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Applied & Computational Mathematics

CDT in Mathematical Modelling, Analysis and Computations (MAC-MIGS)

Reference Number: 2018-MAX-01

Abstract: The EPSRC Centre for Doctoral Training (CDT) in Analysis & its Applications (MIGSAA) accepted its last cohort in September 2018.  However, we now welcome applications to the new CDT in Mathematical Modelling, Analysis and Computations (MAC-MIGS).

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A stochastic framework for microseism generation and for uncertainty quantification in seismic tomography

Reference Number: ACM

Abstract: The study of terrestrial microseismic noise correlations holds a great promise as a means of learning the structure of the Earth’s crust and studying its temporal variations. Apart from the practical applications, seismic tomography is wide open to developing systematic mathematical approaches for robust inverse methods and uncertainty quantification. Microseisms are seismo-acoustic waves excited by nonlinear interactions of ocean waves whose energy is trapped within a wave guide established by the seafloor and the steep gradients in elastic velocities in the crust and upper mantle. The microseism sources illuminating the Earth’s crust are typically not co-spatial with tectonically active regions implying a continuous ability to monitor the crust. Longuet-Higgins (1950) first argued how the interaction between surface gravity waves could lead to the excitation of high phase velocity acoustic components. However, there is no theory for the temporal intermittency and non-Gaussianity which are apparent in the microseism time series. Understanding these non-Gaussian effects is often crucial for carrying out robust full wave field inversion which is essential in petroleum industry. This project will be concerned with developing a stochastic approach to the dynamic microseism generation.

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Adaptive Monte Carlo Methods for Stochastic Differential Equations

Reference Number: 2018-HW-AMS-14

Abstract: Capturing rare events is crucial for accurate risk assessment and its successful management. An example in finance is computing the probability of a large, but rare, loss from a financial portfolio. Approximating expectations involving such rare events is difficult because, when using Monte Carlo, many of the generated samples do not contribute to the final outcome and the expensive samples are effectively wasted. Adaptive sampling methods resolve this issue by spending some minimal computational effort to determine if a sample is of interest and, only if it is, the sample accuracy is then improved by spending further computational effort.  Using concepts from stochastic analysis, probably theory and numerical analysis, this project will look at applying adaptive methods to compute outputs depending on stochastic differential equations rather than simple random variables.

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Approximating persistence times of endemic infections

Reference Number: 2018-HW-AMS-02

Abstract: The spread of infectious disease through a population may be modeled as a stochastic process (typically a continuous-time Markov chain). For infections which are able to persist in the long term (i.e. become endemic in the population), a random variable of interest is the time until eventual extinction of infection.  Programmes exist aimed at global or regional eradication of specific diseases including polio, malaria, measles, onchocerciasis and others; economic planning for such programmes could potentially be helped by good estimates of the expected time to achieve disease extinction.  For relatively simple mathematical models, the expected persistence time may be computed exactly from general Markov process theory.  For more realistic models, this approach is no longer feasible, and approximations must be sought.  This project will use recently-developed methods from statistical mechanics to approximate persistence time for a variety of infectious transmission models, and investigate the effects of disease features (e.g. length of latent period, variability of infectious period, etc) upon this persistence time.

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Bistability of solitary waves in nematic liquid crystals

Reference Number: ACM

Abstract: Optical solitary waves can form in nematic liquid crystals and thermal optical media. An open question of great interest both theoretically and experimentally is whether there is bistability of these solitary waves, that is whether different stable solitary waves with the same power can exist. There is some theoretical indication that this is possible, but the question requires an in depth analysis.

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Bumblebee foraging

Reference Number: 2018-HW-Maths-42

Abstract: Bumblebee populations are in general decline,  a problem not just for the bumblebees but for the entire ecosystem: bumblebees are key pollinators of wild flowers and commercial crops alike. Consequently, it is important to understand the factors that allow bumblebee colonies to persist. Survival of a colony is fundamentally dependent on its foraging abilities: collecting nectar and pollen from the scattered floral resources that surrounds their nest. Gathering as quickly and efficiently as possible is a must: flying is energetically demanding and longer times outside the nest will also increase the risk of predation. Field studies suggest that bumblebees have a capacity to memorise their foraging trips and, in the initial few trips, optimise their routes in a manner reminiscent to solving the traveling salesman problem. The aims of this PhD project will be to use mathematical modelling in order to understand how an individual bumblebee can learn the resources of their landscape and optimise their foraging trips, and how the colony itself can adapt to maximise its exploitation of a changing landscape.

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Collective chemotaxis

Reference Number: 2018-HW-Maths-41

Abstract: Chemotaxis is a widely occurring phenomenon in which cells or animals detect and follow chemical gradients in their environment: the same positive process that allows an animal to follow a pheromone trail to a potential mate has a negative association in the tendency of cancerous cells to invade healthy tissue by following molecular signals. Consequently, mathematical modelling of chemotaxis behaviour has developed into a large area of active research with applications in areas ranging from microbiology to ecology, via developmental biology and cancer. Much of chemotactic modelling has considered it to be an individual response: an individual detects a chemical gradient and moves accordingly. However, recent experiments suggest a phenomenon of collective chemotaxis: a population of cells collectively migrates with an efficiency greater than the sum of the individual elements of the population. The aims of this project will be to explore how existing mathematical models can be adapted to describe such phenomena, and consequently how this can effect aspects such as the rate of invasion in models for cancerous growth.

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Colliding Black Holes and Neutron Stars

Reference Number: ACM

Abstract: The mysterious gamma-ray bursts are the largest explosions since the big bang. They happen on average once a day and last for only a few seconds. One scenario to explain these cataclysmic events involves the collisions of neutrons stars and black holes. The detailed numerical calculations are done on powerful computers to try to simulate the collisions and then explain these bursts. The project would involve first to get acquainted with the computer code and the astrophysical background, and then to extend the code to include new physical effects and/or explore the parameter space.

