A **world-class PhD training programme** which equips students with the skills and confidence to **lead their discipline**.

## World-leading Research

**MAC-MIGS staff carry out research at the cutting-edge of their disciplines. Their research interests encompass a broad great variety of mathematical methods and models. **

Thanks to this excellence and diversity, we are able to supervise high-quality PhD projects across many areas and enable our students to deliver internationally leading research.

## Research Areas

### Mathematical Modelling

Financial models, quantum models , molecular dynamics, interfacing neural networks with physical models, coarse-grained particle systems, lattice-Boltzmann models, and PDE models of fluids and solids, weather and climate models, interacting agents, biological processes, data-driven model reduction, data-analytic methods including Bayesian inference , optimisation, gradient flows, and data assimilation.

### Analysis Research

Continuum models (fluids and solids), harmonic analysis, propagation of singularities, stochastic analysis (including stochastic processes, stochastic ordinary/partial differential equations and stochastic approximation algorithms), inverse problems, data assimilation and filtering.

### Computational research

Numerical analysis (for ordinary and partial differential equations, and stochastic differential equations), finite elements, computational image analysis, medical imaging, parallel computing, high dimensional optimisation, machine learning algorithms, and professional software for scientific applications (molecular simulation, reservoir modelling, gas dynamics, machine learning, neural network models).

**There are also practitioners involved in chemistry (e.g. for molecular and quantum models, materials science), engineering (particle and granular models, processes, materials), informatics (machine learning, artificial intelligence), biology (systems biology, cell modelling) and physics (condensed matter, soft matter, density functional theory).**

## Programme Supervisors

**The supervisors’ main areas of expertise are indicated by the following tags:**

ANAL = Analysis, BIO = Biology, COMP = Computation, DS = Data Science, FIN = Financial Modelling and Analysis, FLU = Fluid Dynamics, MAT = Materials, MD = Molecular Dynamics, MESO = Mesoscale Modelling, OPT = Optimisation, PDE = Partial Differential Equations, PHYS = Physics, QM = Quantum Mechanics, STAT = Statistical Analysis, STO = Stochastic Methods, UQ = Uncertainty Quantification.

### Prof Graeme Ackland

### FRSE

Molecular Dynamics and Monte Carlo simulation; molecular dynamics at high pressure and machine learning of molecular interaction forces.

### Prof Rosalind Allen

Computer simulations to study how bacterial biofilms grow and respond to antibiotics. Part of this work involves using advanced algorithms for rare event simulation, such as forward flux sampling, of which I was a developer.

### Dr Yoann Altmann

Statistical signal and image processing, with a particular interest in Bayesian inverse problems with applications to remote sensing and biomedical imaging

### Dr Tibor Antal

Rigorous treatments and approximations of probability models and their applications to bacterial, cancer and virus populations in particular. Topics include branching processes, game theory, spatial models, genetics. Example projects: modelling the evolution of multidrug resistance in bacteria or cancer, modelling cancer initiation, progression, metastasis formation and fitting to clinical data.

### Prof John Ball

### FRS, FRSE

Nonlinear analysis, solid phase transformations, and liquid crystals, compatibility in polycrystals and defects in liquid crystals.

### Dr Lehel Banjai

Development and analysis of numerical methods with emphasis on acoustic and electromagnetic wave propagation. Numerical methods include time-domain boundary integral methods, space-time discontinuous Galerkin methods, hp-FEM, BEM.

### Prof Lindsay Beevers

Hydrological extremes : floods and droughts. Fluvial, sediment transport, morphological and ecosystem modelling: issues for design uncertainty assessment

### Dr Richard Blythe

Stochastic models of many-body systems in physics, biology and social science. Application of inferential techniques to empirical data to understand the fundamental mechanisms at play in these systems.

### Dr Natalia Bochkina

Theoretical guaranties for data-driven statistical inference for large-scale direct and inverse problems. Example project: adaptive statistical inference for inverse problems with unknown heterogeneous variance.

