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2027 Honours Projects in Applied Mathematics

This info below contains descriptions of thesis projects offered for Honours year students in Applied Mathematics. Please note that the list is not exhaustive, and feel free to contact supervisors for other projects in their field.

Honours candidates are strongly encouraged to contact their preferred supervisor as early as possible to discuss potential projects and to make sure they have any requisite background knowledge. More information about the Honours year is available by emailing the Applied Mathematics Honours Coordinator, or via our Honours Year webpage.

Mathematical Modelling

Christopher Angstmann

Modelling with fractional differential equations

Fractional derivatives are a type of nonlocal operator that generalise the concept of a derivative away from integer order. Whilst they are rather esoteric in their initial definition, they have had an increasing place in modelling a wide variety of physical phenomena. By exploring a connection between random walks and fractional derivatives it is possible to derive a wide range of models. This project will develop fractional order PDE models with applications to biomathematics.��

Semi-Markov compartment models

Compartment models are a widely used class of models that are useful when considering the flow of objects or people or energy between different labelled states, referred to as compartments. Recently we have constructed a general framework for fractional order compartment models, where the governing equations involve fractional order derivatives, via the consideration of a semi-Markov stochastic process. This project will explore the generalisation of the framework to incorporate a wider range of nonlocal operators.

Adelle Coster

��Nutrient transport in crowded cellular environments

Gram-negative bacteria have two cell membranes – the outer membrane largely contains passive transporter channels and the inner�� �� membrane active transporters. The periplasmic space separating these membranes is small in volume and is a very crowded environment. Investigations of the flow of nutrients through these membranes in and out of the cell can be investigated using a variety of dynamical systems and computational techniques.

Starch Grain Analysis

Starch grains grown in the interstitial spaces in plants often have characteristic shapes dependent on the species. It is possible to characterise and classify the origin of some populations of grains based on their shapes. Here extensions to common characterisation methods are planned to define further shape measures which might be used to separate the different species – beyond the normal characteristics of size, area, and perimeter. For example, how triangular must a shape be, before it should be classified as a triangle? How can we define these shapes? Can we put some bounds on the uncertainties of our classifications based on the different definitions for the shapes?�� This investigation is supported by data from a large database of microscope images of starch grains of known origin.

Gary Froyland

Operator-theoretic and differential-geometric kernel methods for Machine Learning�� �� �� ����

This project will develop new mathematical and computational approaches to analyse high-dimensional data. Operator-theoretic methods will be explored, including the use of transfer operators, dynamic Laplace operators, and Laplace-Beltrami operators, which extract dominant dynamic and geometric modes from the data. In the theoretical direction, this project will tackle the mathematisation of aspects of machine learning. In a combined theoretical and numerical direction, this project will investigate the construction of these operators from high-dimensional data using dynamic and geometric kernel methods. A possible application is to analysing global scalar fields obtained from satellite imagery such as sea-surface temperature to extract climate oscillations such as the El Nino Southern Oscillation and the Madden-Julian Oscillation. This project will use ideas from dynamical systems, functional analysis, and Riemannian geometry.

Bill McLean

Adaptive time stepping for diffusion and sub-diffusion equations

A spatial discretisation of a parabolic PDE leads to a large, sparse and stiff system of ODEs, which is typically integrated via an implicit time stepping scheme.�� The initial goal of this project is to design automatic step-size control algorithms for such schemes, guaranteeing a desired accuracy tolerance.�� A further aim is to extend these algorithms to evolution equations modelling sub-diffusion.�� Such equations are non-local in time, that is, they are integro-differential equations, typically formulated in terms of fractional-order time derivatives.

Eva Stadler (Kirby Institute 91ɫ��Ƭ)

Mathematical & Data-Driven Approaches to Infectious Disease and Immunity

Apply��mathematical modelling��and��data analytics��to tackle key questions in infectious disease and immunology. Projects explore how immunity develops, how we measure it, and how interventions like vaccines provide protection.��

You’ll work on real-world problems using methods such as��compartmental models,��stochastic simulations, and��statistical inference, applied to diseases like��malaria,��mpox,��SARS-CoV-2, or Ebola.

