Supervisor: Yue Wu

Mesoscale eddies (rotating current systems 10–100 km in diameter) are critical drivers of ocean circulation, transporting heat, nutrients, and pollutants across ocean basins. Identifying when ocean Lagrangian drifters (free-floating sensors) are captured by eddies is essential for understanding eddy transport efficiency - but traditional methods, such as velocity thresholding and geometric curvature analysis struggle with noisy, high-dimensional drifter trajectory data. 

This project leverages path signature features (from rough path theory) to encode the geometric and temporal patterns of drifter trajectories, enabling accurate classification of "eddy-captured" vs. "non-eddy" trajectories. You will work with real-world drifter data from the Northwest Pacific Ocean and compare the performance of signature-based models to classical baselines. The project combines applied mathematics (path signatures), machine learning, and ocean science, with clear data collection, preprocessing, and validation workflows.

Supervisor: Dr Yue Wu

High-dimensional partial differential equations (PDEs) are ubiquitous in ocean science (such as ocean circulation models, climate prediction, wave dynamics) but are notoriously intractable with classical numerical methods (such as finite elements, finite differences) due to the “curse of dimensionality.” Recent breakthroughs in Deep Learning (DL) - including neural networks (NNs), graph neural networks (GNNs), probability representation, and physics-informed neural networks (PINNs) - have enabled scalable solutions to high-dimensional PDEs. This project will allow you to conduct a structured critical review of key papers in this field, or choose one paper to brainstorm, connect DL-PDE advances to ocean science applications, and propose potential extensions for ocean-related PDE problems.

Prerequisites

• Basic knowledge of PDEs
• linear algebra, and calculus
• Familiarity with machine learning fundamentals (neural networks, loss functions)
• Basic programming skills (Python: NumPy, TensorFlow/PyTorch) optional but recommended

Supervisor: Professor David Greenhalgh

It is tempting to think that the distribution of first digits in a dataset will follow a uniform distribution but this is not necessarily the case. For many real datasets these digits follow a different distribution called Benford’s Law. The project will study Benford’s Law and its applications.

During this project you will:

• research the background literature on Benford’s Law, the underlying theory and its applications
• look for datasets for which Benford’s Law does and does not apply
• investigate the theory and application of statistical tests to determine when Benford’s Law applies and when it does not apply, including, but not exclusively, the chi-squared test, the Kolmogorov-Smirnov test, the Mean Absolute Deviation and the Cho Gaines Statistic
• investigate the Benford Analysis package in R
• survey real life applications of Benford’s Law in practice, for example its uses in fraud detection in accounting.

Some familiarity with the idea of hypothesis testing and the computer package R would be helpful.

Supervisor: Professor David Greenhalgh

This project will explore infectious disease models for superinfection in vector borne diseases. Vector borne diseases such as dengue and chikungunya can be modelled using differential equations. This project will explore infectious disease models for superinfection exclusion in vector-borne diseases. The SEI (susceptible-exposed-infectious) differential equation model will be used to model the two vector borne diseases. One of these diseases is a purely vector borne disease and the other is a mosquito borne disease which is pathogenic to humans. Examples of such purely mosquito viruses are CFAV (cell-fusing agent virus), Negev virus and Eliat virus and the pathogenic human viruses are Sindbis virus, Zika virus, Chikungunya and dengue. There are plenty of existing models for vector-borne diseases in the literature and our model would be based on those.

Equations for two competing viruses within the mosquito population will be set up. The rate at which mosquitoes are infected will be different for a completely susceptible mosquito infected by virus 1, a completely susceptible mosquito infected by virus 2, a mosquito infected by virus 1 only being infected by virus 2 and a mosquito infected by virus 2 only being infected by virus 1.

There are already mathematical models for superinfection exclusion, for example using a multi-strain SIR model as a basis and extending this to a host-vector-model using an SIR model for the host and an SI model for the vector. There is also separate work using a host-vector model to assess the impact of superinfection exclusion on vaccination using dengue and Yellow Fever as an example.

When the model has been formulated mathematical techniques such as equilibrium and stability analyses will be used to predict the expected long-term behaviour of the model. Then the package MATLAB or similar package will be used to simulate the model. "

Supervisor: Professor David Greenhalgh

The project will look at the formulation of epidemic models and the concept of the basic reproduction number, R0. It will look at the calculation of the basic reproduction number both by direct calculation and by the next generation matrix method. The project will look at equilibrium analyses of these models and local, and if appropriate, global stability analysis of the disease-free equilibrium. Then the model will look at local stability analysis of the endemic equilibrium. You will write a MATLAB program to integrate the differential equations using realistic parameter values.