### Dr Lyonell Boulton

Computational aspects of spectral theory and applied spectral theory. Example projects: computing resonances of the Schroedinger equation; evolution problems on domains with fractal boundary.

### Dr David Bourne

Applied analysis (calculus of variations, partial differential equations, optimal transportation theory), numerical analysis, and discrete and computational geometry, with applications in materials science and continuum mechanics.

### Dr Michal Branicki

Interface between applied probability, information theory and dynamical systems with applications to Bayesian data assimilation, Bayesian learning, stochastic control, and data-driven dimension reduction in stochastic systems. In particular, probabilistic approach to prediction and uncertainty quantification, as well as data-driven techniques for state estimation and classification problems from large sets of noisy and incomplete data; all of these based on maximising information flow from empirical data to modelled dynamics.

### Dr Dominic Breit

The focus of my research is the existence and regularity theory for nonlinear (stochastic) PDEs as well as their numerical analysis and related function spaces. In particular, I’m interested in compressible Navier-Stokes equations, models for non-Newtonian fluids and equations of p-Laplace type. A sample project is the long-time behaviour of stochastically forced fluid flows.

### Dr Burak Buke

Applied probability and applications in service industry. I use convergence of measures techniques to develop approximations to large systems. Example project: probabilistic matching networks

### Prof Philip Camp

Molecular simulations, complex fluids, free-energy calculations, equilibrium and non-equilibrium dynamics.

### Dr Timothy Cannings

Statistical methodology and theory, with a focus on machine learning problems, such as classification and clustering. Much of my work is motivated by modern developments in technology, which result in new complex data structures and often require new statistical methods.

### Prof Mike Chantler

Interested in using machine learning to extract meaning and value from data and presenting this to users in a way that promotes understanding and trust

### Dr Lyuba Chumakova

Mathematical modelling in the areas of cell biology and geophysical fluid dynamics. In the area of cell biology the research is data driven and relies strongly on large scale computations, in the gfd I use mostly analytical approach. Example project: mathematical models of intracellular transport

### Prof Damian Clancy

Stochastic modelling of the transmission of infectious diseases through populations, looking particularly at endemic infections and the effects of population heterogeneities. This typically involves computer simulation of a specific disease system of interest, followed by rigorous mathematical analysis of aspects suggested by the initial simulation work, and then further simulation to validate the theoretical results (eg checking that theory developed in the large-population limit gives sensible results for realistic finite population sizes).

### Dr Daniel Coutand

Mathematical analysis of free boundary problems involving continuum mechanics models (Euler equations, Navier-Stokes equations, nonlinear elastodynamics): Well-posedness, finite-time singularity formation. Example project: singularity formation and propagation in free boundary problems

### Dr Cathal Cummins

### Dr Fraser Daly

Applied probability, including modelling of random systems, and analytic techniques in probability (eg, for proving limit theorems). Example projects: random graphs and networks; optimal coupling and rates of convergence to stationarity for Markov chains.

### Prof Vincent Danos

Stochastic modelling in the context of biological systems, Foundations of statistical learning. Example projects:efficient quantum machine learning schemes from Boltzmann machines with provably hard classical sampling problems; Compositional stochastic consensus algorithms (with adaptive levels of trust) based on replicated state machines algorithms.

### Dr Chris Dent

I am an industrial mathematician specialising in energy systems analysis, and good practice in the use of modelling in public policy. Example project: coordination of electric vehicle charging.

### Dr Martin Dindos

Fundamental questions (existence, uniqueness and properties) of elliptic and parabolic PDEs

### Dr Goncalo Dos Reis

I do research at the interface of applied mathematics and stochastic analysis. My current applications involve time-consistent optimal control under evolving streams of information and analysis/development of a stochastic model for charging of electric batteries. The former includes applications in data science and so-called machine learning while the former has a strong component of both modeling and probabilistic numerical analysis.

### Prof Dugald Duncan

Numerical analysis of time dependent PDEs and integral equations, including those describing wave propagation and scattering.