Examples of previous PhDs can be found here

Computation and modelling of noise pollution in marine reserves

Reference Number: 2018-HW-Maths-17

Abstract: This research project will examine acoustic wave propagation in the marine environment to predict and estimate the extent of noise pollution in designated marine reserves from shipping lanes and other human activities such as drilling. We will develop stochastic wave equation models in collaboration with the underwater acoustics department at the School of Engineering and Physical Sciences, Heriot-Watt. These will then be analysed and fast numerical methods developed.  Depending on the interests of the PhD candidate, the topic of the PhD can be in a number of directions: modelling of noise pollution in a marine environment, mathematical analysis of stochastic wave equations, and the developments of efficient numerical methods for the solution of these.

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Cut-off reaction-diffusion systems

Reference Number: ACM

Abstract: Reaction-diffusion equations, such as the Fisher or the Nagumo equations, have found widespread use as 'minimal' models in the sciences. Solutions that retain a fixed profile in time and space, known as propagating fronts, are frequently relevant as asymptotic ('limiting') states to which general solutions converge. While reaction-diffusion models have been widely applied in the continuum limit of discrete (many-particle) systems, their utility is often limited due to stochastic effects which need to be considered for finite particle numbers. These effects can be approximated by introducing a 'cut-off' function which deactivates the reaction terms whenever the particle concentration lies below some threshold.The impact of such a cut-off approximation has been studied in detail in a number of propagation regimes; however, an in-depth understanding of the stability and convergence of the resulting front solutions is lacking, as is the investigation of cut-off systems in more than one spatial dimension.

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Data-driven dimensionality reduction in probabilistic prediction

Reference Number: ACM

Abstract: High-dimensional noisy time series associated with the underlying complex dynamics often contain redundant information and can be compactly represented by a dynamical process on a low-dimensional manifold; this is commonly referred to as the ‘manifold hypothesis’ and is related to the concentration of measure phenomenon in high-dimensional data sets. Due to the linear character of classical dimension reduction methods, such as the Principal Component Analysis, they are ill-suited to recover the nonlinear structure of the underlying state manifold. Robust and accurate dynamical predictions, based on reduced-order models extracted from empirical data, require a systematic understanding of how to combine manifold learning methods with analysis & probability techniques for extracting dynamical features from noisy or incomplete time-dependent data. This project will approach this problem from the probabilistic/stochastic viewpoint.

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Data-driven models in molecular dynamics

Reference Number: ACM

Abstract: Multiscale modelling plays an essential role in molecular simulation as the range of scales involved precludes the use of a single, unified system of equations. The most accurate model is quantum mechanics which describes the evolution of a system of nuclei and electrons. When a modest-sized quantum system is discretized for numerical solution, there results an unimaginably large number of equations which can swamp even the most powerful computer systems. A classical model based on potential energy functions for the interaction of atomic nuclei provides a much simplified description, but one that precludes many important effects (breakage of bonds, quantum tunnelling, etc.). Even the classical description must be further 'coarse-grained' to provide an effective scheme for large scale or slow-developing processes that would otherwise remain inaccessible in computer simulation. In a multiscale model, different models are unified by the use of bridging algorithms, numerical and analytical averaging, and reliance on the principles of statistical mechanics. In this project, the goal is to use experimental data in place of simulation data to capture complex local processes and low-level interactions in a molecular system . A system is no longer viewed as being described by a single inter-molecular potential energy surface, but rather by a collection of surfaces which can be locally determined, on-the-fly, from tabulated data. The resulting procedures will engender methodological changes in order to retain statistical properties that are relevant for the simulator. This project has aspects of molecular dynamics, computational statistical mechanics and quantum mechanics. It further relates to machine learning and has applications in materials modelling.

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Defects in liquid crystals

Reference Number: 2018-HW-Maths-07

Abstract: Liquid crystals are materials composed of usually rod-like molecules that have orientational order, so that locally they tend to align in a particular direction. There are various different levels of description possible, for example the Oseen-Frank theory in which the basic variable is the mean orientation, and the Landau – de Gennes theory in which the order is described in a more detailed fashion via a tensor unknown. The project will study defects of various types (point, line and wall) that can occur in liquid crystals, and how they can be described by different models using techniques of nonlinear analysis.

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Dispersion in random flows

Reference Number: ACM

Abstract: When a constituent, e.g. a pollutant, is released in a fluid flow, it disperses through the combined effect of advection by the flow and molecular diffusion. Some time after the release, the dispersion can often be represented as a simple diffusion process, with a diffusivity, termed effective diffusivity, that is enhanced compared to the molecular diffusivity and accounts for the effect of advection. Mathematical techniques, mostly homogenisation methods, have been developed to compute the effective diffusivity for a range of flows of increasing complexity, from simple time-independent flows to time-dependent random flows that model turbulence. The description in terms of an effective diffusivity is however limited to regions of high concentration of the constituent and fails to describe the low-concentration tails. These turn out to be in important in applications, for instance when the constituent undergoes fast chemical reactions that amplify low concentrations. In recent work with Peter Haynes (Cambridge), we have developed a new asymptotic method that predicts the entire concentration profile, including its tails. The technique relies on large-deviation theory and highlights how the statistics in the tails are controlled by rare, extreme events. It has been applied to simple time-independent flows for which explicit results can be obtained in certain limits. This project will extend the large-deviation approach to random flows, starting with a class of spatially correlated, white in time flows. It will combine the analysis of stochastic partial differential equations with numerical simulations, Monte Carlo sampling in particular. The project could evolve in several directions and analyse, for instance, dispersion in complex geometries or in realistic turbulent flows.

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Efficient numerical methods for stochastic differential equations

Reference Number: 2018-HW-Maths-19

Abstract: Many applications give rise to differential equations which include some form of a time dependent noise. For example noise might be included in the model of molecular movement. These equations also arise in filtering - an important SPDE used to reconstruct data from signals. Typically these are stochastic PDEs or stochastic differential equations potentially coupled to PDEs. The aim of this project is to develop novel solution techniques and that can be used to estimate the uncertainty in computed results and to examine the efficiency and convergence of these methods. The project would develop new methods that are adaptive in space and time and may take account of different scales in the problem.

