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Title Equations over groups Algebra and Analysis Dr Martin Edjvet Let G be a group. An expression of the form g1 t … gk t=1 where each gi 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 torsion-free 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 non-zero 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.
Title Cohomology Theories for Algebraic Varieties Algebra and Analysis Dr Alexander Vishik 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 so-called Generalized Cohomology Theories. This includes classical algebraic K-theory, 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 u-invariants of fields. This is a new and rapidly developing area that offers many promising directions of research.
Title Regularity conditions for Banach function algebras Algebra and Analysis Dr Joel Feinstein Banach function algebras are complete normed algebras of bounded, continuous, complex-valued 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 complex-valued 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 non-empty 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, complex-valued 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.
Title Properties of Banach function algebras and their extensions Algebra and Analysis Dr Joel Feinstein Banach function algebras are complete normed algebras of bounded continuous, complex-valued 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 complex-valued 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 non-empty 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, complex-valued 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 Arens-Hoffman or Cole extensions).
Title Meromorphic Function Theory Algebra and Analysis Prof James Langley 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, and specific areas covered by his and their research include those in the list below. Please note that since Professor Langley is close to retirement age he may not be available to supervise further students, and potential applicants are advised to contact him before applying.Zeros of derivatives. There is a long history of work on these questions which can be viewed as natural generalizations of the classical Picard theorem to the effect that an entire function which omits two values is a constant. Langley’s best known result is probably his 1993 proof of Hayman’s 1959 conjecture on the zeros of a meromorphic function and its second derivative. Langley, Bergweiler and Eremenko settled the last outstanding case of a conjecture of Wiman from 1911 concerning the derivatives of real entire functions, and Langley has extended much of this work to meromorphic functions.Differential equations in the complex domain, in particular the growth and oscillation of solutions of linear differential equations. Properties of compositions of entire functions. This area has connections to complex dynamics, in which the famous Julia and Mandelbrot sets arise.Critical points of potentials associated with distributions of charged points and wires, and zeros of related meromorphic functions.
Title Relative presentations and asphericity Algebra and Analysis Dr Martin Edjvet The subject area is Combinatorial Group Theory. The objects of study are relative pesentations which can be viewed as one-relator products. In particular we consider the questions: when is a relative presentation aspherical; what is the order of the group defined by a given relative presentation? This has applications to cyclically presented groups for example. The methods are geometric (borrowing ideas from algebraic topology); algebraic; and may involve computational group theory via GAP or automatic groups. R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer 1977. (Book.)Martin Edjvet and Arye Juhasz, The infinite Fibonacci groups and relative asphericity, Trans London Math Soc 4 (2017) 148-218W.A. Bogley, M. Edjvet and G. Williams, Aspherical relative presentations all over again, to appear in Groups St Andrews (conference proceedings)
Title Compensated convex transforms and their applications Algebra and Analysis Prof Kewei Zhang 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.
Title Endomorphisms of Banach algebras Algebra and Analysis Dr Joel Feinstein 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.
Title Iteration of quasiregular mappings Algebra and Analysis Dr Daniel Nicks 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. Quasiregular mappings of n-dimensional real space generalise analytic functions on the complex plane. Roughly, a mapping is called quasiregular if it locally distorts space by only a bounded amount, so that small spheres are mapped to small ellipsoids. This is more flexible than the situation with analytic functions, where the Cauchy-Riemann equations tell us that infinitesimally small circles are mapped to small circles. There are many similarities between the behaviour of analytic functions and quasiregular mappings. One can therefore attempt to develop a theory of quasiregular iteration parallel to the results of complex dynamics. Such a theory is just beginning to emerge, lying between the well-studied analytic case (where many powerful tools from complex analysis are available) and general iteration in several real variables, which is much less well-understood. The problems studied will be inspired and guided by existing results in complex dynamics. For example, we can ask questions about the ‘escaping set’ of a function – this is the set of all starting points from which the sequence of iterates tends to infinity. One of the challenges we encounter is that as we increase the number of iterations of a quasiregular mapping, the amount of local distortion may become increasingly large. This is very much a pure mathematics project and will appeal to someone who enjoys topics such as real analysis, complex analysis, metric spaces or discrete dynamical systems.
Title Mean field limits of Ginzburg-Landau equations Algebra and Analysis Dr Matthias Kurzke Many quantum physical systems (for example superconductors, superfluids, Bose-Einstein condensates) exhibit vortex states that can be described by Ginzburg-Landau 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 Gross-Pitaevskii equation (a nonlinear Schrödinger equation), one obtains Euler, for the time-dependent Ginzburg-Landau equation (a nonlinear parabolic equation), one obtains a dissipative variant of the Euler equations.The goal of the project is to study the dissipative equations and to understand instabilities and low regularity solutions. M Kurzke, D Spirn: Vortex liquids and the Ginzburg-Landau equation. arXiv:1105.4781S Serfaty: Mean Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau Equations. arXiv:1507.03821M Duerinckx, S Serfaty: Mean-field dynamics for Ginzburg-Landau vortices with pinning and forcing. arXiv:1702.01919
Title Dynamics of boundary singularities Algebra and Analysis Dr Matthias Kurzke 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. M Kurzke: The gradient flow motion of boundary vortices Ann. Inst. H. Poincaré Anal. Non Linéaire. 24(1), 91-112
Title Where graphs and partial differential equations meet Algebra and Analysis Dr Yves van Gennip Many problems in image analysis and data analysis can be represented mathematically as a network based problems. Examples are image segmentation and data clustering. Traditionally these kind of graph based problems have been tackled by combinatorial algorithms, but in recent years interest has grown in applying ideas from variational methods and nonlinear partial differential equations (PDEs) to such problems.This has lead to an interesting mix of theoretical questions (what is the dynamics on the network induced by these methods? which correspondences between PDE quantities and graph quantities can we find, e.g. total variation, graph cut, curvature...?) and practical applications.This project will investigate graph curvature and related quantities and make links to established results in PDE theory.
Title Graph limits for faster computations Algebra and Analysis Dr Yves van Gennip Many problems in image analysis and data analysis, such as image segmentation or data clustering, require the computation of pairwise similarity scores between nodes of a network. When the image or data set is large, the computational cost is very prohibitive. One way to deal with this problem is to compute only a subset of all the pairwise comparisons and use those to approximate the others.Recent developments in the theory of (dynamics on) graph limits offer the hope that this subset can be chosen in such a way that the resulting dynamics of the segmentation or clustering algorithm is a very good approximation of the dynamics on the full graph.This project will investigate this possibility and can be taken in a theoretical and/or application oriented direction.
Title Vectorial Calculus of Variations, Material Microstructure, Forward-Backward Diffusion Equations and Coercivity Problems Algebra and Analysis, Algebra and Analysis Prof Kewei Zhang This aim of the project is to solve problems in vectorial calculus of variations, forward-backward 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.
Title Dynamics of vortex singularities in nonlinear PDE Algebra and Analysis, Algebra and Analysis Dr Matthias Kurzke Nonlinear PDE of Ginzburg-Landau type can be used to model a wide range of phenomena, from ferromagnetic materials and superconductors to quantum field theory. For certain ranges of the Ginzburg-Landau parameter (for self-duality and for point vortices), the equations can be reduced to ODEs. The focus of the present project will be to study the equations of motion in a wider setting, with the aim to extend the range of validity of the reductions and to compare almost singular solutions of the PDEs with simplified ODEs that describe the motion of the singularities. The project will use mostly rigorous analysis and possibly numerical simulation.
Title Model reduction and homogenisation for filtration and adsorption Industrial and Applied Mathematics Dr Matteo Icardi Porous media are ubiquitous in natural and engineered transport processes. When colloids or diffusive particles flows through their complex geometrical structure, non-trivial interactions arise between the advection, diffusion, particle-particle and particle-wall interactions. These processes can be modelled and simulated with computationally intensive three-dimensional simulations. In this project, a combination of rigorous multiscale analytical and numerical techniques will be used to derive and calibrate faster and simple models for filtration and adsorption processes. Extensions to include electrostatic forces and electrochemical reactions will be also considered. The project is part of a wider research effort that sees the collaboration of several UK and international academic partners, and industrial partners in the Automotive and Oil&Gas sector.
Title Crystallisation in polymers Industrial and Applied Mathematics Prof Richard Graham 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 world-leading experimental groups.
Title Dynamics of entangled polymers Industrial and Applied Mathematics Prof Richard Graham 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 high-tech 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 co-operate with researchers from a range of backgrounds would be a real asset in this project.
Title Power converters Industrial and Applied Mathematics Dr Stephen Cox, Dr Stephen Creagh 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. S M Cox and J C Clare Nonlinear development of matrix-converter instabilities. Journal of Engineering Mathematics 67 (2010) 241-259. S M Cox and S C Creagh Voltage and current spectra for matrix power converters. SIAM Journal on Applied Mathematics 69 (2009) 1415-1437. S M Cox Voltage and current spectra for a single-phase voltage-source inverter. IMA Journal of Applied Mathematics 74 (2009) 782-805.
Title Class-D audio amplifiers Industrial and Applied Mathematics Dr Stephen Cox, Dr Stephen Creagh The holy grail for an audiophile is distortion-free reproduction of sound by amplifier and loudspeaker. This project concerns the mathematical modelling and analysis of class-D 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 class-D amplifiers a reality. (Class-D 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 class-D 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 class-D 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. J Yu, M T Tan, S M Cox and W L Goh Time-domain analysis of intermodulation distortion of closed-loop Class D amplifiers. IEEE Transactions on Power Electronics 27 (2012) 2453-2461. S M Cox, C K Lam and M T Tan A second-order PWM-in/PWM-out class-D audio amplifier. IMA Journal of Applied Mathematics (2011). S M Cox, M T Tan and J Yu A second-order class-D audio amplifier. SIAM Journal on Applied Mathematics 71 (2011) 270-287. S M Cox and B H Candy Class-D audio amplifiers with negative feedback. SIAM Journal on Applied Mathematics 66 (2005) 468-488. S M Cox and S C Creagh Voltage and current spectra for matrix power converters. SIAM Journal on Applied Mathematics 69 (2009) 1415-1437.
Title Mathematical modelling and analysis of composite materials and structures Industrial and Applied Mathematics Dr Konstantinos Soldatos Nottingham has established and maintained, for more than half a century, world-wide research leadership in developing the Continuum Theory of fibre-reinforced 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, visco-elastic or even viscous (fluid-type) 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:Linearly and non-linearly elastic, static and dynamic analysis of homogeneous and inhomogeneous, anisotropic, solid structures and structural components;Forming flow (process modelling) of fibre-reinforced viscous resins;Plastic behaviour modelling as well as study of failure mechanisms/modes of fibrous-composites;Mass-growth modelling of soft and hard tissue, such as human nail and hair.  Mathematical modelling and analysis of the behaviour of thin walled, anisotropic structures in terms of high-order, linear or non-linear, ordinary and partial differential equations;Development and/or use of various analytical, semi-analytical and/or numerical mathematical methods suitable for solving the sets of differential equations which emerge in the above research 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 post-graduate student. The candidate’s relevant co-operation is accordingly desirable and, as such, will be appreciated at the initial, but also at later stages of tentative research collaboration.
Title Dynamics of coupled nonlinear oscillators Industrial and Applied Mathematics Dr Paul Matthews 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 three-dimensional 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.
Title Dynamo action in convection Industrial and Applied Mathematics Dr Paul Matthews 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 determinethe conditions under which convection is able to generate a magnetic field,whether rotation is necessary or advantageous for dynamo action,the underlying topological stretching mechanism responsible for the magnetic field generation.The project is also suitable for analytical work, either based on an asymptotic analysis of the equations, or in investigating or proving 'anti-dynamo' theorems.
Title Nonlinear penetrative convection Industrial and Applied Mathematics Dr Paul Matthews 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.
Title Coupling between optical components Industrial and Applied Mathematics Dr Stephen Creagh 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.
Title Uncertainties in multiphase flows through porous media Industrial and Applied Mathematics Dr Matteo Icardi Despite the recent significant developments in Digital Rock Physics (DRP), two-phase flows in complex pore geometries are still not fully predictable, understood, and quantitatively reproducible. This is due to a number of factors including incorrect physical models, insufficient mesh resolution, unknown parameters and pore heterogeneity. The qualitative and quantitative effects of these uncertainties have not been studied yet. In this project, we aim to develop a modelling and simulation workflow to quantify uncertainty and assess the validity of simplified multiphase flow models in digitalised porous media images. Deterministic and Monte-Carlo techniques, together with two-phase flow solvers, will be used to perform a global sensitivity analysis of the problem. The project will see a collaboration of an industrial partner and the Geoenergy Research Centre in Nottingham.