��This project is only available from T3 2027.

This project will be co-supervised by��Eva Stadler (Infection Analytics Program, Kirby Institute, 91ɫ��Ƭ Medicine & Health) and a co-supervisor from Mathematics & Statistics.

Michael Watson

Modelling cell and lipid dynamics in atherosclerotic plaques�� ��

Atherosclerotic plaques are fatty, cellular lesions that form in artery walls and can lead to heart attack or stroke. Plaques are initiated by blood-borne lipid particles (“bad cholesterol”) that become trapped in the artery wall and trigger an immune reaction. Subsequent plaque progression involves a complex interplay between these lipids and the cells that are recruited to the lesion site. Mathematical modelling of plaque cell and lipid dynamics can provide novel insights into the mechanisms of plaque progression that are unobservable in experimental or clinical settings.

Projects in this area will involve the development, analysis, and numerical solution of new mathematical models to provide mechanistic insight into the dynamics of plaque progression. Potential modelling strategies include:

1) continuously structured cell population models (e.g., lipid-structured integro-PDE models of phenotypic switching in plaque smooth muscle cells);

2) spatial differential equation models (e.g., compartment, reaction-diffusion, or multiphase��PDE models to study the emergence of plaque spatial structure);

3) hybrid stochastic-deterministic models (e.g., individual cell-based models of early plaque formation).

Mechanics & hydraulics of atherosclerotic plaque growth

Atherosclerotic plaques form in the thin��intimal layer of the artery wall. Under pressure from the arterial blood flow, plaques that are seeded locally will grow��radially inward and circumferentially around the artery before proceeding to grow��longitudinally along��the artery. Using tools from dynamical systems, ODEs, and classical geometry, this project will develop a minimal model of the biomechanical processes involved in plaque growth. Simulated plaque geometries will be compared to experimental data from laboratory mouse studies with the aim of providing mechanistic insight to the growth and spread of plaques. This project will be joint supervised with Sami Al-Izzi.

Fluid Dynamics, Oceanography and Meteorology

Gary Froyland

Lagrangian Coherent Structures in Ocean, Atmosphere, and Climate Models

The ocean, atmosphere, and climate system display complex nonlinear behaviour, whose underlying evolution rules change over time due to external and internal influences. Mixing processes of in the atmosphere and the ocean are also complex, but carry important geometric transport information. Using the latest models or observational data, and methods from dynamical systems, and spectral theory, this project will identify and track over time those geometric structures that mix least. Known examples of such structures are eddies and gyres in the ocean, and vortices in the atmosphere, however, there are likely many undiscovered coherent pathways in these geophysical flows. Moreover, higher-dimensional systems arising from climate models could also be tackled. There is also the possibility for the project to further develop mathematical theory and/or algorithms to treat one or more specific challenges arising in these application areas.����

Amandine Schaeffer

����������������General physical oceanography, please contact if interested.

Mathematics Education - No projects available in 2027

Dynamical Systems

Upanshu Sharma

Learning reduced variables in a Markov chain

Large stochastic systems may contain thousands of states, even though their important behaviour can often be described using only a few groups of states. This project asks the question: given the transition rates of a Markov chain, how can one automatically group its states so that the resulting smaller chain retains the important dynamics? The student will develop and compare clustering criteria based on rapid mixing within groups, spectral methods and possibly machine-learning optimisation. This project involves probability, linear algebra, optimisation and numerical computation.

Measuring mixing away from equilibrium.

Poincaré and log-Sobolev inequalities are classical mathematical tools for measuring how rapidly a stochastic system approaches equilibrium. Much less is understood when the probability distribution used as a reference is not itself an equilibrium distribution. This project will investigate such inequalities for simple stochastic differential equations, beginning with Ornstein–Uhlenbeck processes and Gaussian measures, where many questions may be approached using explicit matrix calculations. This project involves stochastic differential equations, Gaussian probability, linear algebra and functional analysis.

Can reduced stochastic models remain accurate for long times?