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Electromagnetic surface waves: optical sensing and solar cell applications

Reference Number: ACM

Abstract: Electromagnetic surface waves may be excited at the interface of two dissimilar materials. Typically, the excitation of these waves is extremely sensitive to the morphology of the interface. Consequently, electromagnetic surface waves represent an attractive proposition for optical sensing applications. Indeed, surface-plasmon-polariton waves - which are electromagnetic surface waves associated with the interface of metals and dielectric materials - are harnessed in highly sensitive optical sensing devices currently used in analytical chemistry and biology. In addition, the excitation of electromagnetic surface waves is associated with a sharp increase in absorption of incident light. For this reason, in recent years surface-plasmon-polariton waves have been considered for possible solar cell applications. Beyond simple metals and dielectric materials, electromagnetically-complex materials offer greater scope for the excitation of electromagnetic surface waves. For example, certain complex materials can support multiple modes of electromagnetic surface waves which are promising for both optical sensing and solar cell applications.Projects are available which involve the theory underpinning electromagnetic surface waves at the interfaces of dissimilar complex materials, with a view to optical sensing and solar cell applications.

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Energy backscatter by ocean mesoscale eddies

Reference Number: ACM

Abstract: While there are large scale transport processes within the ocean, the ocean is a highly turbulent fluid, and contains a vigorous field of eddies. These eddies are both formed through instabilities of the mean flow, but can also feed back and modify the mean flows which generate them. This can include energy "backscatter" processes, which can lead to local strengthening of the mean. This project will study simplified models of ocean turbulence, with particular emphasis on the backscatter of energy by mesoscale eddies, and how these processes may be interpreted in terms of coarsening of high resolution data.

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Exponential Asymptotics

Reference Number: ACM

Abstract: Usually asymptotic expansions are divergent, and that means that exponentially small phenomena are hidden in the tail of the divergent series. In exponential asymptotics we make these exponentially small contributions visible. In this way we obtain much more accurate approximations and increase the regions of validity, we obtain the most powerful method to compute the so-called Stokes multipliers, and in many applications explain (and compute) the appearance of oscillatory behaviour when certain Stokes lines are crossed. Topics available are exponential asymptotics for linear or nonlinear ODEs and PDEs with a small parameter.

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Finite elements for wave propagation: Analysis, fast methods and concert halls

Reference Number: 2018-HW-Maths-09

Abstract: This project concerns considers accurate and efficient numerical methods for sound propagation, often steered by hard analysis. Some key challenges of wave computations arise in the context of an instrument placed in a concert hall: High frequencies, nonlinear wave propagation and boundary conditions, complex domains and sound emission.  On the other hand, detailed models to simulate music instruments have been investigated in numerical analysis, from violins to a grand piano. Their simulations of classical music from first principles are truly impressive, and in the longer-term we hope to place these models in realistic surroundings.
Some directions: At high frequencies naive numerical methods require a very fine mesh to capture the rapidly oscillating sound pressure. Tools from harmonic analysis and PDE give rise to new regularity estimates and efficient adaptive mesh refinements. Detailed models of instruments (or other sound sources like tires or high-speed trains) give rise to realistic simulations of sound, based on mathematically challenging multi-physics problems. 

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Flash Sintering

Reference Number: 2018-HW-Maths-27

Abstract: In "flash sintering" a powdery material quickly bonds into a solid object through the action of an electric current. The project will mathematically model the interaction of the electric current and changing temperature during the process.

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Fractional reaction-diffusion equations: Analysis, mathematical biology and computation

Reference Number: 2018-HW-Maths-10

Abstract: Fractional diffusion equations have attracted much recent interest, as they describe diffusion in the presence of long jumps. Analytically, the Laplace operator of the heat equation is raised to a non-integer power, resulting in a nonlocal operator; probabilistically, Brownian motion is replaced by a Levy process. This project studies either the pure analysis and  efficient numerical approximation of nonlinear reaction-diffusion equations for superdiffusing particles, as they arise in applications from the movement of cells or bacteria (chemotaxis), the spread of diseases, or in social networks.  Basic questions have only recently started to be addressed: Can we rigorously derive such equations from microscopic descriptions for the movement of bacteria, cells or particles? Are the resulting equations well-posed for all positive times, or do solutions blow up? Can we rigorously and efficiently compute the solutions, ideally with provable error bounds? Are there interesting solutions (traveling waves, pattern formation), or can we make interesting predictions for physicists or biologists?

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From Photosynthesis to Solar Cells: investigating new theoretical and computational directions

Reference Number: 2018-HW-Maths-30

Abstract: We begin with investigating the basic quantum mechanical principles such as (classical and quantum) information theory and entropy, thermodynamics of irreversible computations, superposition, locality, entanglement, and quantum gates, infinite dimensional analysis (Hilbert Spaces), which all are fundamental in the development of quantum inspired formulations.
Equipped with these tools, we will look into a quantum-like description for photosynthesis. Here, quantum-like means that the mathematical structure is different from conventional quantum mechanics. We will investigate a quantum network formulation for electron transport in organic molecules and describe the photosynthesis by a quantum channel representation. A question of general interest then is whether and how we can transfer these ideas to solar cells to ultimately improve the performance.

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Homeless Modelling

Reference Number: 2018-HW-Maths-26

Abstract: Earlier work has had success in modelling sizes of waiting lists for public-sector housing and of the local homeless population. The research is to be extended to take into account possible causes of homelessness, such as alcohol dependency. Alternatively, or additionally, prison populations could me modeled and related to, for example, sentencing policy. Some coupling could possibly be done between the two models.