### Dr Ahmed El Sheikh

Parameter estimation for subsurface flow problems, UQ, Carbon Caputre, ML

### Prof Martin Evans

Stochastic modelling of systems `out of equilibrium’ including biophysical processes. The research entails finding analytical solutions of mathematical models and performing stochastic simulations of complex systems. Example projects: nonequilibrium stationary states; modelling of intracellular processes

### Prof Teresa Fernandes

Acquatic ecosystems

### Prof Sebastian Geiger

Modelling and simulation of hydrocarbon recovery from carbonate and fractured reservoirs, geothermal energy and CO2 storage.

### Prof Gavin Gibson

### FRSE

Computational methods for fitting and testing stochastic dynamical models with applications mainly in epidemiology.

### Dr Heiko Gimperlein

PDEs and their numerical analysis in the engineering and biological sciences, recently also swarm robotics. In particular nonlocal operators, error analysis of finite and boundary elements, variational analysis of nonlinear PDEs, computational mechanics. Example project: Space-time adaptive discretisation of wave equations.

### Dr Ben Goddard

Modelling, rigorous analysis and numerical simulation for complex, multiscale systems in areas such as quantum chemistry, molecular dynamics and statistical mechanics. I have a strong history of interdisciplinary research with chemists, engineers and physicists.

### Dr Ramon Grima

Stochastic modeling of biological and biochemical systems, specifically cell movement, cell-cell interactions and stochastic gene regulatory networks

### Prof Istvan Gyongy

### FRSE

The theory and numerical analysis of PDEs and stochastic PDEs with applications in stochastic control and nonlinear filtering and in mathematical models arising in physics, engineering and economics.

### Dr Abdul-Lateef Haji-Ali

My research interests include: Uncertainty Quantification, Stochastic Differential Equation, Numerical methods for SDEs and PDEs, Multilevel Monte Carlo, Particle systems, Crowd modelling, Mean-field theory, Sparse Grids, Combination techniques, Multi-index techniques, Inverse problems, risk measures and adaptive sampling.

### Dr Julian Hall

Optimization methods for linear and quadratic programming. Sparse numerical linear algebra for high performance large scale computational optimization. Industrial applications: feed formulation, genomics, telecommunications, petrochemicals, data science and finance.

### Prof Des Higham

Numerical analysis, stochastic computation, network science and applications in machine learning, digital human behaviour, urban analytics, crime and life sciences.

### Prof Jane Hillston

### FRSE

Formal modelling and logic-based model checking for dynamic properties of stochastic concurrent systems. Approximations and efficient analysis techniques for the underlying Continuous Time Markov Chains. Example project: fluid approximations of heterogeneous populations of Markovian agents

### Dr Tim Johnson

I am interested in how large data sets can circumvent the problem of a priori model selection related to stochastic control problems. More broadly I am interested in the fact that mathematical models actively direct social systems, rather than be passive describers of physical systems, and the ethical implications of this. Example project: stochastic control that does not require the specification of the diffusion but infers necessary functions from large data sets.

### Dr Aram Karakhanyan

a. geometric analysis (curvature flows, convexity estimates, analysis of singularities, L^p Minkowski problem), free boundary boundary problems and minimal surfaces (existence, min-max method, classification of global profiles) b. Calabi-Jorgens-Pogorelov type theorems for the Monge-Ampere type equations, exploring the structure of global convex solutions arising win the reflector-antennae and optimal mass transport problems.

### Prof Elham Kashefi

I work on models of quantum computing and their structural relations, exploring new applications, algorithms and cryptographic protocols for quantum information processing device.

### Prof Ruth King

### FRSE

Statistical modelling; efficient computational algorithms; data analysis; applications including ecology; medicine; epidemiology

### Dr Stuart King

Mathematical modelling, and computational mathematics. Past research focussed on modelling of fluid flows, some more recent projects focused on data analysis and image analysis.

