UCL EPSRC Mathematical Sciences Doctoral Studentships Competition

Project: Equivariant extraction of randomness

• Reference: SOO-RANDOMNESS
• Lead Supervisor: Dr Terry Soo, t.soo@ucl.ac.uk
• Department: Department of Statistical Science
• Details: The project will consider various coordinate-free ways of extracting and finding randomness inside another random system.   Sinai's factor theorem (1964) states that an ergodic stationary system of positive entropy has any independent and identically distributed (i.i.d.) system of no greater entropy as a factor.  In particular, as a deterministic function of a bi-infinite sequence of i.i.d. three-sided fair dice roll, we can produce a bi-infinite sequence of i.i.d. fair coin flips; furthermore, the deterministic function is equivariant, so that if the sequence of dice roll is shifted, then its output under the function is given by shifting the original output.  We will develop various version of Sinai's theorem in probability and ergodic theory.

Project: Bayesian inference in survival analysis: new approaches to modelling and computation

• Reference: LIVINGSTONE-SURVIVAL
• Lead Supervisor: Dr Sam Livingstone, samuel.livingstone@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Survival models are ubiquitous in many application areas of data science, in particular in modelling data arising from clinical trials but also increasingly from observational studies, in which data are more freely available but typically also more structured and heterogeneous.  To accurately learn from observational data more nuanced and complex modelling strategies must be taken, which in turn makes robust and reliable inference a challenge.  The first aim of this project are to develop fit-for-purpose survival models for use with modern datasets used in health economic evaluation.  Following this, we will develop robust and scalable inference algorithms for these models, borrowing ideas from and building upon state-of-the-art approaches used in Bayesian computation.  The project will be done in consultation and collaboration with numerous industrial partners, such as researchers at ICON Plc, a global consultancy company working with major pharmaceutical companies and government bodies, to ensure that any methods developed are both fit-for-purpose and fall within regulatory frameworks.  To complete the project bespoke software packages will be developed to provide an end-to-end data science pipeline for robust Bayesian survival modelling.

Project: Stabilised finite element methods for nonlinear inverse problem

• Reference: BURMAN-FEMINVERSE
• Lead Supervisor: Prof. Erik Burman, e.burman@ucl.ac.uk
• Department: Department of Mathematics
• Details: Inverse problem subject to partial differential equations occurs when one tries to reconstruct some quantity of interest using a physical model and measured data. An important example from seismology is the fault slip inversion, where seismograms together with computational pde are used to understand the physical laws of the fault slip process that causes earthquakes. This is a nonlinear inverse problem. At UCL we have developed a theory for computational methods for linear inverse problems and the aim of this PhD project is to extend this to the nonlinear case. This will include training in analysis, numerical analysis and computational development.  We will start with a model situation where a single coefficient (the Robin coefficient) will be reconstructed for a stationary wave equation and internal measurements. In this case we have already developed the necessary theoretical tools. Then we consider a simple model for the fault slip problem: identify an adhesion coefficient on internal boundaries for wave equations. First scalar, then elastic. This is a very challenging problem requiring new results both from the point of view of analysis and computational methods. To approach the problem we start considering one space dimension and time and then extend to higher dimensions.

Project: High resolution finite element methods and boundary layer models for incompressible turbulent flows

• Reference: BURMAN-TURBULENCE
• Lead Supervisor: Prof. Erik Burman, e.burman@ucl.ac.uk
• Department: Department of Mathematics
• Details: In recent results on computational methods for high Reynolds turbulent flows, in collaboration with Spencer Sherwin at Imperial, we have shown computationally that a carefully balanced dissipation acting on the singular fluctuations of the approximation allows for accurate predictions of averaged quantities using large eddy simulation. In parallel we have shown that the celebrated Smagorinsky turbulence model has similar properties, provided a certain approximation space guaranteeing mass conservation is used. In the present thesis we will build on these two results to further the design of efficient and accurate computational methods for incompressible turbulent flows, aiming to bridge the two paradigms of computational turbulence: model (LES) or no-model (UDNS). On the theoretical side we will show low regularity estimates for the linearised case and use boundary layer theory to create (and analyse) models, that allow for a physically accurate coupling between the bulk and boundary flow without resolving the layer numerically. The project will strengthen ongoing collaborations between the UCL groups in Computational Mathematics (Burman), Fluid Mechanics (Johnson) and the IC group in Computational Fluid Mechanics (Sherwin). The student will get training in computational methods for turbulent flows and related analysis relevant for careers in either industry or academia.