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Interacting pattern mechanisms

Reference Number: 2018-HW-Maths-40

Abstract: Self-organisation is an area of huge interest, beautifully illustrated by the problem of embryonic development: how does a single fertilised cell restructure itself into the complex adult organism? Biologists and theoreticians have spent a phenomenal amount of time investigating such basic questions. Two classic theories have been reaction-diffusion ideas and mechanical-based ideas. The former was proposed by the famous mathematician Alan Turing, who (counterintuitively) showed that diffusion could create spatial patterns in a system of reacting and diffusing chemicals and suggested that the pattern could subsequently drive cell differentiation. Mechanical models are based on the forces of cells as they move through and interact with their environment. Recent experimental results have provided evidence that both theories may occur and, intriguingly, may operate simultaneously in certain systems. This project will focus on integrating these different theories for pattern formation and understanding whether the interaction between different models can create more complicated patterning behaviour.

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Interactions between fast and slow dynamics in nonlinear evolution equations

Reference Number: 2018-HW-Maths-01

Abstract: Many biological systems comprise processes that vary on different time scales. For spatially homogeneous processes, the dynamics can be described by multiple-scale ("fast-slow") systems of nonlinear ordinary differential equations (ODEs). The resolution of bifurcations in such systems, and of transitions between oscillations of varying amplitude that are organised by so-called folded singularities, is crucial in biological applications, such as in the understanding of the dynamics of neuronal activation. One recent approach for analysing nonlinear fast-slow ODEs is based on geometric singular perturbation theory, and combines methods from the theory of dynamical systems with the blow-up technique (geometric desingularisation). The principal objective of the project is the extension of geometric singular perturbation theory to partial differential equations (PDEs) of reaction-diffusion type (RDEs) and reaction- diffusion-convection type (RDCEs) which exhibit different scales in both time and diffusivity. RDEs and RDCEs can be considered as dynamical systems that are defned in infnite- dimensional (Banach) spaces. One immediate aim is the identifcation of bifurcations, for some classes of RDEs and RDCEs, that depend on the relation between the two scales. The influence of the microscopic structure of the underlying environment on the qualitative behaviour of solutions will also be considered, as will the relation between a time-scale separation and oscillations in the diffusivities due to environmental fluctuations. The development of effective numerical schemes is crucial in this context, and represents another worthwhile research direction. In the long term, we envision that the influence of noise on the dynamics of such systems may also be accounted for.

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Mathematical Models of Disease Dynamics in Wild Ungulates

Reference Number: 2018-HW-Maths-33

Abstract: The recent geographical and demographic expansion of wild ungulates (e.g. wild boar, red deer) is one of the most noteworthy changes taking place in natural and managed environments in the northern hemisphere. This expansion has important consequences as wild ungulates are reservoirs for several of the most important emerging diseases threatening livestock and human health. This includes viruses such as African swine fever, bacterial infections such as tuberculosis and tick borne diseases such as Crimean–Congo haemorrhagic fever. This project would develop mathematical models to represent ungulate host, vector and pathogen dynamics and assess the critical link between population density and infection prevalence and persistence. The mathematical models will take the form of non-linear differential equations and will be based on classical compartment models of wildlife disease that consider a population split into separate classes representing different disease stages. The models will be used to examine intervention strategies, such as top-predator introduction, ungulate fertility control, vaccination, culling or hunting bans on ungulates. Such intervention strategies can lead to unforeseen population, vector and pathogen dynamics and so mathematical models are essential to understand the complex ecological and epidemiological feedbacks and predict the likely outcome of disease management strategies. This project will be undertaken in collaboration with the Institute for Game and Wildlife Research, Ciudad Real, Spain who conduct field experiments to monitor the dynamics of ungulates and the diseases they harbour.

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Mathematics applied to the environment

Reference Number: ACM

Abstract: Projects applying mathematics to environmental problems and supervised by James Maddison and Jacques Vanneste are available through the

Edinburgh Earth and Environment Doctoral Training Partnership

Mathematics of ultra-fast chemical reactions

Reference Number: ACM

Abstract: Recent experimental advances in light sources and ultrafast lasers have driven the need for the development of theoretical and computational tools to study and understand ultra-fast chemical reactions. In particular, tools are required to bridge the gap between traditional quantum chemistry (describing electrons) and molecular dynamics (describing nuclei). Current computational approaches require vast resources and can describe the dynamics over only very short times. This project will start from existing mathematical models for one-dimensional systems and extend them to higher dimensions, where far more interesting examples can be studied. It combines asymptotics, PDEs and some light numerical components. No knowledge of chemistry is required, but an interest in it, and a desire to work in a truly interdisciplinary field would be advantageous. In particular there are strong links to work by members of the School of Chemistry.

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Mixed-mode dynamics

Reference Number: ACM

Abstract: Mixed-mode dynamics is a type of complex oscillatory behaviour that is characterised by an alternation of oscillations of small and large amplitudes. Mixed-mode oscillations (MMOs) frequently occur in fast-slow ('multi-scale') systems of ordinary differential equations; they are, for instance, found in models from mathematical neuroscience, where they correspond to complicated firing patterns seen in experimental recordings of neural activity.Various mechanisms have been proposed to explain mixed-mode dynamics; however, the relationship between them has not been investigated systematically yet. Moreover, relevant neurological models are typically too high-dimensional to be amenable to mathematical analysis, and have to be reduced efficiently to lower-dimensional normal forms which still capture the essential model dynamics.

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Modelling of cellular adhesion in close collaboration with experimental biologists

Reference Number: ACM

Abstract: All the multicellular organisms, including us, humans, are in the end just a pile of cells. But why we are not falling apart?  Our team studies cell-cell adhesion from an intracellular perspective. This is a vibrant current area of research, where most labs currently study tissue properties from the macroscopic level (tissue tension, etc), but not from the intracellular level. For cells to stick together, a particular protein must be delivered to its biologically relevant locations along the cell boundary (in order to "glue" nearby cells together) and distributed in such a way that the tissue has desired biological properties. How does this work? Is the outcome robust? We study these processes in close collaboration with the experimental biology lab of Natalia Bulgakova at U. Sheffield, with maths modelling guiding the experiments and vice versa. Maths techniques that we use include PDEs, ODEs, and Stochastic modelling. No previous knowledge of biology is required at the beginning of the project, but the student will learn about the relevant bio processes as the project progresses. The results of this highly interdisciplinary project will be of interest to both maths and bio communities.