### Dr Adam Kirrander

Chemical and quantum dynamics, including nonadiabatic dynamics in chemistry and physics and deep learning for potential energy surfaces in chemical dynamics.

### Prof Andrew Lacey

Mathematical modelling, and both approximate solution and analysis of the resulting equations, of industrial problems and societal issues. Example projects: flash sintering–mathematically modelling the interaction of the electric current and changing temperature during electrical fusing of ceramic powders; homeless modelling: The aim is to better model changing homeless levels, accounting for causes such as drug dependency and relation to crime.

### Prof Omar Lagrouche

Soil-structure interaction Elastic / acoustic wave modelling

### Prof Benedict Leimkuhler

### FRSE

Sampling algorithms and their application in various areas, e.g. molecular modelling and data analytics and am developing large software packages (MIST and TATi) which are moving towards the general release. Currently, I am engaging with an engineering firm in Bristol on a data analytics for wind turbine assessment. I am interested in designing and training elementary structures for geometrically constrained inference in physical models. Neural nets can learn maps, but to be relevant for applications, physical law should be encoded in their DNA.

### Prof Finn Lindgren

Developing Gaussian and other stochastic Bayesian process models for environmental and ecological phenomena, including spatial inhomogeneity and complex observation processes. This is tightly coupled with approximate computational methods, eliminating costly MCMC wherever possible. Example project: modelling sea surface temperatures and their observation biases

### Dr Sebastien Loisel

Preconditioning of partial differential equations by domain decomposition. Also: computational methods for calculus of variations. Example project: optimized barriers methods for calculus of variation

### Prof Gabriel Lord

Applied computational mathematics, numerical analysis, modelling, simulation and stochastically forced systems. Example projects:stochastic wave equation and marine reserves; numerical methods for neural fields with noise.

### Dr James Maddison

Geophysical fluid dynamics, focussing on ocean dynamics and mesoscale turbulence. Numerical methods, including the finite element method, with applications in numerical ocean modelling. Partial differential equation constrained optimisation, including adjoint based methods, with geoscientific applications.

### Dr Simon Malham

Nonlinear PDEs, computational spectral theory, geometric integration, stochastic differential equations, PDEs with nonlocal nonlinearities. Example project: the solution of PDEs with nonlocal nonlinearities.

### Prof Mercedes Maroto-Valer

Clean Energy Technology

### Dr Miguel Martinez-Canales

I study condensed matter using expensive Density Functional Theory. I want to extend the size of the systems I can study without losing accuracy using novel machine learning techniques. Example project: accelerated large-system structure prediction via machine learning.

### Dr Julien Michel

Computer-aided drug design and biophysical chemistry with a focus on all aspects of molecular simulations of biological molecules (algorithms, software development, application studies and integration with experiments)

### Prof Alexander Morozov

Active matter; low-Reynolds number flows; swimming of microorganisms; Newtonian turbulence in parallel shear flows; purely elastic flow instabilities

### Prof Iain Murray

Bayesian inference given data in models represented by simulators, where simulations are expensive. Data-driven modeling of high-dimensional probability distributions, useful for data cleaning, anomaly detection, recognition, forecasting, etc.

### Prof Raffaella Ocone

Hydrodynamics of granular materials and particle-laden flow

### Dr Tadahiro Oh

Analysis of nonlinear dispersive PDEs from deterministic and stochastic points of view

### Prof Adri B Olde Daalhuis

Formal series solutions of ODEs/PDEs are often divergent, and the exponentially small terms are needed to obtain a full understanding. Example projects: Transseries and the higher-order Stokes phenomenon; Transition region expansions for turning points.

### Prof Jin Ooi

Discrete particle modelling and data analysis, applied to a wide range of industrial processes including milling, mixing, granulation, silo flow, high speed ballast railtrack etc.

### Dr Michela Ottobre

Stochastic Analysis, Interacting particle systems and applications to biology, Numerical methods for stochastic differential equations, Markov Chain Monte Carlo Methods.