Project: Time-dependent viscoplasticity

• Reference: HEWITT-VISCOPLASTIC
• Lead Supervisor: Dr Duncan Hewitt, d.hewitt@ucl.ac.uk
• Department: Department of Mathematics
• Details: Numerous materials in nature and industry (e.g. muds, pastes, emulsions and granular slurries) exhibit ‘viscoplastic’ behaviour: they flow like a viscous fluid under sufficient force, but clog and jam if the applied stress is below a critical ‘yield’ value. There is an increasing understanding that such materials often also exhibit more complex time-dependent behaviour: their rheology actually evolves over time under different conditions towards one - or more - equilibrium states. This behaviour can cause unexpected and dramatic effects in the flows of these materials that are not fully understood.  The aims of this project are to assess and systematise the disparate array of models that have been proposed to account for this behaviour; to explore their predictions in different canonical flow settings; and so to better understand the mechanics of these materials. The project will involve mathematical modelling and applied analysis, and some numerical simulation.   The PhD student will have a background in applied mathematics, but there is no other requirement besides enthusiasm and interest. As well as gaining a thorough understanding of non-Newtonian fluid mechanics and numerical coding, the student will be trained in employing an analytic approach to problem solving and developing their own independent research.

Project: High frequency scattering with applications to fluids

• Reference: GALKOWSKI-SCATTERING
• Lead Supervisor: Dr Jeff Galkowski, j.galkowski@ucl.ac.uk
• Department: Department of Mathematics
• Details: The proposed project is part of the primary supervisor's broader research program on spectral and scattering theory. This project focuses on the study of high frequency solutions to wave-type equations. The student's work will focus on the study of scattered waves in the presence of trapping. It has been known since the work of Burq that estimates for such waves must suffer losses relative to non-trapping situations. However, the precise energies at which these losses occur are not fully understood. This project will aim to improve our understanding of this phenomena. While undertaking the project, the student will be trained to use several important theories in mathematics. The student will acquire a strong foundation in mathematical scattering theory as well as in microlocal analysis and functional analysis. These tools have recently found applications in an enormous variety of problems including the study of dynamical systems, internal forced waves in fluids, dynamics of free boundary problems, and general relativity to name a few. The precise understanding of the linearised dynamics is of particular importance in the context of nonlinear PDE. In the course of the project, the student will be trained to look for and implement these tools, particularly in the context of mathematical fluid dynamics; an area of expertise of the secondary supervisor.

Project: Vortical wakes in the ocean

• Reference: JOHNSON-WAKES
• Lead Supervisor: Prof. Ted Johnson, e.johnson@ucl.ac.uk
• Department: Department of Mathematics
• Details: We suggest that the dissipation of the ocean energy occurs predominantly at the ocean surface, ocean floor and lateral boundaries rather than in the interior. This is important because it is the balance between forcing and dissipation that sets the strength of the ocean circulation. ERJ has a 3-year project ""NSFGEO-NERC Stirring at the Walls - A dynamical boundary model for the ocean"" which supports MC to develop the mathematics and numerics for a sidewall model for ocean circulation. Recently published work has shown that boundary layer separation in the lee of seamounts may also be a significant contributor to dissipation. The flow patterns presented there are remarkably close to those that appear in classical high Reynolds number flow. This project aims to demonstrate that methods developed for high Re flows by Professor Susan Brown (UCL, Mathematics) can succinctly describe what is seen in these large simulations. FTS is an expert in these methods and MC in numerical methods for solving the reduced problems that are derived. The student will enter a highly active research field, joining a strong team applying sophisticated mathematics and modern numerical techniques to an important problem in climate science.

Project: Optimising the implementation of Value of Information (VoI) algorithms

• Reference: GREEN-VOI
• Lead Supervisor: Dr Nathan Green, n.green@ucl.ac.uk
• Department: Department of Statistical Science
• Details: From decision theory, Value of Information (VoI) analysis is the estimation of expected reductions in loss from learning specific parameters or collecting data of a given design. Decision making in health economic evaluation is a current area making use of VoI. The three VoI statistics of interest are the Expected Value of Perfect Information (EVPI), Expected Value of Partial Perfect Information (EVPPI) and Expected Value of Sample Information (EVSI). However, calculation of these statistics can incur a large, prohibitive computational burden for several parameters and more realistic situations. The key aims of this project are to build-on existing algorithms and implementations for VoI to create faster, more memory efficient, optimised VoI calculcations and software implementations. The student will learn about health economics, clinical trials and cost-effectiveness analysis and acquire skills in coding in R and C, algorithm optimisation, software parallelisation, memory management (e.g. moisation, caching, lazy loading), and data structures and their algorithms. This is a project appropriate for someone with an interest in statistics, computer science and health.