Title Data-driven coarse-graining and multiscale model reduction for ODEs and PDEs Industrial and Applied Mathematics Dr Matteo Icardi Many theoretical tools have been recently developed to reduce the complexity of high-dimensional non-linear ODEs or highly-resolved multiscale PDEs. These have now an enormous importance in computational chemistry, continuum mechanics, fluid dynamics, and dynamical systems in general.  One of these bottom-up formal approaches is the Mori-Zwanzig projection formalism for dynamical systems. At the same time, also data-driven top-down methods, have been widely studied in machine learning and in numerical analysis. In this project, we aim to connect these theoretical and numerical tools to make them applicable for practical applications, such as the molecular dynamics simulation of complex molecule chains, or the relaxation to equilibrium of non-linear reaction-diffusion equations. In the first case, we can rely on the Hamiltonian structure of the full-resolution model, while the latter can be analysed through model decomposition or spectral analysis. The objective is to develop and implement flexible numerical approaches to deal with the model reduction of different model problems, by combining analytical derivations with numerical simulation data.
Title Solitons in higher dimensions Industrial and Applied Mathematics Prof Jonathan Wattis 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 one-dimensional systems; however, in two- and three-dimensional 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 Bose-Einstein 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.
Title Modelling the vibro-acoustic response of complex structures Industrial and Applied Mathematics Prof Gregor Tanner The vibro-acoustic 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 so-called Statistical Energy Analysis (SEA) in the high-frequency limit. Major computational challenges are posed by so-called mid-frequency problems - that is, composite structures where the local wave length may vary by orders of magnitude across the components.Recently, I propsed a set of new methods based on ideas from wave chaos (also known as quantum chaos) theory.  Starting from the phase space flow of the underlying - generally chaotic - ray dynamics, the new method called Dynamical Energy Analysis (DEA) interpolates between SEA and ray tracing containing both these methods as limiting cases. Within the new theory SEA is identified as a low resolution ray tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. I have furthermore developed a hybrid SEA/FEM method based on random wave model assumptions for the short-wavelength components. This makes it possible to tackle mid-frequency problems under certain constraints on the geometry of the structure.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 CDH AG, Germany and Jaguar/Landrover, Gaydon, UK. Wave chaos in acoustics and elasticity, G. Tanner and N. Soendergaard, J. Phys. A 40, R443 - R509 (2007).Dynamical Energy Analysis - determining wave energy distributions in complex vibro-acoustical structures, G. Tanner, Journal of Sound and Vibration 320, 1023 (2009).
Title Impulsively-generated fluid motions Industrial and Applied Mathematics Prof John Billingham In a series of papers over the past few years [1-3], my collaborators and I have studied how the motion of a solid body into an expanse of fluid generates complex and unexpected flow patterns. In this project, you will use numerical and asymptotic methods to study the impulsive motion of a plate into a 2D layer of inviscid fluid. In the case of a 90 degree intersection between the plate and the free surface, a jet is driven up the plate. At other angles, the dynamics are not clear, and will form the subject of this project. The initial development of a jet caused by fluid, body and free-surface interaction. Part 2. An impulsively moved plate, D.J. Needham, J. Billingham and A.C.King, J. Fluid Mech, 2007.The initial development of a jet caused by fluid, body and free surface interaction. Part 3. An inclined accelerating plate, D.J. Needham, P.G. Chamberlain and J. Billingham, QJMAM, 2008.The initial development of a jet caused by fluid, body and free surface interaction. Part 5. Parasitic capillary waves on an initially horizontal surface, J. Billingham, D.J. Needham, E. Korsukova, R.J. Munro, J. Fluid Mech, 2017.
Title Non-local reaction diffusion equations Industrial and Applied Mathematics Prof John Billingham Non-local reaction-diffusion equations arise in many scientific areas, including population ecology and neuroscience. In this project, you will use numerical and asymptotic methods to study these equations. For example, the non-local Fisher equation, [2], with small diffusivity has recently been shown to have travelling wave solutions that consist of a sequence of localized spikes, [3]. Other interesting dynamics that can occur in similar equations, [1], include accelerating and decelerating travelling waves. Possible extensions include nonlocal equations in more than one spatial dimension and non-monotonic kernels. Dynamics of a strongly nonlocal reaction-diffusion population model, J. Billingham, Nonlinearity, 2003, 17The non-local Fisher-KPP equation: Travelling waves and steady states, H. Berestycki, et al, Nonlinearity, 2009, 22.Slow travelling wave solutions of the non-local Fisher-KPP equation with small diffusivity, J. Billingham, submitted to Nonlinearity, 2018.
Title Numerical Upscaling and model identification for Lithium-Ion batteries Industrial and Applied Mathematics Dr Matteo Icardi Lithium-ion battery models for control applications are typically defined by system identification techniques and fail to capture the intrinsic functional dependence of the parameters on the material attributes and to predict irreversible and complex non-linear phenomena such as fast (dis)charge and degradation.  In this project, we propose new mathematical techniques to develop simple and efficient reduced order models, as an alternative to classical equivalent circuit models, to enable the fast, yet accurate, simulation of short and long-term behavior of lithium-ion cells. Starting from the well-known porous electrode theory and Newman’s model, we aim to derive simple differential equations that can retain the interesting features of the full model (e.g., solid diffusion, non-linearities). We also propose a new integrated framework to incorporate both micro- and system scale experimental data into our model via machine-learning approaches. This is collaborative project with the University of Warwick (Mathematics and WMG) and two industrial partners.
Title Machine learning for first-principles calculation of physical properties. Industrial and Applied Mathematics Prof Richard Graham 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 first-principles, in theory allowing physical properties to be derived ab-initio 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 machine-learning technique that, by using a small number of precomputed ab-initio 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.
Title Modelling Thermal Effects within Thin-Film Flows Industrial and Applied Mathematics Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 558 Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 558 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 thin-film 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 engine-relevant configurations. The classical theory of thin-film 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 thin-film 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 non-isothermal effects relevant to a bearing chamber context is provided byThis multi-disciplinary 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. • Kay, E. D, Hibberd, S. and Power, H., (2014); A depth-averaged model for non-isothermal thin-film rimming flow. Int Jnl. Heat and Mass Transfer,70, 1003-1015. doi 10.1016/j.ijheatmasstansfer.2013.11.040 The supervision team will include include Prof H Power, Faculty of Engineering and a project partner at Rolls-Royce 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 Rolls-Royce plc.
Title On mathematical models for high speed non-isothermal air bearings Industrial and Applied Mathematics Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 558 Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 558 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 aero-engines, are designed to work with no contact and very small gaps and applicable to a wide range of industrial applications. Air-riding 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 thin-film flows through the creation of sophisticated and numerical and modelling approaches. BAILEY, N.Y., CLIFFE, K.A., HIBBERD, S. and POWER, H., 2014. Dynamics of a parallel high speed fluid lubricated bearing with Navier slip boundary conditions. IMA Journal of Applied Mathematics, doi:10.1093/immat/hxu053, 1-22.VOSPER, H., CLIFFE, K.A., HIBBERD, S. and POWER, H., 2013. On thin film flow in hydrodynamic bearings with a radial step at finite Reynolds number. Journal of Engineering Mathematics. DOI 10.1007/s10665-013-9627-8, 1-22 The supervision team will include Prof H Power, Faculty of Engineering
Title Electromagnetic compatibility in complex environments: predicting the propagation of electromagnetic waves using wave-chaos theory Industrial and Applied Mathematics, Mathematical Physics Dr Stephen Creagh, Prof Gregor Tanner 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 EM-field 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 ray-dynamics in phase space using the so-called Wigner distribution function (WDF) formalism.  It allows us to replace the wave propagation problem with one of propagating classical densities within phase space.
Title Wave propagation in complex built-up structures – tackling quasi-periodicity and inhomogeneity Industrial and Applied Mathematics, Mathematical Physics Prof Gregor Tanner, Dr Stephen Creagh 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 quasi-periodic 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 high-frequency wave methods – such as the so-called Dynamical Energy Analysis developed in Nottingham - making it possible to compute the vibrational response of structures with arbitrary complexity at large frequencies.
Title Excitability in biology - the role of noisy thresholds Mathematical Medicine and Biology Dr Ruediger Thul, Prof Stephen Coombes 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. Coombes, S, R Thul, J Laudanski, A R Palmer, and C J Sumner. 2011. “Neuronal Spike-Train Responses in the Presence of Threshold Noise.” Frontiers in Life Science 5: 91–105.
Title Spatio-temporal patterns with piecewise-linear regulatory networks Mathematical Medicine and Biology Dr Etienne Farcot A number of fascinating and important biological processes involvevarious kinds of spatial patterns: spatial patterns on animal skins, orthe very regular organ arrangements found in plants (called phyllotaxis)for instance. These patterns often originate at very small scales, andtheir onset can only be seen using very recent microscope and imageanalysis techniques.Among several families of models for biological patterning, one of thesimplest is based on the idea that mobile substances (called morphogens) are acting upstream of their targets, which respond locally to a globallydefined  gradient pattern.In this project one will consider models where targets are themselvesmobile morphogens, potentially regulating their own input. One will study the effect of such spatial feedback on patterning. To do so, one will rely on a class of models which are biologically relevant, tractable analytically, and not much studied yet in a context with spatial interactions. A class of models which meet all this criteria is provided by piecewise-linear differential equations. L.G. Morelli et al. Computational Approaches to Developmental Patterning. Science 336, 187 (2012).L. Glass, S.A. Kauffman. The logical analysis of continuous, nonlinear biochemical control networks. Journal of Theoretical Biology 39, 103-129 (1973).
Title Spine morphogenesis and plasticity Mathematical Medicine and Biology Prof Stephen Coombes, Dr Ruediger Thul 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 sub-cellular 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 spine-heads [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 in-silico 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. [1] Rafael Yuste, 2010, Dendritic spines, MIT Press[2] Y Timofeeva, G J Lord and S Coombes 2006 Spatio-temporal filtering properties of a dendritic cable with active spines, Journal of Computational Neuroscience, Vol 21, 293-306[3] Cian O'Donnell, Matthew F. Nolan, and Mark C. W. van Rossum, 2012, Dendritic Spine Dynamics Regulate the Long-Term Stability of Synaptic Plasticity, The Journal of Neuroscience, 9 November 2011, 31(45):16142-16156[4] R M Henig, 2012, Lifting the black cloud, Scientific American, Mar, p 60-65
Title Multiscale mechanochemical modelling of exercise-induced asthma Mathematical Medicine and Biology Dr Bindi Brook, Dr Reuben O'Dea Inflammation and airway hyperresponsiveness (leading to bronchoconstriction) are hallmarks of asthma. Detailed measurements of exhaled inflammatory markers during rest and exercise provide an indicator of the inflammatory processes that arise during exercise-induced asthma. By combining this data with mechanistic models of cell-signalling, inflammatory cell recruitment and continuum-based nonlinear tissue mechanics models of the airways this project will provide insight into the development of bronchoconstriction during exercise. Model development will inform future experiments and interventions in human subjects.
Title Rare event modelling for the progression of cancer Mathematical Medicine and Biology Prof Richard Graham, Prof Markus Owen Purpose This project will apply cutting-edge 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 cellular-level 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. BackgroundA wide-spread problem in treating cancer is to distinguish indolent (benign) tumours from metastatic-capable 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 quasi-stable 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 state-of-the-art 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.
Title Stochastic Neural Network Modelling Mathematical Medicine and Biology Prof Stephen Coombes, Dr Ruediger Thul 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 integrate-and-fire model, which is described by a non-smooth 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 non-Gaussian noise. One of the main mathematical challenges will be to extend the Master-Stability framework for networks of deterministic limit cycle oscillators to the noisy non-smooth 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 sub-served by distributed brain networks [3]. [1] S Coombes, R Thul and K C A Wedgwood 2012 Nonsmooth dynamics in spiking neuron models, Physica D, DOI: 10.1016/j.physd.2011.05.012[2] S Coombes, R Thul, J Laudanski, A R Palmer and C J Sumner 2011 Neuronal spike-train responses in the presence of threshold noise, Frontiers in Life Science, DOI: 10.1080/21553769.2011.556016[3] J Hlinka and S Coombes 2012 Using computational models to relate structural and functional brain connectivity, European Journal of Neuroscience, to appear
Title Cell signalling Mathematical Medicine and Biology Prof John King 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 well-suited to mathematical study. Experience with the study of nonlinear dynamical systems would provide helpful background for such a project.
Title Modelling DNA Chain Dynamics Mathematical Medicine and Biology Prof Jonathan Wattis Whilst the dynamics of the DNA double helix are extremely complicated, a number of well-defined modes of vibration, such as twisting and bending, have been identified. At present the only accurate models of DNA dynamics involve large-scale 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 larger-scale dynamics, such as the unzipping of the double helix, to be studied.