### Dr Diego A. Oyarzun

Biological networks; nonlinear dynamics; optimisation; stochastic dynamics; dynamic optimisation; optimal control; network theory; systems biology; synthetic biology; control theory; data science

### Prof Kevin Painter

Mathematical modelling in medicine, biology and ecology, including pattern formation during skin morphogenesis, neural crest migration and cancer invasion.

### Dr John Pearson

Modelling and computational methods for inverse problems/PDE-constrained optimisation/control problems, for problems of scientific and engineering interest. Applications are optimal transport techniques include fluid dynamics and imaging, chemical and biological systems, and medical imaging.

### Prof Beatrice Pelloni

Analysis of nonlinear PDEs, particularly of fluid dynamics, including uniqueness of solutions of geophysical fluid dynamics and numerical schemes for small dispersion limits of PDEs.

### Dr Marcelo Pereyra

Mathematical theory, methods and algorithms to solve large-scale inverse problems related to mathematical and computational imaging, such as medical imaging and astronomical imaging problems. I am particularly interested in new Bayesian analysis and computation approaches, and in developing deep connections between modern Bayesian, variational, and Machine Learning approaches to data science. Example projects: efficient Bayesian computation in high-dimensional bilinear inverse problems; combining infinite-dimensional Markov chain Monte Carlo and Deep Learning techniques for imaging inverse problems.

### Prof Gareth Peters

Methodology and applications in statistical modelling and machine learning in many application areas in risk, insurance and applied finance. Example projects: relationship between agricultural productivity, local climate models and commodity futures price dynamics; telematics and machine learning; crypto-currency volatility modelling

### Dr Oana Pocovnicu

Mathematical analysis of nonlinear dispersive PDEs. These are PDEs that model wave propagation phenomena in a variety of fields such as plasma physics, nonlinear optics, Bose-Einstein condensation.

### Dr Nikola Popovic

I work at the interfaces between mathematics, (stochastic) systems biology, neuroscience, and (veterinary) medicine. My approach relies on a combination of analysis and numerical simulation; specific techniques include (geometric) singular perturbation theory, geometric desingularisation (“blow-up”), and asymptotic analysis, as well as low-rank approximation and coarse-graining.

### Dr Mariya Ptashnyk

Nonlinear partial differential equations, homogenisation (deterministic and stochastic), multiscale numerical methods, from discrete to continuum, bifurcation analysis & pattern formation. Multiscale modelling of biological systems and analysis and numerical simulations of mathematical models. Applications include transport processes in and mechanical properties of biofilms, biological tissues, cellular signalling processes, plant root and shoot growth, interactions between plant roots and soil, plant-soil-atmosphere system.

### Dr Audrey Repetti

Research at the interface of optimisation theory (convex, nonconvex, and stochastic), Bayesian inference, and applications to high dimensional inverse problems. I am particularly interested in designing new optimisation methods, with theoretical guaranties, efficient to solve “real word” problems encountered in data science. Applications include computational imaging (e.g. in areas such as astronomy or biomedical), statistical graph processing (e.g. for computer vision), etc.

### Dr Valerio Restocchi

Modelling and simulation of complex socio-economic systems, Interaction and propagation on social networks, Network science, Human behaviour modelling, Financial markets and Econophysics, Cryptocurrencies, Agent-based simulations.

### Dr Max Ruffert

Simulations of merging stars and the analysis of underwater echolocation click data.

### Dr Sotirios Sabanis

Explicit numerical algorithms for nonlinear random systems of (typically) high dimension and their interplay with data science techniques. Examples of these algorithms are stochastic approximation/stochastic gradient methods, explicit numerical schemes for stochastic (partial) differential equations, parameters and MCMC algorithms (TULA = Tamed Unadjusted Langevin Algorithm).