Project: Causal inference with machine-learning to address modelling challenges in single-cell genomics

• Reference: BARTLETT-SINGLECELL
• Lead Supervisor: Dr Tom Bartlett, thomas.bartlett.10@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Causal inference with machine-learning has grown rapidly in popularity in recent years among computational-statistics researchers. The logical framework of causal inference coupled with the predictive power of machine-learning (ML) has proved to be a powerful combination, but fitting this sort of model require sample-sizes much larger than are often available in molecular biology studies. Single-cell analysis has also grown rapidly in popularity in recent years among cell biology researchers. The very different characteristics of these new types of data have required the development of new analyses methodologies for the new data. But the increase in sample-size that is available to statisticians from n cells rather than from n tissue-samples also opens up entirely new possibilities for modelling cell-biological data. In particular, the recent advances in causal inference with ML would be practical to use with these single-cell data-sets, and these methods would be expected to be particularly adept at uncovering important patterns for the first time in these data. Specifically, the excellent performance of causal inference with ML in the presence of unknown and/or unmeasured confounders is likely to lead to improvements, for example in inferring changes in gene-regulatory behaviour between cell-types of the same tissue. This will lead to an improved understanding of the mechanisms behind the gene-expression patterns that define cellular identity and function.

Project: Statistically explainable GAN inversion

• Reference: XUE-GAN
• Lead Supervisor: Dr Jinghao Xue, jinghao.xue@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Generative adversarial networks (GANs) are a deep learning framework that learns a mapping, from a latent space to a data space, to generate new data points (often images) with the same probability distribution as that of the training data. Its reverse dual, GAN inversion, aims to invert a given image back into the latent space of a pretrained GAN model, such that editing of an image can be simply achieved by editing its code in the latent space. However, to achieve desired image editing (e.g. style transfer, imagination realisation or fine-grained modification), it is pivotal to understand, model and infer the statistical structure and randomness of the latent space of a pretrained GAN model. This is the aim of this project, a new and interdisciplinary topic able to inspire numerous exciting image-editing innovations and applications.   To achieve the aim, the student will investigate from three perspectives: firstly to discover independent latent factors of the desired attributes by exploring nonlinear dimension reduction of the latent space; secondly to model interpretable fine-grained controls with some intermediate priors from domain knowledge as regularisation; and finally to infer from the latent space a common subspace for multimodal synchronisation between images, text and audios.

Project: Problems in Ramsey theory

• Reference: LETZTER-RAMSEY
• Lead Supervisor: Dr Shoham Letzter, s.letzter@ucl.ac.uk
• Department: Department of Mathematics
• Details: In this proposal I will mention two future research directions that I hope to pursue together with my future PhD student. Both directions are in Ramsey theory. Ramsey's classical theorem (1930) asserts that whenever the edges of a complete graph G on N vertices are red-blue coloured, one can find a monochromatic (namely, all-red or all-blue) copy of a given graph H, provided that N is sufficiently large with respect to H.   An interesting direction of research explores the structure of graphs G whose every red-blue colouring contains a monochromatic copy of H, and which are *minimal* with respect to this property. In particular, one could explore how small the minimum degree of such G could be. While a lot is known about this problem, there are many interesting open questions.   Another direction with quite a different flavour explores the number of monochromatic copies of H in a red-blue coloured complete graph G. More specifically, one could ask for the maximum number of edge-disjoint monochromatic copies of H. Together with Gruslys, we have recently answered this question for the case where H is a triangle. Many variants of this question are open, yet some may be approachable with our techniques.

Project: Explicit arithmetic of curves

• Reference: DOKCHITSER-ARITHMETICCURVES
• Lead Supervisor: Prof. Vladimir Dokchitser, v.dokchitser@ucl.ac.uk
• Department: Department of Mathematics
• Details: The aim of the project is to take advantage of the ongoing surge in algorithmic number theory, as witnessed by the computer algebras Sage and Magma, and the LMFDB database. These have made it viable to experiment numerically with arithmetic objects that have been previously inaccessible. Specifically, the student will investigate higher genus curves and their L-functions and local invariants (solubility, Galois representations, Tamagawa numbers), and develop an explicit theory of their motivic pieces. For comparison, the analogous objects for genus 1 curves (""elliptic curves"") have been key in some of the strongest developments towards the Birch-Swinnerton-Dyer conjecture, which is one of the central challenges in modern mathematics.  The student will learn the local and global theory of elliptic curves, higher genus curves and their Jacobians, the use of computer algebra packages, and the use of numerical investigations for proving theoretical results. The student will actively participate in a London-Bristol research study group and regularly contribute talks, interact with the primary advisor's research team, attend the London Number Theory seminar, and benefit from graduate lecture courses available through the LTCC and LSGNT networks.