Title Multiscale modelling of vascularised tissue Mathematical Medicine and Biology Prof Markus Owen 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 blood-borne 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 tissue-level 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.
Title Self-similarity in a nanoscale island-growth Mathematical Medicine and Biology Prof Jonathan Wattis 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 self-similarity. 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 size-distribution. The relationship of equations with other models of deposition, such as reactions on catalytic surfaces and polymer adsorption onto DNA, will also be explored.
Title Sequential adsorption processes Mathematical Medicine and Biology Prof Jonathan Wattis 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 one-dimension 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.
Title Robustness of biochemical network dynamics with respect to mathematical representation Mathematical Medicine and Biology Dr Etienne Farcot In the recent years, a lot of multi-disciplinary efforts have beendevoted to improving our understanding of the dynamics of interactionsbetween the many types of molecules present in biological cells. Thishas led to a widespread viewpoint where networks of genes, proteins andother biochemical species are considered at once, as complex dynamicalsystems from which the global state of cells emerge.Several mathematical formalisms are used to represent these systems,from discrete or boolean models to differential equations. One strikingfact, especially regarding models of developmental processes, is that anumber of relevant properties of these networks can be capturedsimilarly by all these formalisms, like for instance the property ofbistability.One possible interpretation of this independence with respect toformalism is that biological regulatory systems are most often extremelyrobust.The project will start by developing parallel models – using differentformalisms – of actual biological networks whose behaviour is known.Elaborating on these examples the theoretical and practical implicationsof this notion of robustness will be explored.
Title Neurocomputational models of hippocampus-dependent 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.

Humans and other animals can readily remember significant places and associated events and return to these places as appropriate. From an experimental point of view, studies of the neuro-psychological mechanisms underlying place learning and navigation offer unique opportunities, because similar tests can be used in rodent models and human participants. Studies in rodent models have led to a detailed understanding of the neuro-psychological mechanisms of place memory, and the importance of the hippocampus for place learning and navigation in humans and other animals is well-established. In this project, we aim to develop quantitative models describing how neurons in the hippocampus and associated brain areas give rise to place learning and navigation, and construct an in silico model for testing ideas about functional mechanisms. The project brings together behavioural neuroscience expertise on hippocampal function and place learning (Bast, Psychology) with expertise in mathematical and computational neuroscience (Coombes, Mathematical Sciences) to understand rapid place learning. A particular emphasis will be on the hippocampal learning-behaviour translation: how place information (as encoded, for example, by hippocampal place cells) is related to decision making processes and, ultimately, translated into motor behaviour (for example, by way of interactions with prefrontal and subcortical circuits). From a mathematical perspective the project will develop new neurocomputational models of hippocampus-dependent place learning and navigation using tools from stochastic optimal control, reinforcement learning theory, dynamical systems and computational neuroscience.

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/apply-online.aspx

## Funding Notes

Summary: 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 auto-soliton scattering: interface analysis in a neural field model Mathematical Medicine and Biology Prof Stephen Coombes, Dr Daniele Avitabile 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 spatio-temporal 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 integro-differential 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 auto-solitons, whilst strong scattering solutions will be investigated using the scattor theory that has previously been developed for multi-component reaction diffusion systems [2]. [1] C R Laing 2005 Spiral waves in nonlocal equations, SIAM J. Appl. Dyn. Sys., Vol 4, 588-606.[2] Y Nishiura, T Teramoto, and K-I Ueda. Scattering of traveling spots in dissipative systems. Chaos, 15:047509(1â€“10), 2005. 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 Mathematical Medicine and Biology Dr Daniele Avitabile, Prof Stephen Coombes Insects have evolved diverse and delicate morphological structures in order tocapture the inherently low energy of a propagating sound wave. In mosquitoes, thecapture of acoustic energy, and its transduction into neuronal signals, is assistedby the active mechanical participation of actuators called scolopidia.When a sound wave reaches the head of a mosquito, the antenna oscillates under theaction of the external pressure field (passive component) and of the force provided bythe mechanical actuators (active component). The latter is particularlyrelevant for sexual recognition: when a male mosquito hear the flyby of a female, hisantennal oscillation are greatly amplified by the scolopidia. In other words, theantenna of a male is tuned very sharply around the frequency and intensity of afemale flyby.Recent studies have shown that mosquitoes of either sex use both their antenna andtheir wing beat to select a partner: understanding how their hearing system workscould help us controlling the population of species that carry viral diseases.Even though some models of mosquitoes hearing systems have been proposed in the past,a number of key questions remain unanswered. Where do the mechanical actuators gettheir energy? How do they twitch? How is the mechanical motion of the antennatransformed into an electric signal? Do neurones control the mechanical motion? Howdoes the brain of a mosquito process the neural information and distinguish varioussources of sound? Is the sexual recognition entirely based on sound perception, or isit also influenced by olfactory signals? Is the antenna sensitive to sounds fromdifferent directions? AVITABILE, D, HOMER, M, CHAMPNEYS, AR, JACKSON, JC and ROBERT, D, 2010. Mathematical Modelling Of The Active Hearing Process In Mosquitoes Journal Of The Royal Society Interface. 7(42), 105-122CHAMPNEYS, AR, AVITABILE, D, HOMER, M and SZALAI, R, 2011. The Mechanics Of Hearing: A Comparative Case Study In Bio-Mathematical Modelling Anziam Journal. 52(3), 225-249
Title Nonsmooth dynamical systems: from nodes to networks Mathematical Medicine and Biology Prof Stephen Coombes, Dr Ruediger Thul 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 piece-wise linear models of excitable tissue, integrate-and-fire 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 mode-locked states of oscillatory systems, as well as bifurcation diagrams for excitable systems. This will rely heavily on the construction of so-called 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 spatio-temporal states that can be generated in realistic neural networks. [1] M di Bernardo, C Budd, A R Champneys,and P Kowalczyk. Piecewise-smooth Dynamical Systems: Theory and Applications. Applied Mathematical Sciences. Springer, 2008.[2] P Ashwin, S Coombes and R Nicks 2016 Mathematical frameworks for oscillatory network dynamics in neuroscience, Journal of Mathematical Neuroscience, 6:2[3] S Coombes and R Thul 2016 Synchrony in networks of coupled nonsmooth dynamical systems: Extending the master stability function, European Journal of Applied Mathematics, Vol 27(6), 904-922[4] F Sorrentino, L M Pecora, A M Hagerstrom, T E Murphy, and R Roy 2016 Complete characterization of the stability of cluster synchronization in complex dynamical networks, Science Advances, 2:4, e1501737
Title Pattern formation in biological neural networks with rebound currents Mathematical Medicine and Biology Prof Stephen Coombes 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 hyperpolarisation-activated channels. For the case of grid cells in the medial enthorinal cortex this gives rise to a so-called I_h current. Many other cells types also utilise rebound currents for firing, and in particular thalamo-cortical relay cells do so via slow T-type 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 spatio-temporal 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 integrate-and-fire 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 cross-fertilisation of ideas with those being developed in the engineering community for impact oscillators and piece-wise linear systems. This PhD will translate and develop mathematical methodologies from non-smooth 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 non-smooth 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.
Title Mechanistic models of force adaptation in airway smooth muscle cells - application to asthma Mathematical Medicine and Biology Dr Bindi Brook 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 force-generating capacity of ASM via a cell-level 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. Adaptation of airway smooth muscle to basal tone relevance to airway hyperresponsiveness, Bosse et al, American Journal of Respiratory Cell Molecular Biology,Vol 40. pp 13–18, 2009The role of contractile unit reorganization in force generation in airway smooth muscle, B S Brook and O E Jensen, Mathematical Medicine and Biology, 2013. doi:10.1093/imammb/dqs031
Title Synchronisation and propagation in human cortical networks Mathematical Medicine and Biology Dr Reuben O'Dea 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.This project aims to address these deficiencies via improved computational models with which to investigate thoroughly the influence of the geometry and connectivity of the human brain on epileptic seizure progression and initiation, and the development of novel network measures with which to characterise epileptic brains. Such investigations will be informed by exhaustive patient datasets (such as recordings of neural activity in epilepsy patients and age-matched controls), and will be used to study (i) improved diagnostic strategies, (ii) the influence of treatment strategies on seizure progression and initiation, and (iii) the identification of key sites of epilepsy initiation.
Title Multiscale modelling of cell signalling and mechanics in tissue development and cancer Mathematical Medicine and Biology Prof John King, Dr Reuben O'Dea 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 intra-cellular 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 spatio-temporal scales, from intracellular signalling cascades to tissue-level 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 well-suited to mathematical study.This project will develop a predictive framework, coupling key signalling pathways to cell- and tissue-level mechanics, to elucidate key developmental mechanisms and their interaction. Investigations will include both multiscale computational approaches, and asymptotic methods for model reduction and analysis. Importantly, model development, analysis and experimental validation will be enabled via close collaboration with Dr Robert Jenkins (Francis Crick Institute, a multidisciplinary biomedical discovery institute dedicated to understanding the scientific mechanisms of living things), thereby ensuring the relevance of the investigations undertaken.Experience of mathematical/numerical techniques for ODEs and PDEs, the study of nonlinear dynamical systems, or mathematical biology more generally would be an advantage; prior knowledge of the relevant biology is not required.
Title Patterns of synchrony in discrete models of gene networks Mathematical Medicine and Biology Dr Etienne Farcot One of the greatest challenges of biology is to decipher the relation between genotype and phenotype. One core difficulty in this task is that this relation is not a map; the proteins whichare produced thanks to the information contained in the genome are themselves used to control which parts of the genome are being used in a given situation. To understand the effectof this feedback between genes and their product it is crucial to consider the dynamics of this process.The term 'gene network' refers to a set of genes which regulate each other; understanding the dynamics of gene networks is thus crucial to decipher the genotype-phenotype relation. Mathematical models of gene networks have been proposed since the 1960's, among which the class of Boolean models has proved very successful. Because of the discrete nature of these models, the effect of time is often described using representations inspired by manufactured computing device, where the genes are updated in parallel, or in series. However, the updating scheme of genes could in principle be much more general. In this project, one will investigate the effect of such a generalization. One will consider arbitrary update schemes, both deterministic and stochastic, notably in relation to the dynamics of continuous models of gene networks.
Title Cell cycle desynchronization in growing tissues Mathematical Medicine and Biology Dr Etienne Farcot A very general phenomenon is the fact that coupled oscillators tend to naturally synchronize [1]. This simple fact takes many forms observable in real life: synchronization of applause after a concert, of neural cells, of flashing fireflies, and many other. A complete understanding of this phenomenon, depending on the particular dynamics of individual oscillators or the nature of their coupling, is still an on-going topic of mathematical research.However general, the synchronization of coupled oscillators is not a universal rule. In this project, one will study a situation where it indeed does not seem to occur: the division cycles of cells in a growing tissue does not seem to be synchronized, as observed in recent data form plant tissues. A plausible explanation is that the divisions of cell induce a change in the coupling between cells, which is mostly due to physical or chemical exchanges between neighbouring cells. Relying on simplified representations, one will consider the effect of growth, whereby the coupling structure of a system changes in time, on synchronization in populations of oscillators. [1] S.H. Strogatz and I. Stewart. Coupled oscillators and biological synchronization. Scientific American 269 (6) 102-109 (1993).
Title Mechanical models of growth-factor activation in asthmatic airway remodelling. Mathematical Medicine and Biology Dr Bindi Brook, Dr Reuben O'Dea, Prof Markus Owen Airway remodelling in asthma has until recently been associated almost exclusively with inflammation over long time-scales. Recent evidence suggests that stretch-induced activation of a pro-remodelling growth factor from the extracellular matrix during bronchoconstriction (as a result of airway smooth muscle contraction) plays an important role in airway remodelling. Traditional top-down continuum-mechanics approaches fail to account for the movement of cells relative to extracellular matrix and the corresponding adhesion-mediated opening of latent stores of the growth factor. Multiscale mechanics models of the airway tissue accounting for both cells, ECM and their interactions, will be developed in this project, building on a previous model of cell-ECM interactions [1]. The models will be informed by ongoing experiments of stretch-induced TGFbeta activation in precision-cut lung slices at the tissue scale, as well as atomic force microscopy experiments at the sub-cellular scale. There will also be the opportunity to design new experiments based on model results. Linda Irons, Markus R. Owen, Reuben D. O’Dea and Bindi S. Brook. Effect of Loading History on Airway Smooth Muscle Cell-Matrix Adhesions, Biophysical Journal 114, 2679–2690, 2018
Title From molecular dynamics to intracellular calcium waves Mathematical Medicine and Biology Dr Ruediger Thul, Prof Stephen Coombes 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 fire-diffuse-fire (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 life-threatening condition?) as well as to elucidate fundamental concepts in cell signalling.