### Prof Guido Sanguinetti

Statistical analysis of large biological datasets. Inverse problems in stochastic processes

### Dr Lucia Scardia

Calculus of variations and partial differential equations, homogenisation (deterministic and stochastic), harmonic analysis. Applications in material science: plasticity and dislocations, non-local aggregation problems, nonlinear elasticity, fracture and damage.

### Dr Linus Schumacher

Collective behaviour, Bayesian inference, data-driven modelling, cell-level models, stem cells, developmental biology, regenerative medicine

### Prof Jonathan Sherratt

### FRSE

Mathematical modelling of spatiotemporal patterns in ecology, biology and medicine. Example projects:deconstructing models for vegetation patterning in semi-arid environments; mathematical modelling of cell adhesion in wound healing.

### Dr David Siska

Theoretical, computational and applied aspects of stochastic analysis, control and partial differential equations. Example projects:SPDEs in elastodynamics and regularisation by noise.

### Prof Noel Smyth

Nonlinear waves in fluids and in optics, in particular solitary waves and undular bores (dispersive shock waves). Most of my work in nonlinear optics is performed in conjunction with experimental groups.

### Prof Amos Storkey

Machine Learning, Inference and Probabilistic Models: Deep learning and neural networks, machine learning markets, Hamiltonian Monte-Carlo, learning and inference in stochastic differential systems, stochastic optimization, game theory approaching in ML,

### Dr George Streftaris

Bayesian modelling and inference in stochastic processes for partially observed populations. Inter-disciplinary work involving statistical methodology applied in epidemiology, morbidity, actuarial mathematics and life sciences.

### Dr Lukasz Szpruch

Probabilistic representations to construct efficient computational methods for high dimensional problems. Topics include non-linear, non-local PDEs, Mean-Field models, Particle Systems, Monte Carlo Methods, Deep Neural Networks, Statistical Sampling, Game Theory, Stochastic Control and Reinforcement learning.

### Dr Aretha Teckentrup

My research interests are at the interface of numerical analysis, statistics and data science. I am particularly interested in uncertainty quantification in simulation with complex computer models, with recent research focussing on multilevel sampling methods, Bayesian inverse problems and Gaussian process emulators. Example projects: numerical analysis of Gaussian process emulators;efficient sampling methods for Bayesian inverse problems.

### Prof Jacques Vanneste

### FRSE

Fluid dynamics, mostly applied to oceanography, using geometric, asymptotic, stochastic and numerical methods. Example projects:mixing properties of multiphase flows inferred from PEPT data; geometric methods in oceanography.

### Dr Anna Lisa Varri

Theoretical astrophysics • Stellar dynamics • Phase space complexity • Gravitational N-body problem • Vlasov-Poisson systems • Bayesian methods for astrophysical data

### Dr Sara Wade

My research interests lie on the intersection between statistics and machine learning, with a strong interest in Bayesian nonparametrics and developing novel methodology and computationally efficient inference for complex and high-dimensional data. Example project: nonlinear scalar-on-image regression models

### Prof Andrew White

The use of deterministic and stochastic models to understand ecological and infectious disease systems. The models have applications to species conservation and disease managment strategies. Example projects: mathematical models of biological invasions; mathematical models of wildlife disease reservoirs.

### Dr Anke Wiese

Stochastic analysis, its application to mathematical and computational finance, and the analysis of stochastic differential systems and their algebraic structures. Example project: numerical methods for financial market models

### Dr Mark Wilkinson

My interests lie in the analysis of PDE. More specifically, I am interested in the kinetic theory of gases (in particular, the Boltzmann equation), dynamical billiards, atmospheric fluid dynamics (in particular, the semi-geostrophic equations), and liquid crystal theory (in particular, Q-tensor theory).

### Dr David Woolf

Marine Renewable Energy, Air-sea exchange of gases and particles, Global carbon cycle

### Prof James Wright

Euclidean harmonic analysis and its interactions with a variety of surrounding subject areas; integral geometry, number theory and Lie theory.

### Dr Kostas Zygalakis

Numerical aspects of stochastic processes with applications to computational statistics, machine learning and computational biology