Complex stochastic systems are often replaced by simpler, lower-dimensional models which are easier to analyse and simulate. A reduced model may accurately reproduce the original system over short time intervals, or may have the correct equilibrium distribution, but neither property guarantees that it remains accurate over long times. In particular, it may fail to reproduce temporal correlations, relaxation rates and other quantities that depend on how the system remembers its past. This project will investigate when a reduced stochastic model captures both the long-time behaviour and the time correlations of the original system. The student will begin with linear stochastic differential equations, where many quantities can be calculated explicitly, and will look for criteria guaranteeing long-time accuracy as well as examples where apparently reasonable reductions fail. The project may combine rigorous analysis with numerical experiments. This project involves stochastic differential equations, probability, linear algebra and numerical simulation. It is motivated by model reduction problems in molecular dynamics and other multiscale stochastic systems.

Wolfgang Schief

Topics in Soliton Theory

Solitons constitute essentially localised nonlinear waves with remarkable novel interaction properties. The soliton represents one of the most intriguing of phenomena in modern physics and occurs in such diverse areas as nonlinear optics and relativity theory, plasma and solid state physics, as well as hydrodynamics. It has proven to have important technological applications in optical fibre communication systems and Josephson junction superconducting devices.��

Nonlinear equations which describe solitonic phenomena (`soliton equations’ or `integrable system’) are ubiquitous and of great mathematical interest. They are privileged in that they are amenable to a variety of solution generation techniques. Thus, in particular, they generically admit invariance under symmetry transformations known as Bäcklund transformations and have associated nonlinear superposition principles (permutability theorems) whereby analytic expressions descriptive of multi-soliton interaction may be constructed. Integrable systems appear in a variety of guises such as ordinary and partial differential equations, difference and differential-difference equations, cellular automata and convergence acceleration algorithms.��

��There exist deep and far-reaching connections between integrable systems and classical differential geometry. The geometric study of integrable systems has proven to be very profitable to both soliton theory and differential geometry. Moreover, integrable systems play an important role in discrete differential geometry. The term ‘discrete differential geometry’ reflects the interaction of differential geometry (of curves, surfaces or, in general, manifolds) and discrete geometry (of, for instance, polytopes and simplicial complexes). This relatively new and active research area is located between pure and applied mathematics and is concerned with a variety of problems in such disciplines as mathematics, physics, computer science and even architectural modelling. Specifically, theoretical and applied areas such as differential, discrete and algebraic geometry, variational calculus, approximation theory, computational geometry, computer graphics, geometric processing and the theory of elasticity should be mentioned.��

Soliton theory constitutes a rich source of Honours topics which range from applied to pure. Specific topics will be tailored towards the preferences, skills and knowledge of any individual student.

Gary Froyland

Topics in dynamical systems, ergodic theory, or differential geometry

Ergodic theory is the study of the dynamics of ensembles of points, in contrast to topological dynamics, which focusses on the dynamics of single points. A number of theoretical Honours projects are available in dynamical systems, ergodic theory, and/or differential geometry, aiming at developing new mathematics to analyse the complex behaviour of nonlinear dynamical systems. Depending on your background, these projects may involve mathematics from Ergodic Theory, Functional Analysis, Measure Theory, Riemannian�� Geometry, Stochastic Processes, and Nonlinear and Random Dynamical Systems.

Differential and spectral geometry with applications to fluid mixing

Techniques from differential geometry and spectral geometry (via Laplace-type operators) have recently been shown to be particularly effective for analysing complex dynamics in a variety of theoretical and physical systems. This project will focus on developing and extending powerful techniques to extract important geometric and probabilistic dynamical structures from fluid-like models. If desired, application areas include the ocean (an incompressible fluid) and the atmosphere (a compressible fluid). This project will involve dynamical systems and differential/spectral geometry.

Stability of linear operator cocycles

Classical perturbation theory yields continuity of the spectrum and eigenprojections of compact and quasi-compact linear operators. The situation is dramatically different when one creates a cocycle of different operators, driven by some ergodic process. This dramatic�� difference even occurs in finite-dimensions (cocycles of matrices). This project will discover theory for which one can expect continuity of the corresponding spectral objects, namely Lyapunov exponents and Oseledets spaces. The project will use mathematics from probability and statistics, functional analysis, and connects to dynamical systems and ergodic theory.