Title Waves on a folded brain Mathematical Medicine and Biology Dr Daniele Avitabile, Prof Stephen Coombes 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.In more detail the project will consider models of neural activity represented by non-local integro-differential equations posed on both idealised and human realistic cortical structures.  The former will allow the development of analytical tools to understand the role of tissue heterogeneity and disorder in sculpting wave dynamics, such as the recently developed interface approach [1].  The latter will extend this so-called neural field approach [2] using cortical meshes from human connectome databases, making extensive use of spectral and finite element methods.  This applied mathematical project will be facilitated by interaction with colleagues from the Sir Peter Mansfield Imaging Centre. As well as exposing the PhD student to rich neuroimaging data-sets collected locally using cutting edge magnetoencephalography techniques, the project will contribute to our understanding of cortical waves in the functioning of the human brain. [1] S Coombes, H Schmidt and I Bojak 2012 Interface dynamics in planar neural field models, Journal of Mathematical Neuroscience, 2:9.[2] S Coombes, H Schmidt and D Avitabile 2014 Spots: Breathing, drifting and scattering in a neural field model, Neural Fields, Ed. S Coombes, P beam Graben, R Pottiest and J J Wright, Springer Verlag
Title Modelling macrophage extravasation and phenotype selection Mathematical Medicine and Biology Prof Markus Owen 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 micro-environment 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. This project will consider in some detail the mechanisms that regulate monocyte extravasation and macrophage phenotype selection. Initially, mathematical models will be formulated as systems of ordinary differential equations describing transitions between monocyte subpopulations (for example, those fully adherent to the vessel wall, and those that are actively moving through the wall), regulated by various signalling and adhesion molecules. Research on phenotype selection will determine whether the dynamics can be manipulated by subsequent external intervention. For example, if the system is bistable, it may be possible to force a switch from a deleterious to a beneficial phenotype. Relevant signal transduction pathways will be modelled in detail, and the law of mass-action will be used to derive systems of ODEs. Where possible, model reductions based on a separation of timescales will be used to simplify the system, and analytical and numerical approaches will be used to characterise steady state structure and bifurcations as tissue conditions vary. A Mantovani, S Sozzani, M Locati, P Allavena, and A Sica: Macrophage polarization: tumor-associated macrophages as a paradigm for polarized M2 mononuclear phagocytes. Trends Immunol., 23:549--555, (2002).
Title Next generation neural field models on spherical domains Mathematical Medicine and Biology Dr Rachel Nicks 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, short-term (working) memory and binocular rivalry. A typical formulation of a neural field equation is an integro-differential 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. S Coombes, R Nicks, S Visser and O Faugeras (2017) Standing and travelling waves in a spherical brain model: the Nunez model revisited, (In press)S Coombes and Á Byrne, (2016) Next generation neural mass models. In: Lecture Notes in Nonlinear Dynamics in Computational Neuroscience: from Physics and Biology to ICT Springer. (In Press.)S Coombes, P Beim Graben and R Potthast, (2014). Tutorial on Neural Field Theory. In: S Coombes, P Beim Graben and R Potthast, eds., Neural Fields: Theory and Applications Springer. 1-43
Title Multiscale modelling of signalling microdomains Mathematical Medicine and Biology Dr Ruediger Thul, Prof Stephen Coombes A key role of cells is to translate external signals into appropriate cellular responses. For example, when cells that line blood vessels experience weak stimulation, they initiate the expression of certain genes, while for strong stimuli, they begin to move. How cells accomplish such diverse responses is still an open question. What has transpired, though, is that so called microdomains are vital for cellular decision-making.Microdomains are small parts of a cell where molecular mediators and switches are concentrated in close proximity. This is advantageous since cell signalling intrinsically relies on molecules interacting, and if they are close to each other, chances are higher that signal transduction is successful. In many cases, these signalling pathways rely on small molecules that diffuse through the microdomain and hence can carry information from one molecular partner to the next. To appreciate the full potential of the signalling micordomains, it is crucial to have a comprehensive understanding of the dynamics of these diffusible messengers.In this project, we will use a combination of semi-analytical and numerical techniques to develop three dimensional models of signalling microdomains. In particular, we will investigate how the intracellular calcium concentration changes in space and time within microdomains, and how these changes affect signal transduction. Gaining deeper insights into microdomains is key for understanding for understanding healthy physiology such as fertilisation and muscle contraction as well as diseases such as immunodeficiency and neurological disorders. The model will be informed by experiments conducted at Oxford and Penn State University.
Title Exploiting network symmetries for analysis of dynamics on neural networks Mathematical Medicine and Biology, Industrial and Applied Mathematics Dr Rachel Nicks, Prof Stephen Coombes, Dr Paul Matthews 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 (phase-amplitude reduced or piecewise-linear) 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. M Golubitsky and I Stewart (2016) Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics Chaos 26, 094803P Ashwin, S Coombes and R Nicks (2016) Mathematical frameworks for network dynamics in neuroscience. Journal of Mathematical Neuroscience. 6:2.B. D. MacArthur, R. J. Sanchez-Garcia and J.W. Anderson (2008) Symmetry in complex networks, Discrete Applied Mathematics 156 (18), 3525-3531F Sorrentino, L M Pecora, A M Hagerstrom, T E Murphy, and R Roy (2016) Complete characterization of the stability of cluster synchronization in complex dynamical networks. Science Advances. 2, e1501737–e1501737.
Title Analysing and interpreting neuroimaging data using mathematical frameworks for network dynamics Mathematical Medicine and Biology, Mathematical Medicine and Biology Prof Stephen Coombes Modern non-invasive probes of human brain activity, such as magneto-encephalography (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 spatio-temporal 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 coupled-oscillator theory and phase-amplitude 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 time-series output of oscillator models on networks). The project will focus in particular on developing techniques for the analysis of dynamics on “multi-layer 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. P Ashwin, S Coombes and R Nicks (2016) Mathematical frameworks for network dynamics in neuroscience. Journal of Mathematical Neuroscience. 6:2.J Hlinka and S Coombes (2012) Using Computational Models to Relate Structural and Functional Brain Connectivity, European Journal of Neuroscience, Vol 36, 2137—2145M J Brookes, P K Tewarie, B A E Hunt, S E Robson, L E Gascoyne, E B Liddle, P F Liddle and P G Morris (2016) A multi-layer network approach to MEG connectivity analysis, NeuroImage 132, 425-438
Title Mathematical modelling of macromolecular capillary permeability Mathematical Medicine and Biology, Scientific Computation Dr Reuben O'Dea, Dr Matthew Hubbard 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 diffusion-convection 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.
Title Modelling lymphatic vessel pumping Mathematical Medicine and Biology, Scientific Computation Dr Bindi Brook, Dr Matthew Hubbard The efficacy of lymphatic vessels for collecting and pumping lymph from interstitial tissues is key in returning fluid back to the cardiovascular system. Impaired pumping can lead to debilitating conditions such as lymphedema (often as a result of lymph node removal during breast cancer surgery). Upscaling current models of lymphatic pumping to large networks is currently computationally inefficient. This project aims to develop network models of lymphatic vessels that accounts for the complex active mechanics as well as valve behaviour of these vessels to provide deeper insights into how mechanical therapies can be developed to alleviate lymphedema. C. D. Bertram, C. Macaskill, M. J. Davis, J. E. Moore Jr.. Contraction of collecting lymphatics: organization of pressure-dependent rate for multiple lymphangions, Biomechanics and Modeling in Mechanobiology, 2018. https://doi.org/10.1007/s10237-018-1042-7
Title Parameter estimation and uncertainty quantification in a multidisciplinary study of asthmatic airway remodelling Mathematical Medicine and Biology, Statistics and Probability Dr Bindi Brook, Dr Reuben O'Dea, Dr Theodore Kypraios Inflammation, airway hyperresponsiveness and airway remodelling are key characteristics of asthma, but it is unclear how they are interconnected. A recent comprehensive experimental in vivo asthma mouse study quantifying structural changes and how they relate to the inflammatory state in the airway, has generated an unprecedented amount of data to inform a mechanistic model airway remodelling in asthma [1].  Some parameter sets are well-defined from experimental data but others provide high levels of uncertainty in parameter value and model selection. This project seeks to understand how uncertainties in parameter estimation and model selection influence prediction, adding significant value to ongoing experiments, and dramatically increase the predictive power of the mechanistic model. M.R. Hill, C.J. Philp, C.K. Billington, A.L. Tatler, S.R. Johnson, R.D. O’Dea, B.S. Brook. A theoretical model of inflammation- and mechanotransduction- driven asthmatic airway remodelling, Biomechanics and Modeling in Mechanobiology https://doi.org/10.1007/s10237-018-1037-4, 2018
Title Optimising experiments for developing ion channel models Mathematical Medicine and Biology, Statistics and Probability Dr Gary Mirams, Dr Simon Preston 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].Aim: we have been developing novel experimental designs to provide more information about ion channel behaviour from shorter experiments. We would like to improve our techniques – to describe the ion current and also to characterise drug binding to ion channels (which can physically block them and reduce the current that flows to zero, sometimes leading to fatal heart rhythm changes). It is difficult to measure the rate at which drug/ion channel binding occurs and whether it occurs when the channels are open, closed, or both. These factors may be crucial in determining whether novel pharmaceutical compounds are likely to have side effects or not, and there is a need to develop efficient ways to measure them.Approach: this project will involve computational biophysical modelling (efficient numerical solution of nonlinear ODE systems); the application of statistical techniques to quantify our uncertainty in model parameters and model equations/structure; and some wet-lab laboratory electrophysiology experiments. We will design more information-rich experiments to reduce our uncertainty in the models we develop [4] and work closely with labs to test out experiments we design and improve them.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. A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” J. Physiol., vol. 117, pp. 500–544, 1952.D. Noble, A. Garny, and P. J. Noble, “How the Hodgkin – Huxley equations inspired the Cardiac Physiome Project,” vol. 11, pp. 2613–2628, 2012.M. Fink and D. Noble, “Markov models for ion channels : versatility versus identifiability and speed,” Philos. Trans. A., vol. 367, no. 1896, pp. 2161–79, Jun. 2009.G. R. Mirams, P. Pathmanathan, R. A. Gray, P. Challenor, and R. H. Clayton, “White paper: Uncertainty and variability in computational and mathematical models of cardiac physiology.,” J. Physiol., Mar. 2016. Please see Gary Mirams' research homepage for more information.
Title Quantum Thermal Engineering Mathematical Physics Prof Gerardo Adesso Notice: Undefined index: pmzlac in /maths/www/html/postgraduate/projects/index.php on line 560 Notice: Undefined index: pmzlac in /maths/www/html/postgraduate/projects/index.php on line 560 , In essence, quantum thermodynamics aims at unveling the connections between quantum physics and thermodynamics. Even if statistical mechanics teaches us that thermodynamics is only an effective theory emerging in the limit of large particle numbers, an individual quantum system can, in fact, convert heat to work or produce refrigeration in pretty much the same way as a car engine or a household refrigerator.Understanding the extent of this analogy would help us answer fundamental questions such as: Which ingredients are necessary for thermodynamic behaviour on an individual quantum system?, as well as more pragmatic questions such as: Can quantum effects help to bend the laws of quantum thermodynamics? Even if they cannot, can quantum thermodynamic devices assist in the deployment of quantum technologies?Quantum thermodynamics was born side by side with the theory of open quantum systems, back in the late 1970s, and developped slowly for over two decades. Only very recently, has the field started to attract an ever-increasing interest from a wide variety of communnities, encompassing condensed matter physics, quantum information theory, statistical mechanics, or quantum many-body physics.Most of the existing restuls in the field are limited to the canonical scenario of systems weakly coupled to equilibrium environments. In this research programme, we shall use instead cutting-edge tools for the treatment of open quantum systems srontgly coupled to their environments—i.e. exact methods, such as quantum Langevin equations, as well as perturbative techniques such as the polaron transformation, the reaction coordinate mapping, collisional models, dissipation into finite baths, etc.—.The goal of this programme is to generalise the existing results in the literature about the performance limitations of quantum thermodynamic cycles to the strong coupling regime. In turn, understanding the thermodynamics of strongly coupled open quantum systems would ultimately allow to formulate system-bath coupling schemes tailor-made so as to e.g. maximise the energy efficiency of a heat devise, or its power output, or both.The successful candidate should have a strong background in quantum mechanics and, ideally, also on thermodynamics, statistical mechanics, fluid mechanics, quantum optics, and/or molecular physics. Good analytical skills and high motivation are essential. No specific programming/numerical skills are required, although familiarity with e.g. Matlab, Mathematica or Python would be desirable. https://arxiv.org/abs/1305.2268https://arxiv.org/abs/1505.07835https://arxiv.org/abs/1508.06099
Title Coherent state path integrals in many-body physics Mathematical Physics Dr Alexander Ossipov Path integrals were originally introduced by Richard Feynman as an alternative formulation of a single particle quantum mechanics [1]. The advantage of this approach is that it deals with the classical action calculated on all possible quantum trajectories instead of non-commutative operators. The coherent state path integrals generalise this idea and can be applied to more complicated Hamiltonians such as many-body interacting systems [2]. Despite a wide range of successful applications of the coherent state path integrals in many areas of modern theoretical physics, usually they are calculated only perturbatively. The aim of this project is to develop a new non-perturbative approach for evaluation of the coherent state path integrals applied to many-body systems such as the Bose-Hubbard model. [1] H. Kleinert, "Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets", Singapore: World Scientific (2004).[2] A. Altland and B. D. Simons, "Condensed Matter Field Theory", Cambridge University Press (2010).