Project: Combining networks of infinite server queues with spatial modelling for the design of service systems

• Reference: UTLEY-QUEUES
• Lead Supervisor: Prof. Martin Utley, m.utley@ucl.ac.uk
• Department: Department of Mathematics (Clinical OR Unit)
• Details: Infinite server queueing models offer flexibility for modelling the time-varying distribution of offered load at each node within a network through which distinct classes of customer describe intersecting paths. Classes of customer can be characterised by the stochastic paths taken, the pattern of arrivals to the network (appointment driven or walk-in) and the distributions of service time that apply at each node. However, little has been done on using these models to inform the arrangement of such networks of process within buildings.  The starting point will be analytical infinite server models for obtaining the time varying distribution of transitions between processes (arc use). The student will then explore approaches for evaluating allocations of processes to physical spaces within a structure that guides or inhibits movement from space to space. This will draw on the architectural discipline of space syntax, and analysis of transit times and crowding of physical conduits such as corridors or stairwells that bear the flows along arcs. The scope for using these fast, analytical models to guide simulation of capacitated systems would then be explored.  Application will focus on the design of health facilities resilient to future pandemic requirements for social distancing and one-way systems within buildings.

Project: Multiscale transport in spatially heterogeneous bacterial biofilms

• Reference: DALWADI-BIOFILMS
• Lead Supervisor: Dr Mohit Dalwadi (about to move to UCL from Oxford, mohit.dalwadi@maths.ox.ac.uk)
• Department: Department of Mathematics
• Details: The majority of bacteria live in biofilms, communities of bacterial cells embedded within heterogeneous extracellular matrices (ECM). These ECM provide important benefits to cells in biofilms, including chemical protection through microscopic interactions with relevant chemicals, such as antibiotics and signalling molecules. To develop new treatments for antibiotic-resistant biofilm infections in diseases such as cystic fibrosis, it is crucial to understand the relationship between microstructural properties of the ECM and mass transport properties of the biofilm at the colony scale.  In this project, the student will develop new mathematical tools to understand the role of biofilm microstructure in the colony-scale properties of biofilms. This will involve adapting classic mathematical homogenization to systematically derive upscaled transport equations that account for heterogeneity in microstructure. These derived methods will be more widely applicable to general mass transport problems. The student will analyse the population-level equations they derive to characterize the relationships between microstructure and colony-scale dynamics.  The student will be trained in continuum mechanics, asymptotic analysis, and scientific computing. This project is aligned with the lead applicant’s research programme in developing new mathematical tools for problems involving multiscale mass and fluid transport, and the results will provide an excellent base for future research proposals.

Project: Physical and genetic design of bacterial signalling via multi-scale modelling

• Reference: PEARCE-SIGNALLING
• Lead Supervisor: Dr Philip Pearce (about to move to UCL from Harvard, philip_pearce@hms.harvard.edu)
• Department: Department of Mathematics
• Details: The design and construction of synthetic bacterial populations can yield the solution to complex societal problems including the discovery and production of new drugs and biofuels. Synthetic populations can be controlled via their intercellular signalling systems, in which cells secrete and sense molecules called autoinducers. Autoinducers diffuse between cells, and are therefore affected by physical aspects of the environment including external flow and extracellular proteins. In this project, we will develop a multi-scale mathematical framework to characterise and design the physical and genetic aspects of bacterial signalling at the levels of molecules, cells and populations. The framework will combine molecule-level Langevin simulations for autoinducer diffusion with cell-level differential equation models for genetic circuits, to derive effective population-level partial differential equations. The project will provide new mathematical insight into the connections between molecular interactions and effective continuum equations in soft matter, with relevance to transport problems throughout biology. The student will be trained in computational and analytical skills, and interdisciplinary collaboration. This project will form the basis for future research in the lead applicant’s research programme on connecting molecular properties to macroscopic dynamics, and will open up a collaboration with Prof. Chris Barnes in the Division of Biosciences.