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Modelling of interactions between immune and tumour cells and analysis of fractional reaction-diffusion equations

Reference Number: 2018-HW-Maths-54

Abstract: Experimental evidences indicate that immune cells move in an unrestricted search pattern if they are in the pre-activated state, while they tend to stay within a more restricted area upon activation induced by the presence of tumour antigens. This suggest that the movement of inactive cell can be modelled by standard diffusion, whereas the movement of active immune cell is better described by a fractional diffusion operator. In this project, we will develop mathematical models for tumour-immune competition that take explicitly into account the difference in movement between inactive and activated immune cells, and we will analyse the properties of the solutions to systems of coupled nonlinear reaction-diffusion and fractional reaction-diffusion equations.

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Modelling, Analysis, and Numerics for Stochastic Multiscale Systems

Reference Number: 2018-HW-Maths-29

Abstract: The starting point will be to gain a basic understanding of the appearance and emergence of randomness in seemingly deterministic systems. To this end, we will make use of Statistical Mechanics, Analysis, PDE theory, Thermodynamics, and numerical discretisation strategies. Key problems, that we will investigate, are how to reliably coarse grain non-equilibrium systems and to develop systematic, analytical and numerical methods to solve stochastic homogenization problems which arise in science, engineering, and data science.

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Multiscale diffusion processes in swarm robotics

Reference Number: 2018-HW-Maths-11

Abstract: Biological organisms have found efficient strategies to move and disperse in search of randomly located targets, like food. ​S​warm robotics ​aims at ​​developing bio-inspired ​computational​ models​ usually combined​ with machine learning techniques to ​allow robots ​to autonomously ​explore areas for tasks like search-rescue missions, monitoring, surveillance​, ​forest fire combat​ and agricultural applications, just to name a few​.A main mathematical approach for understanding the biological ​organisms ​movement or robot​ ​movement relates it to diffusion problems.​ ​This project derives macroscopic diffusion equations for the dispersion of robot ​swarms. Their pure and numerical analysis leads to efficient search strategies in applications. Diffusion equations are long-studied for biological or physical systems, such as the Keller-Segel model for cell movement in the presence of chemical cues. Robotic systems open up new possibilities of interaction, coordination and control. They lead to challenges in the mathematical analysis and the development of algorithms. In particular, the macroscopic description allows the efficient optimisation of movement rules and real-time learning and evaluation of strategies.

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Multiscale modelling and analysis of interactions between plant roots and soil

Reference Number: 2018-HW-Maths-24

Abstract: Plants grow deep roots to acquire water and nutrients more efficiently. However, growth in the deeper soil can be a considerable mechanical challenge for plants’ soft and pliable primary tissues and there is limited knowledge on the biophysical limitations to growth under high mechanical stresses from the deep soil. Experimental results indicate that growth under mechanical pressure may be limited by mechanical instabilities at the root tip and these instabilities results from interactions with soil particles taking place at the microscopic scale (individual particles causing micro-deflection of the roots). This finding offers valuable insights into how plants may sense and overcome extreme growth environments, but there is no framework to grasp the complexity of the interactions taking place between a root and the many individual soil particles from the surrounding soil. The objective of this interdisciplinary project is to develop a theoretical framework to better understand limitations to root elongation under high levels of soil mechanical resistance. The project will focus on multiscale modelling of interactions of plant roots with the granular media at the particle level, and will build on data of root growth in pressure chambers obtained in previous and ongoing experiments. The numerical simulations of mathematical models will be first tested against experimental data and then used to analyse the effect of microscopic interactions between roots and soil particles on growth response at the macroscale. The project will combine mathematical modelling, analysis and development of efficient numerical schemes for coupled microscopic-macroscopic description of interactions between soil particles and plant roots.

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Multiscale modelling and analysis of mechanical responses in biological tissues

Reference Number: 2018-HW-Maths-23

Abstract: The development and function of biological tissues – should it be within a human or a plant - involve many interconnected processes that occur at different spatial and temporal scales. Thus, multiscale mathematical modelling and analysis are needed for a better understanding of how the different scales and the various structural components at the same scale all interact, as well as the analysis of the compensatory mechanisms and complex microscopic structures of biological tissues. In this interdisciplinary research project we will consider multiscale modelling and analysis of mechanical interactions between cells within a biological tissue, especially when exposed to external mechanical forces. For example, plant stems, such as tree trunks, are constantly reacting to their physical environment. When challenged mechanically (i.e. being bent or tilted), plants generate more wood cells (secondary xylem) at the specific locations by enhancing the cell division in the area. Multiscale modelling of mechanical properties and growth of biological tissues, analysis and numerical implementation of model equations, as well as comparison of numerical solutions to experimental data for plant tissues, obtained in the Nakayama group, will constitute main objective of the project. In mathematical models for interactions between external processes, mechanics and growth we shall also consider a coupling between changes in mechanical properties and inter-cellular processes (signalling processes, chemistry), which will allow us to determine mechanisms controlling tissue level responses to external or internal stimuli. A better understanding of mechanisms controlling cellular responses to external forces will help us, for example, to improve the growth and development of plants in specific physical environments (e.g. hard soil, windy areas, steep slopes).

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Multiscale modelling of intercellular signalling processes using Bulk-surface PDEs

Reference Number: 2018-HW-Maths-53

Abstract: Intercellular signalling is the primary mechanism by which cells react with and respond to their environments. Such signalling therefore is crucial to all cell biological processes. The main mechanisms involved are transport of signalling molecules and membrane bound receptors and reactions between these species. Mathematical models of intercellular signalling processes comprise coupled systems of nonlinear partial differential equations defined in bulk domains (intercellular or intracellular spaces) and surfaces (cell membranes). In applications we wish to understand the influence of cell scale signalling processes on tissue level phenomena and thus a multiscale modelling approach is required. Using techniques from homogenisation theory we will rigorously derive two-scale models for biological phenomena posed on tissue scales that incorporate cell signalling. We will then consider the analysis and approximation of the two-scale models.