Title Models of Quantum Geometry Mathematical Physics Prof John Barrett Non-commutative 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 non-commutative 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 non-commutative geometry. This non-commutativity relates to the "internal space" i.e. a geometric structure at every point of space-time, and reveals itself in the non-abelian gauge groups, the Higgs and their couplings to fermion fields.The new idea is to use the non-commutative geometry also for space-time itself, which one hopes will eventually give a coherent explanation of the structure of space-time 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/ Jean Petitot: Noncommutative geometry and physics. https://arxiv.org/abs/1505.00132John W. Barrett, Lisa Glaser: Monte Carlo simulations of random non-commutative geometries. https://arxiv.org/abs/1510.01377John W. Barrett: Matrix geometries and fuzzy spaces as finite spectral triples. https://arxiv.org/abs/1502.05383Walter D. van Suijlekom: Noncommutative Geometry and Particle Physics. http://www.springer.com/gb/book/9789401791618
Title Hydrodynamic simulations of rotating black holes Mathematical Physics Dr Silke Weinfurtner 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). Carlos Barceló and Stefano Liberati and Matt Visser, "Analogue Gravity", Living Rev. Relativity 14, (2011), 3. URL: http://www.livingreviews.org/lrr-2011-3 W. G. Unruh, “Experimental Black-Hole Evaporation?” Phys. Rev. Lett. 46, 1351 – Published 25 May 1981. Silke Weinfurtner, Edmund W. Tedford, Matthew C. J. Penrice, William G. Unruh, and Gregory A. Lawrence, “Measurement of Stimulated Hawking Emission in an Analogue System”, Phys. Rev. Lett. 106, 021302 – Published 10 January 2011 Mauricio Richartz, Silke Weinfurtner, A. J. Penner, W. G. Unruh, “Generalised superradiant scattering”, Phys. Rev. D80:124016,2009Mauricio Richartz, Angus Prain, Stefano Liberati, Silke Weinfurtner, "Rotating black holes in a draining bathtub: superradiant scattering of gravity waves ", arXiv:1411.1662 [gr-qc]
Title Acceleration, black holes and thermality in quantum field theory Mathematical Physics Dr Jorma Louko 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 may 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, arXiv:1605.01316 and arXiv:1610.08455, and the 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.Theory supporting analogue spacetime laboratory experiments for detecting Hawking radiation and related effectsl. See arXiv:1807.04584 and references therein.
Title Quantum learning for large dimensional quantum systems Mathematical Physics Dr Madalin Guta Statistical inference and learning play an increasing role in Quantum Engineering and Quantum Metrology. The efficient statistical reconstruction of quantum states is a crucial enabling tool for current quantum engineering experiments in which multiple qubits can be prepared in exotic states. However, standard estimation methods such as maximum likelihood become practically unfeasible for systems of merely 10 qubits, due to the exponential growth in size of the Hilbert space. The aim of this project is to develop mathematical theory and investigate new methods for learning quantum states of large dimensional quantum systems. This stems from ongoing collaborations with Theo Kypraios and Ian Dryden (Statistics group, Nottingham), Cristina Butucea (Univesite Paris Est), Michael Nussbaum (Cornell), Jonas Kahn (Toulouse) and Richard Kueng (Caltech). In [1,2] we proposed and analysed faster estimation methods with close to optimal accuracy. The first goal is to better understand the behaviour of the estimators with respect to different measurement scenarios. Next, we would like to equip them with reliable confidence regions (error bars) which are crucial for experimental applications. Going beyond "full state tomography" 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. Possible directions to be explored include models based on matrix product states, neural networks, quantum time series, compressed sensing [3] and the study of the asymptotical structure of the statistical models [4]. The project will involve both theoretical and computational work at the overlap between quantum information theory and modern statistical inference. [1] M. Guta, J. Kahn, R. Kueng, J. A. Tropp, Fast state tomography with optimal error bounds, ArXiv: 1809.11162[2] C. Butucea, M. Guta, T. Kypraios, Spectral thresholding quantum tomography for low rank states, New Journal of Physics, 17 113050 (2015), ArXiv:1504.08295[3] D. Gross, Y. K. Liu, S. Flammia, S. Becker and J. Eisert, Physical Review Letters 105 150401 (2010) Arxiv:0909.3304[4] C. Butucea, M. Guta, M. Nussbaum, Local asymptotic equivalence of pure quantum states ensembles and quantum Gaussian white noise, Annals of Statistics, 46, 3676-3706 (2018) ArXiv:1705.03445 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 Mathematical Physics Dr Madalin Guta 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 input-output theory of quantum open systems [2] offers a clear conceptual understanding of quantum dynamical systems and continuous-time 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 input-output 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 information-disturbance trade-off which in the context of quantum dynamical systems becomes identification-control trade-off.The first steps in this direction were made in [4] which introduce the concept of asymptotic quantum Fisher information for "non-linear" 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 many-body open systems and high precision metrology for dynamical parameters (see arXiv:1411.3914). [1] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Universtiy Press (2000)[2] C. Gardiner, P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer (2004)[3] K. Zhou, J.C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, (1995)[4] M. Guta, Quantum information Fisher information and asymptotic normality in system identification for quantum Markov chains, Physical Review A, 83, 062324 (2011) Arxiv:1007.0434; M. Guta, J. Kiukas, Equivalence classes and local asymptotic normality in system identification for quantum Markov chains, Commun. Math. Phys. 335, 1397–1428 (2014)[5] M. Guta and N. Yamamoto, System identification for linear passive quantum systems Short version of the archive paper appeared in Proc. 52 IEEE CDC Conference Florence 2013 arXiv:1303.3771v2 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 Using artificial intelligence to design quantum optics experiments Mathematical Physics Prof Gerardo Adesso Notice: Undefined index: pmzpk in /maths/www/html/postgraduate/projects/index.php on line 560 Notice: Undefined index: pmzpk in /maths/www/html/postgraduate/projects/index.php on line 560 , In order for technologies to benefit from the inherent power of quantum mechanics, quantum states with specific properties must be engineered for particular applications. This project explores the use of an evolutionary algorithm – a subset of artificial intelligence which mimics natural selection – that has been previously developed to design quantum states for performing quantum-enhanced measurements [1]. PhD students will have the opportunity to extend this work in one or more of the following veins:i)                 Extend this technique to design quantum states and quantum experiments for a wide range of applications, including quantum computing, quantum cryptography, high precision measurements, and tests of fundamental physics. This strand will involve developing an understanding of the structure of quantum states – and the state space they live in – in order to design new quantum states with novel properties.ii)                Incorporate realistic experimental imperfections, such as photon loss and imperfect detectors, into the evolutionary algorithm. This will enable the algorithm to find quantum states that can then be made in the laboratory, including in our collaborators’ labs in Bristol, Oxford, Stuttgart (Germany), and MIT (Boston, USA). This strand of the project will involve working closely with experimentalists to enhance and optimise their current experiments, and to design new experiments that they will subsequently perform.iii)              As the algorithms used in i) and ii) become more complicated, more advanced artificial intelligence methods will need to be utilised and developed in order to make the simulations tractable. The use of machine learning (in particular reinforcement learning) and genetic algorithms will be explored, and the student will have the opportunity to collaborate with artificial intelligence researchers. Knott, P. A. "A search algorithm for quantum state engineering and metrology." New Journal of Physics 18.7 (2016): 073033. [ http://iopscience.iop.org/article/10.1088/1367-2630/18/7/073033 ]
Title Developing new relativistic quantum technologies Mathematical Physics Prof Ivette Fuentes 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 Unruh-Dewitt 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 non-trivial structure of spacetime. We will look for new protocols which exploit not only quantum but also relativistic resources for example, the non-local quantum correlations present in relativistic quantum fields. T. Downes, I. Fuentes, & T. C. Ralph Entangling moving cavities in non-inertial frames Physics Review Letters, 106, 210502 (2011)D. E. Bruschi, I. Fuentes, & J. Louko Voyage to Alpha Centauri: Entanglement degradation of cavity modes due to motion accepted in Physical Review D Rapid CommunicationsN. Friis, D. E. Bruschi, J. Louko & I. Fuentes Motion generates entanglement Submitted to Physical Review D
Title Homotopical algebra and quantum gauge theories Mathematical Physics Dr Alexander Schenkel 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 quasi-isomorphisms 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 (AQFT), 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 AQFT with homotopical algebra and which is expected to be a suitable mathematical framework for quantum gauge theories.Specific problems for a PhD project include model-independent developments in hoAQFT or explicit contructions of examples of quantum gauge theories within this framework. [1] M. Benini, A. Schenkel and R. J. Szabo, Homotopy colimits and global observables in Abelian gauge theory, Lett. Math. Phys. 105, no. 9, 1193 (2015) [arXiv:1503.08839 [math-ph]].[2] M. Benini and A. Schenkel, Quantum field theories on categories fibered in groupoids, Commun. Math. Phys. 356, no. 1, 19 (2017) [arXiv:1610.06071 [math-ph]].[3] M. Benini, A. Schenkel and U. Schreiber, The stack of Yang-Mills fields on Lorentzian manifolds, Commun. Math. Phys. 359, no. 2, 765 (2018) [arXiv:1704.01378 [math-ph]].[4] M. Benini, A. Schenkel and L. Woike, Operads for algebraic quantum field theory, arXiv:1709.08657 [math-ph].[5] M. Benini, A. Schenkel and L. Woike, Homotopy theory of algebraic quantum field theories,' arXiv:1805.08795 [math-ph].
Title Quantum optomechanics: radiation pressure at the single photon level Mathematical Physics Dr Tommaso Tufarelli Optomechanics investigates the interaction of quantised light (photons) with microscopic vibrating objects such as mirrors, dielectric membranes or levitated nanoparticles [1]. Such interaction takes place via radiation pressure, a phenomenon initially predicted by Johannes Kepler in 1619 in the context of astronomy, and nowadays observed even at the single-photon level. Among other applications, optomechanics embodies a promising experimental platform to probe quantum effects in massive objects, and hence investigate the classical/quantum boundary. The student will initially review a widely used effective Hamiltonian for cavity optomechanics (the "linear model") [2], which is analytically solvable and predicts the generation of nonclassical states of light as well as light-matter entanglement. Subsequently, he/she will delve into the more rigorous canonical quantization of an optomechanical system [3], and assess the crucial limitations of the more basic model. Unfortunately, the more rigorous "mircoscopic" Hamiltonian is currently intractable both analytically and numerically, so that novel approximation techniques will need to be developed to improve the basic model while retaining computability. An example of a preliminary study in this direction can be found in Ref. [4].The broad objective of the project will be to develop new optomechanical models that strike an optimal balance between reliability and tractability. These will then be used to verify and refine a number of theoretical predictions that have been made in the literature, tipically based on the linear model alone. Such predictions pertain a variety of applications of optomechanical systems, ranging from quantum information science to gravitational wave detection and Planck-scale physics [5]. At the same time, an improved theoretical description will give us an opportunity to explore new physical effects and applications of these systems.A futher ambitious goal of the project will be to develop a rigorous open quantum system model for optomechanics, improving the phenomenological approaches that are currently used in the literature.Depending on the inclinations of the student, more emphasis can be put on either analytical or numerical work (e.g. via Matlab, Python or Mathematica). [1] M. Aspelmeyer et al., Rev. Mod. Phys. 86, 1391 (2014) [2] S. Bose et al, Phys. Rev. A 56, 4175 (1997)[3] C. K. Law, Phys. Rev. A 51, 2537 (1995)[4] https://arxiv.org/abs/1711.06688[5] arXiv:1708.05659.
Title Many-body localization in quantum spin chains and Anderson localization Mathematical Physics Dr Alexander Ossipov Properties of wave functions in many-body systems is very active topic of research in modern condensed matter theory. Quantum spin chains are very useful models for studying quantum many-body physics. They are known to exhibit complex physical behaviour such as quantum phase transitions. Recently, they have been studied intensively in the context  of many-body localization. The key idea of this project is to explore the similarity between Hamiltonians of the spin chain models and the Anderson models on d-dimensional hypercube. In such models, single particle wave functions can be localized in space due to the famous phenomenon of the Anderson localization. Understanding of the relation between many-body localization and Anderson localization, quantum phase transitions in spin chains and the Anderson metal-insulator transition will be the main topic of this project.