Project: Multi-scale modelling of blood flow in sickle cell disease

• Reference: PEARCE-SICKLE
• Lead Supervisor: Dr Philip Pearce (about to move to UCL from Harvard, philip_pearce@hms.harvard.edu)
• Department: Department of Mathematics
• Details: In sickle cell disease (SCD), red blood cells (RBCs) stiffen in deoxygenated conditions, increasing effective blood viscosity and eventually causing complete occlusion of blood vessels, if left untreated. To guide the development of genetic and pharmacological treatment strategies, we need to understand the biophysical processes that lead to vessel occlusion, and how blood properties after treatment affect occlusive risk. To this end, in this project we will quantitatively characterise the factors that contribute to the clogging dynamics of suspensions of deformable and rigid particles in microchannels. Using a fluid-structure interaction code framework called BioFM, we will connect distributions of RBC stiffnesses to effective blood material properties and clogging dynamics. The results will provide new mathematical insight into the connection between heterogeneous particulate properties and emergent non-Newtonian continuum dynamics in non-Brownian suspensions, with applications throughout soft and biological matter. The project will include collaboration with Profs. David Wood (University of Minnesota) and John Higgins (Harvard Medical School), allowing the student to access newly emerging experimental data. Furthermore, the student will learn highly transferable coding, simulation and analytical skills. The project fits into the lead applicant’s wider research programme on multi-scale modelling of complex biological systems. The lead supervisor will be Dr Philip Pearce (ECR), an expert in modelling of complex biological systems with a track record of research on modelling blood flow, and experience of informal  supervision of PhD students. This will be the lead applicant’s first PhD project supervised at UCL. The secondary supervisor will be Prof. Helen Wilson, an expert in non-Newtonian flows with significant experience of supervising PhD students. Biological expertise and data will be provided by Prof. John Higgins (HMS) and Prof. David Wood (Minnesota), and assistance with the BioFM code will be provided by its developer Dr. Timm Krueger (Edinburgh).

Project: An Information-Theoretic Foundation of Deep Learning Algorithms

• Reference: RODRIGUES-DEEPLEARNING
• Lead Supervisor: Dr Miguel Rodrigues, m.rodrigues@ucl.ac.uk
• Department: Department of Electronic and Electrical Engineering
• Details: Artificial intelligence technologies powered by deep learning have been leading to remarkable progress in automated decision-making tasks, with potential to transform various industrial sectors such as healthcare, finance, insurance, and more.  However, deep learning algorithms are essentially black-boxes, with classical theories unable to explain their performance due to their inability to capture the interplay between various elements associated with the learning process.   It is therefore recognized that the ability to develop a new mathematical foundation can considerably advance our understanding of state-of-the-art deep learning algorithms, catapulting in turn the development of more reliable, robust, and transparent artificial intelligence technologies  This PhD project builds upon emerging advances lying at the intersection of information theory, learning theory, and machine learning, to develop a new foundation for learning machines. This PhD project involves:  (1) Developing of an information-theoretic framework allowing to quantify the interplay between key quantities and phenomena associated with the machine learning process and (2) Leveraging the framework to cast insight onto the performance of and develop more reliable deep learning algorithms   The PhD student will develop new mathematical skills / ideas lying at the interface of various fields, along with programming skills in languages / frameworks such as Python or PyTorch. This project pioneers a mathematical foundation to artificial intelligence systems, leveraging principles from information theory, learning theory, and machine learning  The supervisory team therefore encompasses experts in the fields of information theory and processing (Prof Rodrigues and Dr Toni, EEE Dept) and learning theory / machine learning (Prof Shawe-Taylor, CS Dept), with a track-record delivering outstanding research in their areas.  The student will attend regular meetings with the elements of the supervisory team, including junior colleague (Toni, Lecturer)  The student will have the opportunity to engage with and benefit from the wider UCL ecosystem in the area of artificial intelligence.