Machine-learning dynamical systems

This project explores the use of machine learning in either (i) prediction of dynamical systems or (ii) in the construction of efficient linear operator representations of the dynamics. In the latter case, the project will focus on those linear operators that are generated by the�� dynamical system and which allow a spectral analysis of the dynamics.

Transfer operator computations in high dimensions

Many real-world dynamical systems operate in phase spaces that are very high dimensional and/or unknown. For example, the dynamics of ocean-atmosphere circulation at various spatial and temporal scales (e.g. from local weather to global climate) is invariable extremely high dimensional. On the other hand, there is increasing availability of spatial datasets from e.g. satellite imagery, which provide high resolution spatial images as “movies” in time. One can hope to construct dynamics of a projected system from the dynamics of these images, which are themselves operating in a high-dimensional space (dimension >= number of pixels in the image).�� This project will investigate recent ideas in constructing transfer operator for high-dimensional systems, and use ideas from dynamical�� systems, stochastic processes, functional analysis, and Riemannian geometry.

Lagrangian coherent structures in haemodynamics

Haemodynamics (the dynamics of blood flow) is believed to be a crucial factor in aneurysm formation, evolution, and eventual rupture. Turbulent motion near the artery wall can weaken already damaged arteries, as can oscillations between turbulent and laminar flow.�� �� Simulations of 3D blood flow is either derived by (i) computational fluid dynamics (CFD) from patient-specific mathematical models obtained from angiographic images or (ii) laser scanning of real flow through a patient- specific physical plastic/gel cast. In this project,�� joint with Prof. Tracie Barber (91ɫ��Ƭ Mech. and Manufact. Engineering), you will develop and apply new mathematical techniques for�� low analysis, based on dynamical systems and spectral methods to separate and track regions of turbulent and regular blood flow. Prof. Barber will provide the realistic flow data from her laboratory, from both CFD simulations and physical casts. There is also the opportunity to further develop mathematical theory to solve problems specific to haemodynamics.

Chunxi Jiao

Bubbling in the presence of noise

Singularities can arise in physical systems such as ferromagnetics and liquid crystals. The corresponding blow-up phenomenon of these models follows from that of harmonic map heat flow. In dimension two, the blow-up is known as bubbling, with backward bubbling closely related to non-uniqueness of weak solutions. At microscopic scales, systems are susceptible to thermal noise which is often modelled by space-time stochastic processes and can contribute to the total energy. This project aims to explore how random noise affects the bubbling, using techniques from PDE and stochastic analysis.�� ��

Sami Al-Izzi

The dynamics of robotic metamaterials

From venus flytraps to cilia, nature uses the flexing and snapping of slender structures for fast motion and cyclic shape change. Many of these systems are internally energized by active processes, and are a workhorse for propulsion, locomotion and actuation across biology and bioinspired engineering. This activity, coupled with mechanical nonlinearities, underpins a much richer space of possible dynamics than the corresponding energy conserving systems. Increasingly there has been the aim to use these minimal active mechanical units, often energised by a nonreciprocal interaction, as the basis for developing decentralised information-processing materials or robotic metamaterials. There is however, a lack of well-developed design principles founded in a detailed mathematical analysis of minimal model systems.

This project will involve studying simple models of these active materials, that correspond to a minimal robot metamaterial, in the form of active rigid linkages. There are several possible angles from which to attack the problem but, broadly, they utilise tools from dynamical systems theory and computational methods to understand the behaviour of such systems (with the potential to learn tools from asymptotic methods and calculus of variations). A potential goal of the project could be to make predictions and develop design principles that could be tested by collaborators in robotics engineering (Dr. Jack Binysh - University of Birmingham, Prof. Corentin Coulais - University of Amsterdam).