Title Entanglement of non-interacting fermions at criticality Mathematical Physics Dr Alexander Ossipov Entanglement of the ground state of many-particle systems has recently attracted a lot of attention. For non-interacting 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 many-body system. The ground state entanglement is well understood now for free fermions, whose wavefuctions are simple plane waves. However, there are almost no analytical results available in the case where wavefuctions are non-trivial.This project will explore the ground state entanglement at the quantum critical point of the metal-insulator transition, where single particle wavefunctions are known to have self-similar fluctuations, characterised by non-trivial fractal dimensions.
Title Gravity and differential forms Mathematical Physics Prof Kirill Krasnov It turns out that solutions of 4D gravity can be lifted to solutions of a certain theory of differential 3-forms in 7 dimensions. This suggests a new perspective on 4D gravity and its problems, such as non-renormalisability. The aim of this project is to further study the geometry in 7 dimensions, in its relation to 4D geometry. Another aim is to understand the quantum properties of diffeomorphism invariant theories of differential forms in dimensions 7 and 6, and in particular understand how such theories renormalise at one and possibly two loops. This project lies at the intersection of differential geometry and quantum field theory. https://arxiv.org/abs/1611.07849https://arxiv.org/abs/1705.01741
Title Gravity at all scales Mathematical Physics Prof Thomas Sotiriou Various projects are available on the interplay between any of the following areas: quantum gravity, alternative theories of gravity, strong gravity and black holes.The description of phenomena for which gravity is important and also are in the realm of quantum physics requires a quantum gravity theory. Developing candidate theories to the extent that they can be confronted with observations is a very challenging task. In their classical limit these theories are mostly expected to deviate from General Relativity. In this sense, classical alternative gravity theories can be the interface between quantum gravity theory and classical phenomenology.The gravitational interaction is much less explored in regimes where gravity is strong, such as in the vicinity of black holes or veer compact stars. These system can be thought of as natural laboratories for gravity.The overall scope is to follow a synthetic approach which will combine results about the behavior of gravity at all different scales - from the quantum to astrophysical and cosmological system - in order to provide new insights.See also http://thomassotiriou.wix.com/challenginggr for further info
Title Quantum control of non-linear interactions Mathematical Physics Dr Tommaso Tufarelli Photons do not directly interact with each other, but effective interactions between them can be obtained by exploiting the mediation of matter. Perhaps the most common systems in which these effective interactions are achieved are nonlinear crystals. The sophistication of modern experiments, however, allows us to consider the mediation of quantum systems such as single atoms and optomechanical devices, where light is coupled to the vibrational degrees of freedom of a mesoscopic mirror via radiation pressure.For example, in typical experimental conditions, the interaction between light and a moving mirror (both of them being harmonic oscillators) is approximated well by "bilinear" Hamiltonians, such that when a photon is absorbed (emitted) an elementary excitation (phonon) of the oscillating mirror is created (annihilated) [1]. Truly nonlinear contributions that describe e.g. the conversion between two photons and one phonon are typically negligible. Such non-linear interactions (i.e. non-bilinear Hamiltonians) are, however, of great importance for many applications of photonics. To give a paradigmatic example, it has been shown that purely quadratic Hamiltonians are not sufficient to achieve universal quantum computation: a form of ‘nonlinearity’ is always required [2].By exposing a quantum system to a periodic control field one can change its properties substantially. For example, if a vibrating mirror or atom is manipulated appropriately, then its interactions with other systems may be modified and controlled to a significant extent. In turn, this may permit us to effectively enhance the non-linear photonic processes mediated by these systems. The goal of the project is indeed to identify optimally designed control sequences that realise robust non-linear interactions between photons.You would start the project by studying an interaction between two modes of resonator (cavity) mediated by an atom that is modeled by a ground state and two excited states (known as V-system). Each of the atomic transitions is close to resonance with a cavity mode. The goal of this sub-project would be to identify time-dependent control fields for the atom that results in nonlinear effective Hamiltonians for the light fields. While the project will involve some numerical work, this will be complemented and guided by various analytical techniques for the approximate description of time-dependent Hamiltonians.In the rest of your PhD project you will investigate a variety of quantum systems, and assess their capability of mediating photonic interactions. Optomechanical systems will likely be your object of study, and it is probable that other promising research directions will have arisen during the first few months. Since oscillating mirrors are mesoscopic objects, they typically feature substantial dissipation that can not be neglected. You would thus need to study the quantum control of dissipative systems, which requires a more general theoretical framework as compared to the purely Hamiltonian dynamics discussed earlier [3, 4]. Depending on your preliminary results and inclinations, you may identify a variety of long-term goals for this project. These may include the creation, verification and exploitation of light-matter entanglement, or the use of the achieved non-linear interactions for fundamental tests of quantum mechanics (e.g. the study of macroscopic superpositions and their implications in gravitational collapse models) [3, 4]. [1] M. Aspelmeyer et al., Rev. Mod. Phys. 86, 1391 (2014) [2] A Mari, J. Eisert, https://arxiv.org/abs/1208.3660[3] F. Haddadfarshi and F. Mintert, ArXiv e-prints (2015), 1510.05814.[4] F. Haddadfarshi, J. Cui, and F. Mintert, Phys. Rev. Lett. 114, 130402 (2015) This project will involve close collaboration with Dr. Florian Mintert of the Controlled Quantum Dynamics theory group, Imperial College London.
Title Scattering approach to topological insulators and superconductors Mathematical Physics Dr Alexander Ossipov Topological insulators and superconductors are one of the hottest topics in the modern condensed matter theory. They represent a new phase of matter characterised by a rather unusual quantum-mechanical behaviour of electrons. Topological insulators are insulating materials, but they conduct electricity via special surface states. The surface states are topologically protected due to quantum symmetries and cannot be destroyed by disorder. Nevertheless the role of disorder is essential for understanding the properties of topological insulators. A novel approach to disordered quantum system based on the scattering formalism has been proposed recently [1]. The aim of the project is to generalise and apply this approach to topological insulators and superconductors. [1] A. Ossipov, "Scattering approach to Anderson localisation", Phys. Rev. Lett. 121, 076601 (2018).
Title Topological Resonances on Graphs Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann 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 well-known mechanisms for resonances. In this project these signatures will be analysed in detail. GNUTZMANN, S., SMILANSKY, U. and DEREVYANKO, S., 2011. Stationary scattering from a nonlinear network Physical Review A. 83, 033831
Title Quantum Chaos in Combinatorial Graphs Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann 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. Idan Oren, Amit Godel and Uzy Smilansky Trace formulae and spectral statistic for discrete Laplacians on regular graphs (I) J. Phys. A: Math. Theor. 42 (2009) 415101Idan Oren, Amit Godel and Uzy Smilansky Trace formulae and spectral statistic for discrete Laplacians on regular graphs (II) J. Phys. A: Math. Theor. 43 (2010) 225205
Title Quantum Searching in Random Networks Mathematical Physics, Industrial and Applied Mathematics Prof Gregor Tanner, Dr Sven Gnutzmann The project deals with properties of quantum networks, that is, of networks on which unitary (wave-) evolution takes place along edges with scattering at the vertices. Such systems have been studied in the context of quantum information as well as in quantum chaos. It has been noted that a quadratic speed up of quantum random walks on these networks overclassical random walks can be found on certain graphs with applications for search algorithms and search engines. The speed up has often be traced back to wave interference effects due to symmetries in the quantum propagation. Recently, it has been shown, that quantum searching can also been undertaken on random graphs, that is, graphs for which connections between edges are given only wth a certain probability - so called Erdös-Rényi graphs. We will explore this new set-up for quantum searching and develop statisticsal models for the arrival times and success probabilities as well extend the model to realistic graph set-ups. The project will be a good mix of graph theory, quantum mechanics and will require both analytical and numerical skills. Quantum walks and quantum search on graphene lattices, Iain Foulger, Sven Gnutzmann, and Gregor Tanner, Phys. Rev. A 91, 062323 (2015)Quantum Search on Graphene Lattices, Iain Foulger, Sven Gnutzmann, and Gregor Tanner, Phys. Rev. Lett. 112, 070504 (2014)Wave Communication across Regular Lattices ,Birgit Hein and Gregor Tanner, Phys. Rev. Lett. 103, 260501 (2009)Spatial Search by Quantum Walk is Optimal for Almost all Graphs, Shantanav Chakraborty, Leonardo Novo, Andris Ambainis, and Yasser Omar, Phys. Rev. Lett 116, 100501 (2016)
Title Supersymmetric field theories on quantum graphs and their application to quantum chaos Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann 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 mean-field 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 field-theoretic approach. GNUTZMANN, S, KEATING, J.P. and PIOTET, F., 2010. Eigenfunction statistics on quantum graphs Annals of Physics. 325(12), 2595-2640GNUTZMANN, S., KEATING, J.P. and PIOTET, F., 2008. Quantum Ergodicity on Graphs PHYSICAL REVIEW LETTERS. VOL 101(NUMB 26), 264102GNUTZMANN, S. and SMILANSKY, U., 2006. Quantum graphs: applications to quantum chaos and universal spectral statistics Advances in Physics. 55(5-6), 527-625GNUTZMANN, S. and ALTLAND, A., 2005. Spectral correlations of individual quantum graphs. Physical Review E. 72, 056215GNUTZMANN, S. and ALTLAND, A., 2004. Universal spectral statistics in quantum graphs. Physical Review Letters. 93(19), 194101
Title Pseudo-orbit expansions in quantum graphs and their application Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann 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 so-called pseudo-orbits. In this project these methods will be develloped further and applied to questions related to quantum chaos and random-matrix theory. Daniel Waltner, Sven Gnutzmann, Gregor Tanner, Klaus Richter, A sub-determinant approach for pseudo-orbit expansions of spectral determinants, arXiv:1209.3131 [nlin.CD] Ram Band, Jonathan M. Harrison, Christopher H. Joyner, Finite pseudo orbit expansions for spectral quantities of quantum graphs, arXiv:1205.4214 [math-ph]
Title The ten-fold way of symmetries in quantum mechanics. An approach using coupled spin operators. Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann About 50 years ago Wigner and Dyson proposed a three-fold symmetry classification for quantum mechanical systems -- these symmetry classes consisted of time-reversal invariant systems with integer spin which can be described by real symmetric matrices, time-reversal invariant systems with half-integer spin which can be described by real quaternion matrices, and systems without any time-reversal symmetry which are described by complex hermitian matrices. These three symmetry classes had their immediate application in the three classical Gaussian ensembles of random-matrix 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 anti-commutators.The classification was completed by Altland and Zirnbauer who have shown that there are essentially only seven further symmetry classes on top of the Wigner-Dyson classes leading to what is now known as the 'ten-fold way'. All symmetry classes have applications in physics. The new symmetry classes are realised by various cases of the Dirac equation and the Bogoliubov-de Gennes equation. For a long time people have thought of these symmetries only in the context of many-body physics or quantum field theory. However there are simple quantum mechanical realisations of all ten symmetry classes which in terms of two coupled spins where the classification follows from properties of the coupling parameters and of the irreducible SU(2) representations on which the spin operators act. This project will explore these simple representations in the quantum mechanical and semiclassical context. One goal will be to understand the implications of the quantum mechanical symmetries for the corresponding classical dynamics which appears in the semiclassical limit of large spins. M.R. Zirnbauer, Riemannian symmetric superspaces and their origin in random matrix theory, J. Math. Phys. 37, 4986 (1996)A. Altland, M.R. Zirnbauer, Non-standard symmetry classes in mesoscopic normal-/superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997)S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. I. Construction and numerical results, Physical Review E 69, 056219 (2004)S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. II. Semiclassical approach. Physical Review E 69, 056220 (2004)
Title Nonlinear waves in waveguide networks Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann Many wave guides (such as optical fibres) show a Kerr-type 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 non-trivial 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 Y-junctions (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). Sven Gnutzmann, Uzy Smilansky, and Stanislav Derevyanko, Stationary scattering from a nonlinear network, Phys. Rev. A 83, 033831 (2011)
Title The statistics of nodal sets in wavefunctions Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann 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 3-dimensional 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. Galya Blum, Sven Gnutzmann, Uzy Smilansky, Nodal domains statistics- a criterion for quantum chaos, Phys. Rev. Lett. 88, 114101 (2002)Alejandro G. Monastra, Uzy Smilansky and Sven Gnutzmann, Avoided intersections of nodal lines, J. Phys. A. 36, 1845-1853 (2003)G. Foltin, S. Gnutzmann and U. Smilansky, The morphology of nodal lines- random waves vs percolation, J. Phys. A 37, 11363 (2004)Yehonatan Elon, Sven Gnutzmann, Christian Joas and Uzy Smilansky, Geometric characterization of nodal domains: the area-to-perimeter ratio, J. Phys. A 40, 2689 (2007) S. Gnutzmann, P. D. Karageorge and U Smilansky, Can one count the shape of a drum?, Phys. Rev. Lett. 97, 090201 (2006)
Title Coherent states, nonhermitian Quantum Mechanics and PT-symmetry Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann 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 non-hermitian 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 PT-symmetric 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. S.Gnutzmann, M Kus, Coherent states and the classical limit on irreducible SU3 representations, J. Phys. A 31, 9871 (1998)E.-M. Graefe, M. Höning, H.J. Korsch, Classical limit of non-Hermitian quantum dynamic - a generalized canonical structure, J. Phys A 43, 075306 (2010)E.-M. Graefe, R. Schubert, Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians, J. Phys. A 45, 244033 (2012)C.M. Bender, M. DeKieviet, S.P. Klevansky, PT Quantum Mechanics, Philosophical Transactions of the Royal Society A 371, 20120523 (2013)
Title Geometry and analysis of Schubert varieties Number Theory and Geometry Dr Sergey Oblezin 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 infinite-dimensional setting (loop groups and, more generally, Kac-Moody groups), and such generalizations provide new constructions in infinite-dimensional 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. A.Gerasimov; D.Lebedev; S.Oblezin, Parabolic Whittaker functions and topological field theories I. Commun. Number Theory Phys. 5 (2011), no. 1, 135–201
Title Orthogonal polynomials in probability, representation theory and number theory Number Theory and Geometry Dr Sergey Oblezin 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. I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition.I.G.Macdonald, Affine Hecke algebras and orthogonal polynomials.A.Gerasimov; D.Lebedev; S.Oblezin, Baxter operator formalism for Macdonald polynomials. Lett. Math. Phys. 104 (2014), no. 2, 115–139
Title Number theory in a broad context Number Theory and Geometry Prof Ivan Fesenko 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 L-function 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 K-theory, 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.