Project: Modelling and optimisation of creased membranes for deployable space antennas

• Reference: BOSI-MEMBRANES
• Lead Supervisor: Dr Federico Bosi, f.bosi@ucl.ac.uk
• Department: Department of Mechanical Engineering
• Details: Creased membranes are ubiquitous in the natural and man-made world, ranging from insect wings to origami metamaterials and aerospace structures. In particular, space antennas are essential satellite systems, allowing a variety of space missions. The increasing demand for payload weight reduction and collecting surface maximisation has led to the development of membrane space antennas. These flexible systems are folded using origami techniques to reduce the storage volumes and be transported into space. However, the introduction of inelastic creases deeply affects the shape configuration and stability of the deployed membranes. In this framework, the influence of the creases on the membrane response is not well understood, resulting in the mathematical challenge of modelling such deformable systems and the fascinating engineering opportunity of realising high-performance space antennas. Therefore, the proposed project seeks (i) a fundamental understanding of the mechanics of creased thin films through accurate visco-elastoplastic modelling, (ii) the optimisation of their origami pattern to develop high-efficiency membrane space antennas, and (iii) the design and realisation of a novel experimental setup to assess the deployment and shape accuracy of lightweight thin-film structures. The research objectives will be achieved using a combination of advanced mathematical modelling, computational analyses, and experimental tests.

Project: Multiscale models for carbon recycling through synthetic biology

• Reference: DALWADI-CARBON
• Lead Supervisor: Dr Mohit Dalwadi (about to move to UCL from Oxford, mohit.dalwadi@maths.ox.ac.uk)
• Department: Department of Mathematics
• Details: In a typical carbon recycling set-up, carbon dioxide is bubbled through a fluid containing genetically modified microorganisms. These will ideally consume the gas and produce high-value chemicals, such as medicines or biofuels. Modelling gas dissolution and transport within bioreactors is an incredibly complex multiphase problem, since bubbles are sliced by impeller blades hundreds of times per second within swirling liquids. However, the extreme lengthscale and timescale ratios involved mean that a careful analysis can systematically reduce this complexity.  In this project, the student will use and adapt multiscale methodology to build computationally efficient models for gas transport, accounting for bubble slicing within a bioreactor. Since this will involve tracking the size distribution of bubbles within the bioreactor, it will involve developing novel methods to perform multiple scales analyses on coupled systems of integro-differential equations. In collaboration with an industrial partner, the student will derive and analyse general reduced-order models for gas transport to characterise and optimise operating conditions for gas fermentation.  The student undertaking this project will be trained in continuum mechanics, asymptotic analysis, and scientific computing. This project is aligned with the lead applicant’s research programme in developing new mathematical tools for problems involving multiscale mass and fluid transport.

Project: Development of New Effective Models for Energy-Efficient Lattice Materials using Mathematical Homogenisation

• Reference: TAN-LATTICE
• Lead Supervisor: Dr PJ Tan, pj.tan@ucl.ac.uk
• Department: Department of Mechanical Engineering
• Details: This project concerns a new class of engineered materials that mimics the hierarchical lattice construction found in nature. The aim is to derive quantitative relations to link the different length scales by energy equivalence concepts and homogenisation techniques based on the general theorems of gamma-convergence. When several scales are present in space and time, the approach is first to construct micro-scale models, using appropriate representative volume elements, and then to deduce macro-laws and constitutive relations that relate effective behaviour to micro-scale geometry and physics by exploiting separation of length scales. The perturbation method of multiple scales is often employed; however, application of homogenisation to problems involving fracture, localised instability and/or microstructural imperfections within an appropriate multi-scale framework remains a considerable challenge and will be addressed in this proposed project. A range of approaches will be employed, including a periodic length scale that varies as a function of the global length variable, as well as coupled macro-scale models that incorporate different micro-scale features in different zones of the macro-domain. The student will be trained in the techniques of applied mathematics, continuum solid mechanics and computational mechanics; all applied to the study of a new class of practical lightweight and energy-efficient materials.

Project: Statistical Causal Machine Learning for Multiple Outcomes

• Reference: MANOLOPOULOU-CAUSAL
• Lead Supervisor: Dr Ioanna Manolopoulou, i.manolopoulou@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Flexible statistical learning methods for causal inference of individual treatment effects (i.e. the effect of treatment on each individual) have seen significant advances recently, partly driven by the availability of large, observational studies (eg. electronic health records). However, almost all methods focus on modelling a single treatment outcome, whereas in fact treatment is usually chosen on the basis of two outcomes, typically the main outcome of interest versus adverse side-effects of treatment. The motivating application is a large observational study from NHS Scotland on incidence and treatment of heart disease, where choice of treatment is based, for example, on a balance between the risk of a heart attack against the risk of bleeding due to anti-coagulant therapy. The PhD student will extend Bayesian Causal Forests to build a composite joint model for the two adverse outcomes to infer individual-level estimates of the effect of treatment. Subject to approval, the methods will be implemented on a sample of ~50,000 patients admitted between 2013 and 2016 to Scottish hospitals with symptoms of heart disease, in collaboration with Dr Catalina Vallejos and Prof Nick Mills from the University of Edinburgh. This work naturally fits into H3 and H4 of “Outcomes and Ambitions”.