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Numerical analysis and algorithms for wave propagation

Reference Number: 2018-HW-Maths-39

Abstract: The propagation of acoustic, electromagnetic and elastic waves is a remarkable source of mathematical and computational challenges with a huge array of applications. Here, a number of projects in various directions are available. One the greatest challenges in this field is the accurate computation of waves at high frequencies.  Two projects available in this direction is the development of  hybrid numerical-asymptotic time domain boundary integral methods and modified finite element methods each designed to minimize the computational costs associated to high-frequencies by using our knowledge of the underlying physics of wave propagation. Further projects include numerical non-linear wave scattering, waves and non-local operators with applications in therapeutic medicine, fluid-structure interaction  etc.  Depending on the project, the research will be done in collaboration with external groups in the UK, US, and Europe (Vienna, Rome, Tubingen).

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Numerical analysis of Bayesian inverse problems

Reference Number: ACM

Abstract: In areas as diverse as climate modelling, geosciences and medicine, mathematical models and computer simulations are routinely used to inform decisions and assess risk. However, the parameters appearing in the mathematical models are often unknown, and have to be estimated from measurements. This project is concerned with the inverse problem of determining the unknown parameters in the model, given some measurements of the output of the model. In the Bayesian framework, the solution to this inverse problem is the probability distribution of the unknown parameters, conditioned on the observed outputs. Combining ideas from numerical analysis, statistics and stochastic analysis, this project will address questions related to the error introduced in the distribution of the parameters, when the mathematical model is approximated by a numerical method.

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Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint operators

Reference Number: 2018-HW-Maths-31

Abstract: The aim of this PhD project is to device strategies for computing rigorous sharp bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods (Galerkin methods). The theory of computation for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart however a systematic procedure for computing their spectrum remains as a largely unsolved open problem. During this PhD, the student will examine general strategies for numerically estimating spectra of J-self-adjoint operators. Particular directions will involve the following. (a) Computation of enclosures of spectra for operators arising in the theory of rotating viscous fluids. (b) Computation of sharp enclosures for the complex isotonic harmonic oscillator. (c) Impact in the study of time evolution problems for J-self-adjoint operators.

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Numerical methods for Brownian dynamics simulations with applications in physics and biology

Reference Number: ACM

Abstract: Brownian dynamics is the method of choice in the simulation of polymers, complex fluids and biomolecules at intermediate scales. In the standard model, a stochastic differential equation is formulated to model the interaction forces between bodies and the effective stochastic interactions with a solvent bath. To model hydrodynamic interactions, the equations of motion must incorporate multiplicative stochastic noise, which just means that noise enters into the equations multiplied by a function of positions. (In difficult cases this function may need to be computed on-the-fly, from atomistic simulations.) The incorporation of multiplicative noise complicates the design of numerical methods and the analysis of numerical error. In this project, the existing schemes for performing Brownian dynamics will first be exhaustively compared and assessed in terms of different quantitative measures of error. Model problems will be used to evaluate the performance of different schemes. Analytical methods will be used to estimate the errors in different regimes and to suggest new numerical methods which can improve the computational performance and accuracy obtained in simulation. Examples for demonstrating the performance of methods will be found in applications in polymer physics and biological systems.

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Numerical methods for non-local phenomena

Reference Number: 2018-HW-Maths-12

Abstract: Recently there has been an explosion of interest in the mathematical and numerical analysis of non-local phenomena, driven by a number of important applications including therapeutic medicine, porous media of fractal character and human travel. These models often contain fractional time derivatives, fractional Laplace operators, or combinations of both. The non-locality in space or time of fractional operators poses difficult questions for the development of numerical schemes, and this project aims to address these issues. This is a fast moving field, hence details can only be known at the start of the project. However, it will be either in the direction of integral operators, finite element methods or time-stepping. Collaboration on this topic is planned with groups in the UK, Chile and Italy.

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Numerical simulation and neural modelling

Reference Number: 2018-HW-Maths-18

Abstract: Neural field equations are being used more and more to extract data from real reading - for example from fMRI scans. To be of use we need fast and convergent algorithms for their numerical simulation. This project would examine the stochastic neural field and how to compute structures, such as waves, through the field.
At the other end of the scale, with colleagues in biology we have the opportunity to take a large amount of cell movement data and to develop models for molecular movement and interactions. The project would apply new techniques for estimating parameters and could be used to inform new biological theories and experiments.

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Ocean Modelling with Mimetic Finite Element Methods

Reference Number: ACM

Abstract: Mimetic finite element methods are a type of numerical discretisation for partial differential equations which obey discrete analogues of continuous properties of partial differential operators -- for example that the curl of a gradient should be zero, or that the divergence of a curl should be zero. The ability of a numerical method to represent these mathematical properties can often be related to the ability of the method to represent physical balances in discretisations of equations, such as the important physical balances which are present in the equations which govern ocean dynamics. This project will study the application of mimetic finite element methods to the numerical simulation of problems in ocean dynamics, with particular emphasis on problems which are relevant for the large-scale ocean.

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Optical vortices

Reference Number: ACM

Abstract: Optical vortices, the optical equivalent of the familiar fluid vortex, have found widespread use in many fields where they are used to optically trap small objects such as cells. This project would investigate the interaction of optical vortices and the interaction of optical vortices with refractive index changes. The idea behind this is to control the position of optical vortices by optical methods.

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Optimal construction of statistical interpolants

Reference Number: ACM

Abstract: Many problems in science and engineering involve an unknown complex process, which it is not possible to observe fully and accurately. The goal is then to reconstruct the unknown process, given a small number of direct or indirect observations. Mathematically, this problem can be reformulated as reconstructing a function from limited information available, such as a small number of function evaluations. Statistical approaches, such as interpolation or regression using Gaussian processes, provide us with a best guess of the unknown function, as well as a measure of how confident we are in our reconstruction. Combining ideas from machine learning, numerical analysis and statistics, this project will address questions related to optimal reconstructions, such as the optimal choice of the location of the function evaluations used for the reconstruction.