Title Computational methods for elliptic curves and modular forms Number Theory and Geometry Dr Christian Wuthrich 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 Mordell-Weil 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 p-adic theory of modular forms and the so-called 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.
Title Variants of automorphic forms and their L-functions Number Theory and Geometry Dr Nikolaos Diamantis 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 L-functions, which are the subject of some of the most important conjectures of Mathematics. In recent years, investigations into the theory of automorphic forms have led into the study of variants of automorphic forms and of their L-functions, such as quasi-modular forms, harmonic Maass forms, mock modular forms, higher order modular forms and multiple Dirichlet series. In most cases, the motivation for introducing these objects was not just to generalize the classical automorphic forms and their L-functions, but to obtain novel tools to address already stated number theoretic problems. The techniques associated with these new objects in turn raise new interesting questions  and highlight connections beyond the original motivating problems. For example, the theory of harmonic Maass forms and modular forms has been used to resolve problems in partitions of numbers, and higher order modular forms have been applied to Percolation Theory problems in Physics.As these techniques have only recently been discovered, they lead to a number of very interesting open questions, e.g. how to construct mock modular forms encoding specific partition functions, how to determine the arithmetic nature of high-order forms or how to use the theory of multiple Dirichlet series to bound moments of the Riemann zeta function. Questions of this type, are highly relevant both for the outstanding problems in classical automorphic forms and for the further development of the new subjects themselves. Therefore, many of these questions are very appropriate for a PhD project.
Title Thermal characterisation of the building fabric under uncertainty Scientific Computation Dr Marco Iglesias The built environment is responsible for 45% of all UK carbon emissions with approximately 27% attributed to the domestic sector and 18% to non-domestic buildings. Reducing the energy demand in the built-environment is thus essential for the UK decarbonisation policy which legislates an 80% reduction of its 1990 green house gas emissions by 2050. The existing housing stock is a primary target for reductions of the energy demand since it is estimated that up to 85% of existing buildings will be standing by 2050. An accurate characterisation of the thermal performance of the existing housing stock in the UK is thus needed to inform large-scale cost-effective policies for retrofit intervention that can effectively contribute towards achieving those decarbonisation targets. Unfortunately, existing approaches for the in-situ characterisation of the building fabric (including ISO standards) cannot accurately characterise the thermal performance of buildings in the presence of thermal bridge effects that arise from heterogeneities, irregularities and/or abrupt changes and discontinuities in the thermophysical properties of the building fabric. In particular, these approaches cannot capture thermal bridge effects due to fabric degradation and moisture condensation which are likely to be found in existing dwellings.This challenging research will develop novel thermal imaging algorithms capable of characterising, with an accurate measure of uncertainty, the thermal performance of the building fabric in the presence of a general class of thermal bridge effects. This project will build upon state-of-the-art Bayesian algorithms for inverse problems that have been successfully applied for tomographic inversions in the context of groundwater flow [1], electrical impedance tomography [1], resin transfer moulding [2], and the characterisation of thermophysical properties of walls [3,4]. The techniques developed in this project will be validated with real experiments. Although highly ambitious, this proposed research has enormous potential to revolutionise current approaches for in-situ characterisation of the thermal performance of buildings thereby enhancing the predictive capabilities of existing housing stock models. [1] Iglesias, M.A. 2016. “A Regularizing Iterative Ensemble Kalman Method for PDE-Constrained Inverse Problems.” Inverse Problems 32 (2):025002. [2] Iglesias, M.A., M. Park, and M. Tretyakov. 2017. “Bayesian Inverse Problems in Resin Transfer Molding.” Inverse Problems, 34(10), 105002, 2018[3] Iglesias, M.A., Sawlan Z., Scavino M., Tempone R., and C. Wood. 2018. “Bayesian Inferences of the Thermal Properties of a Wall Using Temperature and Heat Flux Measurements.” International Journal of Heat and Mass Transfer 116 (Supplement C):417–31.[4] De Simon,L. M. A. Iglesias, B. Jones and C. Wood. 2017. “Quantifying Uncertainty in Thermal Properties of Walls by Means of Bayesian Inversion.” Energy and Buildings, 177, 220-245, 2018
Title Bayesian geometric inverse problems Scientific Computation Dr Marco Iglesias The aim of this project is to develop computational Bayesian techniques for the solution of geometric inverses problems that arise in a wide range of applications such as subsurface geophysics, manufacturing engineering and the built environment. Examples of geometric inverse problem that will be addressed with the techniques developed in this project are: (i) inference of fracture networks and/or conduits in Karst aquifers during the injection of CO2 for its geologic storage; (ii) detection of defects in reinforced preform during the resin infusion process in the fabrication of composite materials; (iii) inference of internal structures (e.g. cavities) in building structures such as walls with the aim of improving estimates of energy consumption. These problems have an underlying (forward) model described by Partial Differential Equation(s) (PDE) with input parameters associated to some (unknown) physical property of interest; the inverse problem is to infer this property from noisy observations of the solution of the PDE. In the context of problems (i)-(iii), sophisticated geometric parameterizations are often required to enable an accurate and realistic characterization of these properties (e.g. channelized structures in Karst networks). This project will develop computational hierarchical Bayesian methodologies to infer those geometry-constrained properties within an infinite-dimensional Bayesian framework for PDE-constrained inverse problems.
Title Discontinuous Galerkin Finite Elements for Moving Boundary Problems Scientific Computation Dr Matthew Hubbard Many physical and chemical processes, typified by those related to fluid flow, can be modelled mathematically using partial differential equations. These can usually only be solved in the simplest of situations, but solutions in far more complex cases can be approximated using numerical and computational techniques. Traditional approaches to providing these computational simulations have typically modelled the evolution of the system by approximating the equations on a uniform mesh of points covering a domain with a fixed boundary. However, many situations (consider the spreading of a droplet, for example), naturally suggest a domain which evolves with the flow, while the main focus of interest in others (say the movement of a shock wave up and down an aeroplane wing) is in following the motion of a sharp internal feature. For accuracy and efficiency a computational method should not only approximate the partial differential equations appropriately, but also move the computational mesh in a manner which follows such features.Recent research has developed a finite element approach to the adaptive approximation of time-dependent physical problems involving moving boundaries or interfaces. It has been deliberately designed to preserve inherent properties (such as conservation principles and invariances) of the underlying partial differential equations and hence of the system the mathematics is intended to represent. Extremely promising results have been obtained for a wide range of problems in one and two space dimensions, but the applicability of the approach is still limited (as are all moving mesh methods) by the potential for the computational mesh to tangle''.The aim of this project will be to develop an alternative approach, derived within the same framework, which takes advantage of the additional flexibility inherent in the discontinuous Galerkin finite element framework. This has the potential to reduce the occurrence of mesh tangling and to greatly improve the robustness of the method when modelling problems involving complex, interacting features and when using different monitor to govern the movement of the mesh. M.J.Baines, M.E.Hubbard, P.K.Jimack, Velocity-based moving mesh methods for nonlinear partial differential equations, Commun Comput Phys, 10(3):509-576, 2011.M.J.Baines, M.E.Hubbard, P.K.Jimack, A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries, Appl Numer Math, 54:450-469, 2005.
Title Foundations of adaptive finite element methods for PDEs Scientific Computation, Algebra and Analysis Dr Kris van der Zee Foundations of adaptive finite element methods for PDEs(Or- Why do adaptive methods work so well?) 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 degrees-of-freedom 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: * How can the theory be extended to, for example, nonsymmetric problems, nonlinear problems, or time-dependent problems? * What about nonstandard discretisation techniques such as, discontinuous Galerkin, isogeometric analysis, or virtual element methods? Depending on the interest of the student, several of these issues (or others) can be addressed.Also, the student is encouraged to suggest a second supervisor, possibly from another group! J.M. Cascon, C. Kreuzer, R.H. Nochetto, and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 26 (2008), pp. 2524-2550
Title Partitioned-domain concurrent multiscale modelling Scientific Computation, Mathematical Medicine and Biology Dr Kris van der Zee Partitioned-domain concurrent multiscale modelling(Or- How does one get cheap, but accurate, models?)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 multiscale-modeling type is known as partitioned-domain concurrent modelling. This type addresses problems that require a fine-scale 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 fine-scale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem.Challenges for students: * How can one couple, e.g., discrete (particle) systems with continuum (PDE) models? * Or a fine-scale PDE with a coarse-scale PDE?* How can one decide on the size and location of the fine-scale domain?* Is it possible to proof the numerically observed efficiency of concurrent multiscale models? * Can the multiscale methodology be applied to biological growth phenomena (e.g., tumours) where one couples cell-based (agent-based) models with continuum PDE models?Depending on the interest of the student, several of these issues (or others) can be addressed.Also, the student is encouraged to suggest a second supervisor, possibly from another group! J.T. Oden, S. Prudhomme, A. Romkes, and P.T. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput. 28 (2006), pp. 2359-2389
Title Phase-field modelling of evolving interfaces Scientific Computation, Mathematical Medicine and Biology Dr Kris van der Zee Phase-field modelling of evolving interfaces(Or – How does one effectively model and simulate interfacial phenomena?)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 sharp-interface descriptions, such as parametric and level-set methods, and those with diffuse-interface descriptions, commonly referred to as phase-field models.Challenges for students:* Can one develop a phase-field model for a particular interfacial phenomenon? * What are the foundational laws underpinning phase-field models? * What is the connection between sharp-interface models and phase-field models? * Can one design stable time-stepping schemes for phase-field models? * Or efficient adaptive spatial discretisation methods?Depending on the interest of the student, one of these issues (or others) can be addressed.Also, the student is encouraged to suggest a second supervisor, possibly from another group! H. GOMEZ, K.G. VAN DER ZEE, Computational Phase-Field Modeling, in Encyclopedia of Computational Mechanics, Second Edition, E. Stein, R. de Borst and T.J.R. Hughes, eds., to appear
Title Uncertainty quantification in geoelectrical imaging and monitoring Scientific Computation, Scientific Computation Dr Marco Iglesias Over the past decade, geoelectrical imaging has become the leading technology for continuously monitoring the shallow subsurface volumetrically and in real time. This technology plays a crucial role in assisting industrial and governmental stakeholders in addressing some of the most pressing societal challenges that impact on the subsurface, such as unconventional energy sources, carbon sequestration, waste management and groundwater contamination. However, current geoelectrical imaging techniques do not allow appropriate quantification of the uncertainty intrinsic to (1) the subsurface and (2) conventional image reconstruction methods based on deterministic inversion. The absence of uncertainty quantification in the subsurface has profound detrimental effects on evidence-based decision-making, the assessment and management of risks associated with subsurface hazards, the design of cost-effective remediation strategies and the improvement of stakeholder and public acceptance in the context of potentially controversial uses of the subsurface (e.g. unconventional hydrocarbons, CO2 storage). This project will develop Bayesian methodologies for geoelectrical imaging with the ultimate aim of inferring and quantify uncertainty in subsurface properties in the presence of realistic geologies. The project will be focused on applications that include carbon capture and storage, unconventional hydrocarbons and underground gas storage
Title Geometric integration of stochastic differential equations Scientific Computation, Statistics and Probability Prof Michael Tretyakov For many applications (especially, in molecular dynamics and Bayesian statistics), it is of interest to compute the mean of a given function with respect to the invariant law of the diffusion, i.e. the ergodic limit. To evaluate these mean values in situations of practical interest, one has to integrate large dimensional systems of stochastic differential equations over long time intervals.  Computationally, this is a challenging problem.   Stochastic geometric integrators play an important role in long-time simulation of dynamical systems with high accuracy and relatively low cost.  The project involves construction of new efficient numerical methods for ergodic stochastic differential equations and stochastic numerical analysis of properties of the methods. We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a very good background in Probability and has good computational skills. Milstein, G.N. and Tretyakov, M.V. (2004) Stochastic Numerics for Mathematical Physics. Series: Scientific Computation, Springer
Title Numerical methods for stochastic partial differential equations Scientific Computation, Statistics and Probability Prof Michael Tretyakov 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.