Project: Growth-fragmentation processes

• Reference: WATSON-GROWTH
• Lead Supervisor: Dr Alex Watson, alexander.watson@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Growth-fragmentation processes describe systems in which cells grow slowly and fragment suddenly. The field has a wide range of applications, from biophysics models and theories of quantum gravity through to telecommunications. The focus of this research project is the long-term behaviour of these processes, describing the way they settle into equilibrium. With this as a framework for summarising the emergent properties of these systems, the project will consider optimal control of the division mechanism and generalisations to higher dimensions, in tandem with Monte Carlo methods for computation, with cell division applications in mind.

Project: Stochastic models in genetics: using molecular genetic data to learn about population divergence and speciation.

• Reference: HERBOTS-GENETICS
• Lead Supervisor: Dr Hilde Wilkinson-Herbots, h.herbots@ucl.ac.uk
• Department: Department of Statistical Science
• Details: This project is concerned with the application of stochastic models to questions in evolutionary genetics regarding the process of speciation and the role of gene flow between species. In recent years, vast amounts of genetic data have become available; as a result, there is much interest in fast methods to analyse such data. This project aims to develop a computationally efficient maximum-likelihood method to fit alternative evolutionary scenarios to genetic data from two closely related species. The project will build on Costa and Wilkinson-Herbots (Genetics, 2017) but aims to overcome two important limitations when dealing with real data: 1)    Whereas our previous work assumed that the relative mutation rates of different loci are known, in practice these need to be estimated and may be subject to considerable uncertainty. This project will investigate how to incorporate uncertainty about the relative mutation rates into our maximum-likelihood method. 2)    Whereas our previous work assumed that individual DNA sequences are available, data from diploid species are typically ‘unphased’, i.e. it is not known which combinations of nucleotides belong together on each of the two chromosomes within the same individual. This project aims to extend the maximum-likelihood method above to unphased data from diploid individuals.

Project:  Validity of Graphical Causal Modeling

• Reference: SADEGHI-GRAPHICAL
• Lead Supervisor: Dr Kayvan Sadeghi, k.sadeghi@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Inferring causal relationships has always been one of the main objectives of science. Examples in today's world include, but are not limited to, inferring potential causes of cancer, the effect of gene manipulation, and much more. Statistical procedures, known generically as causal models, must be used to infer such causal relationships from observed data. Extensive research has been conducted on defining, interpreting, and applying causal models. Today, a very popular method for inferring causal relationships is based on the use of statistical models over graphs with nodes  that are random variables representing the quantities of interest. Our proposed project will try to explain when one can or cannot utilize graphical causal models, and how one utilizes them correctly. In particular, the candidate investigates what types of probability distributions and statistical models can be represented by graphs; and what the limitations of using graphs for causal inference and testing causal relationships are. To conduct the project successfully, the candidate must be theoretically inclined, and have a good knowledge of statistical modelling and multivariate statistics. A major part of the project aims at proving properties of causal models, but some programming skills will also be needed.

Project: Reynolds' stress closure for stochastically forced turbulent flows on Jupiter

• Reference: ESLER-JUPITER
• Lead Supervisor: Prof Gavin Esler, j.g.esler@ucl.ac.uk
• Department: Department of Mathematics
• Details: For over 70 years the holy grail of geophysical fluid dynamics has been to develop a closure theory for the Reynolds' stress, or eddy-driven momentum flux, applicable to observed turbulent flows. Recent work (Woillez and Bouchet, 2019, J. Fluid Mech.) represents a significant step in this direction for a stochastically forced flow relevant to Jupiter's jets. The aim of the project is to apply classical techniques from applied mathematics, namely multiple-scale asymptotics and boundary layer analysis, to the relevant quasi-geostrophic equations. The work has the potential to clarify and extend the existent treatment and progress towards a full closure theory. The student will first be trained in geophysical fluid dynamics, stochastic partial differential equations and the relevant asymptotic methods. Numerous asymptotic regimes exist, including some in which the equations naturally form internal boundary layers, presenting a rich variety of problems with a good range of complexity, which will allow for the student to make straightforward progress before tackling more challenging mathematics. Results can be verified against numerical results from existing code developed in Esler’s group, the development and adaptation of which will provide an optional supporting strand to the research for a student with a sound background in computational methods.