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Optimal flow control: mathematical models and numerical methods

Reference Number: ACM

Abstract: A key application of mathematical modelling to practical scientific problems arises in the field of fluid flow control, where one wishes to design a physical system in which the flow behaves in some desired or optimal way. Two key challenges result from this approach: (i) it needs to be ensured that the mathematical model accurately reflects the physics of the flow and the desired objective, (ii) computational algorithms must be designed to solve the systems of equations resulting from the model accurately and efficiently. This project aims to address these two challenges for fluid dynamics problems motivated by practical and industrial applications. A possible additional component of the project is to apply this methodology within a high performance computing framework. This project is related to the EPSRC Fellowship.

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PDE-constrained optimization in scientific processes

Reference Number:

Abstract: A vast number of important and challenging applications in mathematics and engineering are governed by inverse problems. One crucial class of these problems, which has significant applicability to real-world processes, including those of fluid flow, chemical and biological mechanisms, medical imaging, and others, is that of PDE-constrained optimization. However, whereas such problems can typically be written in a precise form, generating accurate numerical solutions on the discrete level is a highly non-trivial task, due to the dimension and complexity of the matrix systems involved. In order to tackle practical problems, it is essential to devise strategies for storing and working with systems of huge dimensions, which result from fine discretizations of the PDEs in space and time variables. In this project, "all-at-once" solvers coupled with appropriate preconditioning techniques will be derived for these systems, in such a way that one may achieve fast and robust convergence in theory and in practice. This project is related to the EPSRC Fellowship.

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Pipe Bending

Reference Number: 2018-HW-Maths-25

Abstract: A pipe to be used for sub-sea umbilical purposes starts off with a circular cross section. Before it is laid, it must be rolled up to be put on board the ship which is to deploy it. The rolling process deforms the pipe so that it no longer has its original cross section. This deformation process should be modeled and understood.

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Polycrystal microstructure

Reference Number: 2018-HW-Maths-06

Abstract: Martensitic phase transformations are solid-solid phase transformations of alloys involving a change of shape of the underlying crystal lattice. The different crystallographically possible changes of shape have to fit together at the meso-level, producing beautiful patterns of microstructure that are important for determining material behaviour. Over recent years big advances have been made in applying elasticity models to give information on such microstructures in single crystals using techniques of the multi-dimensional calculus of variations. The project aims to carry over some of this work to polycrystals, for which the crystal axes are different in different grains. It will study in particular the interaction between microstructures in neighbouring grains resulting from the deformation of grain boundaries.

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Probability Models of Cancer Growth

Reference Number: ACM

Abstract: Biology is a fast growing area for applications of probability. Since still in its infancy, there are many unexplored areas and open problems. In particular, there is a great interest in stochastic models of cancers. This PhD project would focus on understanding the most basic and fundamental models of tumor progression. These models include branching processes, other models borrowed from population genetics, or spatial Poisson processes. The work has a light numerical aspect to it, but would focus more on finding exact solutions, and establishing limit theorems. No knowledge of biology is required.

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Quantitative Methods and Models with Applications in Finance and Actuarial Science: Valuation Techniques, Investment Strategies and Dependence Modeling

Reference Number: 2018-HW-AMS-15

Abstract:
Various stakeholders in finance and insurance—such as regulators, investors and managers—rely on quantitative analysis in their decision-making processes. This research project employs quantitative models and methods from probability theory and statistics to tackle problems that are of practical relevance in these fields. Three topics are mainly concerned.  The first topic studies numerical techniques that are useful in financial and actuarial valuation such as option pricing, capital allocation and risk aggregation etc. We aim to propose new efficient computational methods and techniques. The second topic studies investment strategies and behaviors under general risk preference with emphasis on portfolio selection, skewness preference and performance measure etc. The third topic delves into dependence modeling of risks and its applications in finance and insurance. It covers popular research questions such as model uncertainty, systemic risk, high-dimensional risk measure and worst-scenario analysis etc.

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Quantum inspired mathematical formulation for decision-making: computing and algorithms

Reference Number: 2018-HW-Maths-28

Abstract: We begin with investigating the basic quantum mechanical principles such as (classical and quantum) information theory and entropy, thermodynamics of irreversible computations, superposition, locality, entanglement, and quantum gates, infinite dimensional analysis (Hilbert Spaces), which all are fundamental in the development of quantum inspired formulations. Equipped with these tools, we will look into a quantum based description for decision making in two-player games. Here, quantum based means that the mathematical structure is different from conventional quantum mechanics. We will investigate decision processes in games of Prisoners Dilemma type. These processes are gaining increasing interest, since classical (Kolmogorov) probability or classical quantum mechanics seem not to be consistent with statistical data obtained in cognitive experiments.

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Sampling methods in uncertainty quantification

Reference Number: ACM

Abstract: For details on the range of potential topics in this area please contact Aretha Teckentrup.

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Spectral Pollution

Reference Number: 2018-HW-Maths-22

Abstract: This project is about finding certified bounds for eigenvalues in regimes where the classical methods fail catastrophically. The Galerkin method is among the most reliable tools for finding upper bounds for the eigenvalues of selfadjoint operators. It is widely used in applications, ranging from civil and mechanical engineering to thermodynamics and non-relativistic quantum chemistry. Unfortunately the Galerkin method can fail dramatically when applied to the “wrong” eigenvalue problem. This phenomenon, called spectral pollution, is notoriously difficult to predict and it arises in relativistic quantum mechanics, solid state physics and electromagnetism. The purpose of this PhD project, is to characterise on a mathematically rigorous level what spectral pollution is and then examine analytical and numerical techniques for avoiding it. These include the quadratic method, the ZMG method, the factorisation method and the perturbation method. At the end of the project the student will have a broad view on the state-of-the-art in analytical and computational spectral theory with a specific emphasis in the current topic of spectral pollution.