Title Bayesian inversion in resin transfer moulding Scientific Computation, Statistics and Probability Prof Michael Tretyakov Supervisors: Dr Marco Iglesias1, Dr Mikhail Matveev2, Prof Michael Tretyakov1University of Nottingham, University Park, Nottingham NG7 2RD, UK1 School of Mathematical Sciences2 Polymer Composites Group, Faculty of Engineering This project will be based at the University of Nottingham in the School of Mathematical Sciences and the Faculty of Engineering. The use of fibre-reinforced composite materials in aerospace and automotive industries and other areas has seen a significant growth over the last two decades. One of the main manufacturing processes for producing advanced composites is resin transfer moulding (RTM). The crucial stage of RTM is injection of resin into the mould cavity to fill empty spaces between fibres; the corresponding process is described by an elliptic PDE with moving boundaries. Imperfections of the preform result in uncertainty of its permeability, which can lead to defects in the final product. Consequently, uncertainty quantification (UQ) of composites’ properties is essential for optimal RTM. One of important UQ problems is quantification of the uncertain permeability. The objectives of this PhD project include (i) to construct, justify and test efficient algorithms for the Bayesian inverse problem within the moving boundary setting and (ii) to apply the algorithms to real data from composite laboratory experiments.Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a background in PDEs, Probability and Statistics and has exceptional computational skills.For any enquiries please email:  Marco.Iglesias@nottingham.ac.uk or Michael.Tretyakov@nottingham.ac.uk or Mikhail.Matveev@nottingham.ac.uk
Title Computational Finance Scientific Computation, Statistics and Probability Prof Michael Tretyakov Computational Finance is the key element for successful risk management at investment banks and hedge funds and it is also a growing area on the interface between finance, computational mathematics and applied probability. Pricing and hedging financial derivatives, evaluating risks of default for financial product and firms, satisfying requirements of the Basel Accord, etc.  -- all require sophisticated modelling and reliable calibration of the models. These aims cannot be achieved without efficient numerical techniques which form the area of computational finance.  The project will aim at developing new, efficient computational techniques related to finance. Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a good background in Probability and Stochastic Analysis, some knowledge of Finance and has exceptional computational skills.
Title Index policies for stochastic optimal control Statistics and Probability Dr David Hodge Since the discovery of Gittins indices in the 1970s for solving multi-armed 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 multi-armed 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.Potential project topics could include optimal decision making in the areas of queueing theory, inventory management, machine maintenance and communication networks. Keywords: multi-armed bandits, dynamic programming, Markov decision processes
Title Semi-Parametric Time Series Modelling Using Latent Branching Trees Statistics and Probability Dr Theodore Kypraios A class of semi-parametric 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 continuous-time 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.
Title New analytical and simulation tools in clinical oncology Statistics and Probability Dr Gilles Stupfler, Dr Christopher Brignell Cancer drug developers seek to show that satisfactory drug exposure increases the rate of beneficial patient responses, but without accounting for the risk of adverse effects. The student will develop fully parametric sub-distribution hazard models that account for such competing events, handle covariate information such as body weight and are more flexible than the standard proportional hazards model, so that a patient's individual risk of wasting limited life expectancy during treatment can easily be quantified and drug exposure optimised. This project will be led jointly by the School of Mathematical Sciences and the private company BAST Inc Ltd., based in Loughborough, through their scientific director Dr Joachim Grevel. Weekly supervisory meetings with BAST, with at least two years based there, will provide excellent training and career development opportunities. Studentship fully funded by BAST Inc Ltd.
Title Ion channel modelling Statistics and Probability Prof Frank Ball 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.
Title Optimal control in yield management Statistics and Probability Dr David Hodge 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 in-house 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 choice-based 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.
Title Stochastic Processes on Manifolds Statistics and Probability Prof Huiling Le 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.
Title Statistical Theory of Shape Statistics and Probability Prof Huiling Le 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 3-spaces 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 multi-disciplinary area of research which has only become very active recently. Nottingham is one of only a handful of departments at which it is active.
Title Asymptotic techniques in Statistics Statistics and Probability Prof Andrew Wood Asymptotic approximations are very widely used in statistical practice. For example, the large-sample 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. areSaddlepoint and related approximationsRelative error analysisApproximate conditional inferenceAsymptotic methods in parametric and nonparametric Bayesian Inference
Title Computational methods for fitting stochastic epidemic models to data Statistics and Probability Dr Theodore Kypraios, Prof Philip O'Neill Despite recent advances in the development of computational methods for fitting epidemic models to data, many of these methods work best in small-scale settings where the study population is not especially big or the models have relatively few parameters.  There is a need to develop methods which are appropriate to large-scale settings. Furthermore, nearly all existing methods rely on parametric approaches (e.g. models based on specific underlying assumptions), but recent work has shown that Bayesian nonparametric approaches can be successfully adapted to this area. This project involves developing novel computationally efficient methods to fit  both parametric and non-parametric models to data in situations where the existing methods are infeasible. Xu, X., Kypraios, T. and O'Neill, P.D. ( 2016). Bayesian non-parametric inference for stochastic epidemic models using Gaussian Processes. Biostatistics. 17(4), 619-633Kypraios, T. and O'Neill, P.D. ( 2017) Bayesian nonparametrics for stochastic epidemic models. To appear, Statistical Science.
Title Statistical shape analysis with applications in structural bioinformatics Statistics and Probability Dr Christopher Fallaize 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.An example of the unlabelled problem comes from the area of structural bioinformatics, when we wish to compare the 3-d shapes of protein molecules. This is important, since the shape of a protein is vital to its biological function. The landmarks could be, for example, the locations of particular atoms, and the correspondence between atoms on different proteins is unknown. This project will explore methods for unlabelled shape alignment, motivated by the problem of protein structure alignment. Possible topics include development of: i) efficient MCMC methods to explore complicated, high-dimensional distributions, which may be highly multimodal when considering large proteins;ii) fast methods for pairwise alignment, needed when a large database of structures is to be searched for matches to a query structure;iii) methods for the alignment of multiple structures simultaneously, which greatly exacerbates the difficult problems faced in pairwise alignment. Green, P.J. and Mardia, K.V. (2006) Bayesian alignment using hierarchical models, with applications in protein bioinformatics. Biometrika, 93(2), 235-254.Mardia, K.V., Nyirongo, V.B., Fallaize, C.J., Barber, S. and Jackson, R.M. (2011). Hierarchical Bayesian modeling of pharmacophores in bioinformatics. Biometrics, 67(2), 611-619.
Title High-dimensional molecular shape analysis Statistics and Probability Prof Ian Dryden In many application areas it is of interest to compare objects and to describe the variability in shape as an object evolves over time. For example in molecular shape analysis it is common to have several thousand atoms and a million time points. It is of great interest to reduce the dimension to a relatively small number of dimensions, and to describe the variability in shape and coverage properties over time. Techniquesfrom manifold learning will be explored, to investigate if the variability can be effectively described by a low dimensional manifold. A recent method for shapes and planar shapes called principal nested spheres will be adapted for3D shape and surfaces. Also, other non-linear dimension reduction techniques such asmultidimensional scaling will be explored, which approximate the geometry of the higher dimensional manifold. The project will involve collaboration with Dr Charlie Laughton of the School of Pharmacy. Jung, S., Dryden, I.L. and Marron, J.S. (2012). Analysis of principal nested spheres. Biometrika, 99, 551–568.
Title Geometry-based methods for statistics on functions and curves Statistics and Probability Dr Karthik Bharath How can we define an average closed curve on the plane from a random sample of many such curves, such that it is invariant to certain transformations (e.g. rotations)? Answers to such questions have far-reaching impact on analysis of images arising in numerous discplines (e.g. images of brain tumours). Extending statistical methodology from finite- to infinite-dimensional linear and nonlinear settings requires an improved understanding of probability distributions on constrained function spaces. Employing stochastic processes as a tool to study the interplay between probability and geometry, the project will address some fundamental issues that arise in the development of statistical theory and methodology for data in the form of functions and curves. Functional and Shape Data Analysis. (2016). A. Srivastava and E. Klassen. Springer, NY.
Title Analysis and forecasts of long hospital stays Statistics and Probability Dr Gilles Stupfler, Prof Ian Dryden Most hospital stays are very short, typically lasting a few days. This project will take the opposite view of estimating the length of the longest hospital stays, which place a strong material and financial burden on hospital trusts and the NHS. We will develop estimation techniques for a novel extreme value model which will be able to link a variable of interest (here, length of stays) to with a large set of covariates (here, such as age, gender, socioeconomic status…) and that will be able to handle the problem of missing data caused by the random censoring typically present in medical data sets. The method will be applied to real data from the English NHS.
Title Extreme quantile estimation with dependent censoring Statistics and Probability Dr Gilles Stupfler, Prof Andrew Wood Statistical analysis of extremes in missing data frameworks is a recent developing field. It has been exploited to estimate survival times of very strong/weak patients to chronic diseases. The literature on this topic has focused on the case of independent censoring. However, cases when there are suspicions of dependence have been reported several times over the last decades; in medical studies, a common cause of the violation of the independence hypothesis is a sizeable number of patient dropouts. This project will exploit very recent results on extremes with dependent right-censoring to design estimators of extreme quantiles. The methods will be showcased on real sets of medical data.
Title Modelling hospital superbugs Statistics and Probability, Mathematical Medicine and Biology Prof Philip O'Neill, Dr Theodore Kypraios The spread of so-called superbugs such as MRSA and other Antimicrobial Resistant pathogens 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 high-resolution data (e.g. whole genome sequences). Worby C., O'Neill P.D, Kypraios T., Robotham J.V., De Angelis D, Cartwright E.J., Peacock S.J., Cooper B.S. Reconstructing transmission trees for communicable diseases using densely sampled genetic data. Ann Appl Stat. 2016;10(1):395-417Worby, C., Jeyaratnam, D., Robotham, J. V., Kypraios, T., O.Neill, P. D., De Angelis, D., French, G. and Cooper, B. S. (2013) Estimating the effectiveness of isolation and decolonization measures in reducing transmission of methicillin-resistant Staphylococcus aureus in hospital general wards. American Journal of Epidemiology 177 (11), 1306-1313.Kypraios, T., O'Neill, P. D., Huang, S. S., Rifas-Shiman, S. L. and Cooper, B. S. (2010) Assessing the role of undetected colonization and isolation precautions in reducing Methicillin-Resistant Staphylococcus aureus transmission in intensive care units. BMC Infectious Diseases 10(29).
Title Modelling of Emerging Diseases Statistics and Probability, Mathematical Medicine and Biology Prof Frank Ball 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.
Title Structured-Population Epidemic Models Statistics and Probability, Mathematical Medicine and Biology Prof Frank Ball 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 pseudo-likelihood and Markov chain Monte Carlo methods).
Title Bayesian Inference for Complex Epidemic Models Statistics and Probability, Mathematical Medicine and Biology Prof Philip O'Neill, Dr Theodore Kypraios Data-analysis for real-life 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).
Title Epidemics on random networks Statistics and Probability, Mathematical Medicine and Biology Prof Frank Ball 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. Ball F G and Neal P J (2008) Network epidemic models with two levels of mixing. Math Biosci 212, 69-87. Ball F G, Sirl D and Trapman P (2009) Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv Appl Prob 41, 765-796. Ball F G, Sirl D and Trapman P (2010) Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math Biosci 224, 53-73.