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

Ordinary and partial differential equations

(deterministic and stochastic), dynamical systems, harmonic analysis, stochastic analysis and probability, inverse problems, calculus of variations, functional analysis, geometric measure theory, optimal transport. Applications in materials science, fluid mechanics, mathematical biology, chemistry, data analysis.

### 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.

### FRSE

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

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

Mathematical optimization provides guaranteed optimal or near-optimal solutions for various classes of large-scale optimization problems. Much of my research, in collaboration with industry partners, is about improving optimization models and algorithms used in real-world applications. In particular, optimization helps improve the overall performance of the electricity system that is of critical importance to our society. My research contributes to the development of smart grids which combine a traditional electrical power system with a two-way flow of information and energy between suppliers and consumers. This combination can deliver higher integration of wind and solar power, energy savings, and increased reliability and security.

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.

### FRS, FRSE

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

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.

I am interested in the development and application of techniques in mathematical statistics and pure mathematics aimed at answering causal questions in population biomedicine and drug discovery. Examples of these are Targeted Learning and Topological Data Analysis. Applications are often best pursued in a cross-disciplinary setting, so I collaborate with colleagues from the MRC Human Genetics Unit, the Edinburgh Cancer Research Centre, and the School of Informatics.

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

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.

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.

We develop state-of-the-art mathematical and computational models (e.g. molecular dynamics, hybrid/multiscale analysis and machine learning tools) and apply them to investigate open engineering problems and challenges. Including: de-icing/anti-icing aerospace surfaces, rarefied gas flows in ultra tight porous media, interfacial flows and instabilities (nano bubbles, nano droplets, nano films), water filtration through nanostructured membranes, slippery surfaces, ultra-cooling membranes, low-drag marine surfaces.

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

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.

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.

I am interested in applied analysis and numerical analysis with applications in material science. In particular, I study various aspects of crystalline materials including nonlinear elasticity (static or dynamic), crystal defects, and lower dimensional objects (e.g. plate theories).

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.

My work is on singular stochastic partial differential equations (SPDEs) via the theory of Regularity Structures. I am using combinatorial objects called decorated trees for constructing solutions to these equations. Similar structures appear in low regularity numerical schemes for partial differential equations which is another important direction of my research.

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

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

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.

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.

My research area is in Actuarial Science and Financial Mathematics, with broad research directions, including risk sharing, forward preferences and control strategies, life and retirement products, risk aggregation and resources allocation, as well as cyber risk management. In general, I am interested in solving any timely and revolutionary decision-making problems on topics in Actuarial Science and Financial Mathematics, via optimization, stochastic control, machine learning, and data analytics.

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).

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.

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.

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.

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

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.

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

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

I work on the fields of computational statistics, statistical signal processing, data science, and machine learning. I am interested in developing statistical modeling and inference/estimation/forecasting techniques that incorporate uncertainty quantification. My work is on state-space modeling, time-series analysis, Bayesian inference, and approximate inference methods (e.g., Monte Carlo) with different applications including Earth observation, ecology, biomedicine, wireless communications, and sensor networks, to name a few.

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

Acquatic ecosystems

My area of research is logistics and combinatorial optimization, applications where some optimal decisions must be taken from among a finite (but very large) set of potential decisions which cannot be computed in a reasonable amount of time. Examples of these applications that I am working now or I have worked in the past are: warehouse location problems, vehicle routing problems, aircraft cockpit architecture, and matching problems (junior doctor allocation, kidney exchange).

### FRSE

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

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.

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

### 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.

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.

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.

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

### 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.

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.

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.

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

### FRSE

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

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

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

The focus of my research is the data-driven analysis of complex dynamical systems. I am in particular interested in the numerical approximation of transfer operators using only simulation or measurement data. Typical applications include model reduction, system identification, and control (e.g. in areas such as molecular dynamics, fluid dynamics, or quantum mechanics).

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.

Soil-structure interaction Elastic / acoustic wave modelling.

I study methods that can be used to blend mathematical models with observational data: Bayesian and deterministic inverse problems, hierarchical random fields, the theoretical foundations of stochastic optimisation and machine learning, data assimilation, efficient computational methods in data science and uncertainty quantification, and related problems. Areas of application are in medical imaging and engineering.

### 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.

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.

Turbulence and large-scale structure formation in electrically conducting flows, parallel shear flows, boundary layers and thin fluid layers. Low-dimensional modelling of multi-scale non-linear dynamical systems. Applications of functional analysis in fluid dynamics.

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

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.

I study stability properties of stochastic differential equations (SDEs) and their discrete-time counterparts, with a particular emphasis on their convergence to equilibrium. My research on SDEs is directly connected to two areas of applications: 1) Markov Chain Monte Carlo methods that are used for approximate sampling from high-dimensional probability measures in computational statistics and 2) mathematical foundations of machine learning, in particular the theory of stochastic gradient algorithms that are commonly used as tools for training neural networks.

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

Clean Energy Technology.

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.

Combining machine learning and biophysical modelling techniques to understand protein function and and their regulation with a particular interest in antimicrobials.

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).

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

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.

Hydrodynamics of granular materials and particle-laden flow.

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

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.

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.

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

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

The application of ideas and techniques from dynamical systems and machine learning to problems in fluid mechanics. Flow instabilities, transition and turbulence.

My research interests are in using Monte Carlo and optimization methods for statistical problems motivated by applications. I have started my research career in applied probability, which was useful in analysing Markov Chain Monte Carlo and Sequential Monte Carlo methods, as well as the efficiency of optimization methods for stochastic functions. One area of application of such methods Is in the field of inverse problems, in particular chaotic dynamical systems that can be modelled by PDEs and ODEs. In recent years, there have been spectacular advances in optimization methods that scale to much larger problems than previously possible, by clever use of sparsity. I am keen to use them to attack challenging statistical problems.

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.

### FRSE

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.

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.

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.

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.

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.

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.

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.

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

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).

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.

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

### 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.

I am interested in stability and performance analysis of stochastic networks, especially those appearing in queueing theory, communication networks, data centres and energy applications. My work on performance analysis includes limit theorems as well as large deviations. I also work on random graphs, and random processes on them.

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

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.

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,.

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.

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.

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.

My research interests lie on the interface between applied statistics, computational statistics, actuarial science and actuarial data science. In particular, my research primarily, but not exclusively, focuses on statistical learning applications, via estimation methods, such as the Expectation-Maximization (EM) algorithm, and computational aspects of deep learning in actuarial work, including insurance ratemaking, setting appropriate levels of reserves and reinsurance. This field of research has been recently termed Actuarial Learning. Finally, part of my research focuses on classification of green bonds based on machine learning methods.

### 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.

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

I research efficient and reliable probabilistic machine learning “in the wild”, that is when a model has to perform inference and learning in one environment that might not correspond to the one it has been trained on, and for which reasoning over calibrated uncertainties is of primary importance. I also focus on combining classical statistical machine learning with complex reasoning, which is fundamental to enable trustworthy AI. I combine deep learning models with symbolic approaches to make them less data hungry, better at systematic generalization and ensuring their behaviour conforms to our expectations.

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.

I am an assistant professor at the department of actuarial mathematics and statistics, Heriot Watt University. I am interested in stochastic control, optimal stopping, deep learning, and their applications on behavioral economics, finance and insurance, time inconsistent decision making and credit risk.