You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.
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. 
Relevant Publications 

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  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  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  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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