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Statistical mechanics for reaction-diffusion systems

Reference Number: ACM

Abstract: Many systems in chemistry, biology and engineering can be modelled by reaction-diffusion equations in which the populations of a number of different species move and interact or react, leading to the interchange of population or mass between the species. When the populations are large and/or reside in a complex environment (such as particles suspended in a turbulent flow), a powerful approach is to use techniques from statistical mechanics, which describes the 'average' behaviour of such systems. Dynamical density functional theory is one such approach that has met with great success over the past decade or so. This project will extend existing models, which generally describe only the dynamics, to include the reaction terms. The topics covered can be tailored to the interest of the student, covering both rigorous analysis and numerics. Techniques include statistical mechanics, stochastic dynamics, mathematical modelling, homogenisation theory of PDEs, and computational methods such as pseudo-spectral methods and finite elements. This project also has strong links to the work of members of the School of Engineering. 

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Statistical Mechanics for the Brewing Industry

Reference Number: ACM

Abstract: Brewing is perhaps the oldest example of industrial biotechnology.  However, despite continued efforts to scientifically understand the complex processes involved in the transformation of grain into beer, many aspects of brewing remain somewhat of an art.  The central player in the industry is yeast, the cells of which are many orders of magnitude smaller than the brewing vessels in which it does its work.  Brewers require fine control over the behaviour of yeast, needing it to remain suspended in the liquid whilst it produces alcohol, and then for it to sediment out at the end of the process so that it may be easily removed.  Much of the current expertise is a product of hundreds of years of incremental, trial-and-error improvement, with little, if any, input from predictive science.  Clearly there is a need for an accurate, efficient and rational modelling approach that would remove some of the guess-and-test elements and would allow the testing of various aspects such as vessel geometries and agitation on the behaviour of the yeast. The multiscale nature of the problem (with interesting and important effects ranging from the size of yeast cells all the way up to that of the vessels) prohibits the use of standard continuum models, such as the Navier-Stokes equations.  In addition, the unimaginably large number of cells and vast range of timescales involved prevents the use of full-scale numerical simulations.  This project will use a statistical mechanics approach, which couples the microscopic and macroscopic dynamics, based on dynamic density functional theory (DDFT) to study flocculation (clumping) and sedimentation of yeast in brewing vessels, as well as other similar industrial problems.  This project will continue an existing collaboration with WEST Brewing, based in Glasgow.

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Stochastic differential equations, sampling and big data

Reference Number: ACM

Abstract: For details on the range of potential topics in this area please contact Konstantinos Zygalakis.

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Stochastic hybrid modelling of chemical systems

Reference Number: ACM

Abstract: It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modelled by Markov processes and, for such systems, methods such as Gillespies algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse grained schemes, where the fast reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to signicant errors in the simulation. This project will be concerned with developing further and analysing a new hybrid approach (a stochastic dierential equation with jumps) capable of dealing with more general systems.

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Stochastic Modelling of Biological Systems

Reference Number: ACM

Abstract: Due to recent experimental progress, the field of mathematical biology is rapidly growing. There are plenty of biological systems where mathematical models and analysis are needed. Closely interacting with experimentalists, the PhD candidate would formulate and analyze models of cancer progression, virus dynamics, bacterial evolution, and possibly other systems related to molecular motor motion or the origins of life. The project starts with first building and exploring simple model systems, and continues with their study by computer simulations and analytical methods. Knowledge of the biological background is not necessary at the beginning, but one will eventually learn some biology in order to do relevant research.

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The geometry of wave-mean flow interaction

Reference Number: ACM

Abstract: Numerous problems in fluid dynamics involve the separation of a flow between a mean component and fluctuations, often regarded as 'waves'. Examples of this include the separation of the atmospheric flow into its average along latitudes and perturbations, the separation of oceanic flow into a slow time-averaged component and fast surface waves, or the separation of numerically simulated flows into resolved and subgrid parts. The main aim of this separation is to obtain a simplified model for both the mean flow and the waves which accounts for their interactions. It is crucial that this simplified model respect key properties of the parent model such as conservation of energy and momentum and, when relevant, circulation. This places strong constraints on the ways of separating waves from the mean flow, and on the models of wave-mean flow interactions. This project examines how wave-mean flow models that satisfy these constraints can be derived by relying on the geometric foundations of the original fluid equations. A fluid flow can be thought of as a trajectory in an infinite-dimensional space (the space of diffeomorphisms in the simplest case) which minimises a certain action. Applying this viewpoint to the wave-mean flow problem, and using differential geometric tools has proved fruitful in the simple setup of inviscid incompressible fluids. The project will generalise this to consider more complex fluid models that include the effect of density variation and compressibility, among others. The result will be a methodology that can be applied to a broad class of problems in geophysical and astrophysical fluid dynamics.

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Undular bores in nonlinear optical media

Reference Number: ACM

Abstract: Undular bores are a familiar wave form in fluid mechanics, an example being the tidal bore observed in regions of strong diurnal tides, the Severn Estuary being a famous example. Undular bores can also form from optical beams in nonlinear optical media such as colloids and liquid crystals. There is next to no existing theory to describe undular bores in such media, but there has been considerable experimental interest.This project would involve investigating undular bores in nonlinear optical media.

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Uniform Asymptotics for Difference Equations

Reference Number: ACM

Abstract: This field of research is relatively new. For differential equations and integrals it is well-known how to obtain uniform asymptotic expansions. These expansions are valid in large regions, and especially near critical points where the behaviour of the solutions changes dramatically. There are some results in the literature for difference equations, but none of them is as simple or as powerful as what is known for differential equations and integrals. Hence, any new result is worth publishing.

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Wave radiation (and leaky waves) in GFD

Reference Number: ACM

Abstract: In the tropical atmosphere, the upward propagating gravity waves reach the stratosphere and never return. We found that such leakage effects are significant in the atmosphere - a large atmospheric event that spans the full depth of the troposphere (a "mode one" event) can dissipate in less than two days due to the wave leakage. In both the atmosphere and ocean these scenarios are ubiquitous. We discovered a new mathematical approach to study such problems, which includes finding a physically relevant functional basis for the GFD problem at hand. This PhD project involves investigating various GFD problems and evaluating the effect of leakage in them. This would result in novel understanding of GFD problems and have implications to data analysis.

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