There is also the potential for projects with more of an applied differential geometry/PDEs flavour on active surfaces and "odd elasticity". Contact me if this interests you.�� �� �� ����

Optimization

Jeya Jeyakumar

Robust Optimization Methods for Managing risk and quantifying uncertainty in complex decision-making

Project abstract 1 Robust optimization is a relatively new distribution-free method for solving real-world decision-making problems in the face of data uncertainty. Developing mathematical theories and methods for quantifying uncertainty and obtaining solutions with risk prevention guarantees remains a major mathematical challenge. This project aims to study and develop optimization methods to address this challenge.

Stochastic Optimization for data-driven decision-making under uncertain conditions

Project abstract 1: Stochastic optimization, which assumes uncertainty has a probability distribution, is one of the well-established methods for modeling practical decision-making problems in the face of uncertain conditions. Despite widespread use, it has significant limitations for applications to large-scale decision-making problems. This project aims to study a hybrid solution approach by combining the strengths of both distribution-free and stochastic methods with applications.

REQUIREMENTS: Due to the technical nature of these projects, interested students should have completed both MATH3161 and MATH3171 at the HD levels, and should have done mathematical computing courses.

Mareike Dressler

Topics in polynomial optimisation

A polynomial optimisation problem is a special class of nonconvex nonlinear global optimisation in which both the objective and constraints are polynomials. That is, it aims at finding the global minimiser(s) of a multivariate polynomial on a certain set. Polynomial optimisation has a wide range of applications like dynamical systems, robotics, computer vision, signal processing, and economics. Mathematically, it is well-known that solving polynomial optimisation is very hard in general. One of the most powerful approaches for�� handling such problems with rigorous and global guarantees is via algebraic techniques.

A potential honours project would be on the theoretical development and/or practical application of these algebraic techniques.

Gary Froyland

Machine-learning optimal function bases for linear operators

This project will explore the use of machine learning to find optimal basis functions to represent discrete approximations of linear operators.

Computational Mathematics

Alexander Gilbert

Quasi-Monte Carlo methods in uncertainty quantification

Many systems in the physical sciences and engineering involve underlying uncertainty that must be accounted for accurately in order to make��informed decisions. Typically, such systems are modelled by a partial differential equation (PDE), where the uncertainty is present in the model inputs, such as initial conditions, coefficients or��domain. The overall uncertainty in the system can then be quantified by computing statistics of the PDE model output with respect to the (often high-dimensional) uncertain input parameters. Quasi-Monte Carlo methods are a class of quadrature and approximation methods that are designed to compute difficult high-dimensional problems. Recently, they have demonstrated great success in tackling a range of problems in uncertainty quantification. Potential projects would study applying quasi-Monte Carlo methods in uncertainty quantification or computational finance, with scope to focus on a mix of theory and computation.

Frances Kuo

Lifting the curse of dimensionality the "Quasi Monte Carlo" way

High-dimensional computation involving many (or even infinitely many) variables is a frontier in scientific computing, with applications ranging from financial mathematics and groundwater flow to wave propagation, ice-sheet simulation, and precision oncology. In many cases, the dimensionality arises from uncertainty or randomness in the data. High-dimensional problems pose immense challenges because computational costs tend to grow exponentially with the number of dimensions. This project explores how “Quasi Monte Carlo” methods can help to lift this curse of dimensionality and advance the development of predictive digital twins.��Possible research directions include, e.g., using QMC points for training in deep neural networks and Fourier neural operators,��multi-fidelity QMC methods, and QMC for cross validation.

Quoc Thong Le Gia

Physics-Informed Neural Networks for Darcy Flow with Random Permeability

This project investigates the use of Physics-Informed Neural Networks (PINNs) for solving Darcy flow problems with uncertain or random permeability. Darcy's equation is widely used to model fluid flow through porous media, where the permeability of the medium is often heterogeneous and subject to uncertainty. In this project, the permeability will be represented by a random field, for example using a truncated Karhunen--Loève expansion, and PINNs will be developed to approximate the corresponding pressure and velocity fields. The project will begin with deterministic benchmark problems and then extend the methodology to random permeability, with particular attention to estimating statistical quantities such as the mean and variance of the solution. Numerical results will be compared with conventional numerical methods, providing students with experience in partial differential equations, scientific machine learning, numerical computation, and uncertainty quantification.

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