Project: Special cycles and arithmetic intersections on Shimura varieties

• Reference: GARCIA-SHIMURA
• Lead Supervisor: Dr Luis Garcia, l.e.garcia@ucl.ac.uk
• Department: Department of Mathematics
• Details: The project is concerned with understanding  certain geometric spaces of fundamental interest in number theory known as Shimura varieties. These are spaces that are amenable to study using both geometric and number-theoretic tools, leading to fascinating interactions between arithmetic and geometry.   An area that has attracted much interest in recent years is the intersection theory of Shimura varieties. An ambitious program launched by Kudla about twenty years ago relates arithmetic intersections on Shimura varieties to the theory of automorphic forms, which is another central subject in current number theory. This research direction has been highly successful; for example, some of the most recent results on the Birch-Swinnerton-Dyer conjecture (one of the seven mathematics problems singled out by the Clay Mathematics Institute as being a fundamental challenge) and its generalisations use heavily the insights obtained from Kudla’s program.  The student will work on understanding arithmetic intersections on Shimura varieties. For this they will need to acquire the appropriate background in algebraic geometry and number theory. Then they will start working on one of the many open questions in the area (e.g. intersections on p-adic period domains, or contributions from the toroidal boundary).

Project: Statistical inference for continuous variables and critical illness monitoring

• Reference: JENDOUBI-CRITICALILLNESS
• Lead Supervisor: Dr Takoua Jendoubi, t.jendoubi@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Critical illness is defined by the evidence of acute organ failure needing monitoring and/or support, either with drugs or machines. Close monitoring inevitably generates large amounts of data from multiple sources. These data are used to make clinical decisions by the bedside.  Data are generated at different frequencies e.g. vital signs such as heart rate and oxygen levels may be monitored continuously at the bedside. Blood gas analysis may be undertaken every 4-6 hours, other blood tests may be performed 12-24 hourly. Many of the variables are inter-connected and treatments may have predictable effects on some of the variables.  We propose the use of Bayesian multi-level modelling and Bayesian networks using real patient data (>50000 patients from UCLH and >5000 patients from Great Ormond Street Hospital) to generate continuous estimations of intermittently sampled values and model for the missingness mechanism.  These wil be applied to modelling the pH changes continuously and how haemoglobin binds to oxygen, based on data from the ventilator, vital signs, drug used and blood tests.  The work will be part of the UCL CHIMERA hub (www.ucl.ac.uk/chimera) which uses a multi-disciplinary approach to enhance the understanding of human physiology using real patient data.

Project: Multivariable dependence and organ failure inference

• Reference: JENDOUBI-ORGANFAILURE
• Lead Supervisor: Dr Takoua Jendoubi, t.jendoubi@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Critical illness is defined by the evidence of acute organ failure needing monitoring and/or support, with drugs and machines. Close monitoring generates large amounts of data from multiple sources. These data are used to make clinical decisions by the bedside.  Data based decision making currently relies on trend detection or pattern recognition by clinicians. Many of the variables measured and recorded are interdependent. Inferences are made about variables that are technically difficult to measure, e.g.  the use of different pressure measurements to infer flow and resistance patterns. We propose the use of Bayesian non-parametric modelling to a) understand inter-variable dependencies over time, b) detect deviation from expected trajectories. This encompasses Gaussian processes, Bayesian emulation and network analysis.  Data will be available from UCLH (>50000 patients) and Great Ormond Street Hospital (>5000 patients) including the following variables: a) Bedside monitored variables such as heart rate, blood pressure, oxygen saturations;  b) Feedback data from machines such as ventilators; c) Treatment variables such as drug doses; d) Results from blood tests such as blood gas analysis.  The work will be part of the UCL CHIMERA hub (www.ucl.ac.uk/chimera) which uses a multi-disciplinary approach to enhance the understanding of human physiology using real patient data.

Project: Causal Impacts in Network Data

• Reference: SILVA-NETWORK
• Lead Supervisor: Prof Ricardo Silva, ricardo.silva@ucl.ac.uk
• Department: Department of Statistical Science
• Details: Network data, including those coming from social media and other multi-agent environments, is a source of many important problems in data science. This project will investigate ways by which network behaviour is modified by external impacts, such as misinformation campaigns in social networks and other environmental changes. In particular, we will be interested in measures of causal impact and how to disentangle association due to direct interaction between components of this network and possible confounding effects due to common causes affecting group behaviour. To accomplish that, we will investigate methods in causal inference that addresses interactions among agents in a system and measures of influence, along with network science that specialises on feature extraction from network data and assessments of system resilience to changes.