilities of composite manufacturing technologies to be made, overcoming limitations related to part thickness, component robustness and manufacturability as part of a single process chain, whilst yielding significant developments in mathematics and statistics with generic application in the fields of stochastic modelling and inverse problems. The specific aims of this project are: (i) statistical analysis of fibber placements based on textile and composite material data sets; (ii) statistical analysis and stochastic modelling of permeability of textiles and composites; (iii) efficient sampling techniques of stochastic permeability. A student will obtain an excellent grasp of various statistical and stochastic techniques (e.g., spatial statistical methods, use of random fields, Monte Carlo methods), how to apply them, how to work with real data and how to do related modelling and simulation. This knowledge and especially experience are transferable to other applications of statistics and probability. The PhD programme contains a training element, the exact nature of which will be mutually agreed by the student and their supervisors. We require an enthusiastic graduate with a 1^{st} class honours in Mathematics (in exceptional circumstances a 2(i) class degree can be considered), preferably at the MMath/MSc level, with good programming skills and williness to work as a part of an interdisciplinary team. A candidate with a solid background in statistics and stochastic processes will have an advantage. The studentship is available for a period of three and a half years from September/October 2015 and provides a stipend and full payment of Home/EU Tuition Fees. Students must meet the EPSRC eligibility criteria. Informal enquiries should be addressed to Prof. Michael Tretyakov, email: michael.tretyakov@nottingham.ac.uk. To apply, please access: https://my.nottingham.ac.uk/pgapps/welcome/. Please ensure you quote ref: SCI/1262x1. This studentship is open until filled. Early application is strongly encouraged. 
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