Supervisor: Professor Adam Kleczkowski

Coffee is an important produce in Thailand, yet it is under threat from climate change, pests and diseases. With the increase in temperature, the production might not be sustainable or will require structural changes. In this project, we will use data analytics, including regression and Machine Learning, to carry out a spatial analysis of climate suitability for coffee production in Thailand. The student will analyse the weather data, both the past records and future projections, and predict possible shifts in production areas. Coffee is also threatened by pests and diseases, most prominently, a coffee borer. The project will also incorporate climate suitability maps for the insect occurrence and estimate the risk of infestation.

Please note that this is an Advanced Data Science project.

Supervisor: Professor Adam Kleczkowski

Ash dieback, Hymenoscyphus fraxineus, is a fungal disease that has been inflicting devastating impacts on the UK landscapes and biodiversity since its first detection in 2012. As there is no promising treatment or prevention measure, the best hope for the long-term future of the UK's ash trees lies in identifying tolerant or resistant trees for breeding new generations. This project combines unique field data sets with modelling approaches. In this project, the student will first analyse data for a selection of provenances and blocks, identifying the disease progress in each individual tree. The next step will involve constructing and analysing a Markov chain model, to identify the natural variability between trees.

This is an Advanced Mathematical/Computational Modelling and Data Science project.

Supervisor: Professor Adam Kleczkowski

Large parts of agriculture in Asia and Africa are threatened by existing as well as emerging Pests and Diseases, a trend further exacerbated by climate change. Chemical control is widely used for P&Ds, but the rise of microbial resistance and health and environmental concerns mean that finding resistant cultivars becomes a priority. This project will use a combination of data analysis and modelling to assist in non-invasive detection of viruses in plants at early stages of infection. The modelling will be applied to Sterility Mosaic Disease (SMD), an economically important pathogen of legumes (pigeonpea), in collaboration with partners in India. We will use a combination of statistical analysis and modelling (including a hierarchical Bayesian approach) to capture within- and between-cultivar variability in the response to SMD.

This is an Advanced Data Science and Mathematical/Computational Modelling project.

Supervisor: Dr Jack Laverick

This is a family of projects using the StrathE2E end-to-end ecosystem model. You will work with one model implementation, either:

• the Norwegian Sea
• the Celtic Sea
• The Azores
• Senegal
• Ascension Island
• Saint Helena
• the Southern Brazilian Shelf
• the Southern Benguela Current

You will explore a simulation set from their chosen region produced from a Sobol sensitivity analysis of fishing pressure applied to ten simultaneously harvested functional groups. This set of model output allows for the calculation of interactions, which produced strategic trade-offs for managers. You will:

• visualise the interaction surfaces of multiple fisheries acting on specific functional groups
• screen management strategies for desirable outcomes in terms of ecosystem health and harvest
• describe the constraints and trade-offs between different fisheries when selecting strategies which achieve target outcomes
• explore how these strategies and trade-offs evolve under climate change from the 2010s to the 2060s

Supervisor: Dr Debasish Das

Moseley et al. (2020) show that physics-informed neural networks (PINNs) can be used to solve the acoustic wave equation by training a neural network whose output is constrained to satisfy the governing partial differential equation together with initial and boundary conditions. Their work demonstrates that a network trained with only limited early-time data can accurately predict the full space–time wavefield and generalise across different source locations and heterogeneous media. Rasht-Behesht et al. (2022) extend this idea to geophysical wave propagation and full waveform inversion, showing how PINNs can model forward wave motion and also recover unknown wavespeed structures from sparse observations while enforcing the underlying physics.  

Proposed project

The aim of this project is to learn the classical wave equation and implement a physics-informed neural network solver for 1D and 2D wave propagation, following the methodology of Moseley et al. (2020) and Rasht-Behesht et al. (2022). You will first derive the wave equation, specify appropriate initial and boundary conditions, and implement a reference finite-difference solver. You will then build a PINN that approximates the solution u(t,x) (and u(t,x,z) in 2D) by minimising a loss function consisting of a data misfit term and a wave-equation residual term. The final stage of the project will benchmark the PINN against the finite-difference solution and investigate how accuracy and training stability depend on wave frequency, number of training points, and boundary-condition enforcement. As an extension, you may attempt a simple inverse problem by recovering a spatially varying wavespeed in 1D from sparse sensor measurements, following the inversion framework of Rasht-Behesht et al. (2022).

References

Moseley, B., Markham, A., & Nissen-Meyer, T. (2020). Solving the wave equation with physics-informed neural networks. arXiv:2006.11894.
Rasht-Behesht, M., et al. (2022). Physics-Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversion. Journal of Geophysical Research: Solid Earth."

Supervisor: Dr Debasish Das

Drazin (1977) presents a classical stability analysis of a finite-amplitude internal gravity wave in a uniformly stratified (Boussinesq) fluid, showing that small perturbations can undergo parametric growth and deriving the corresponding instability conditions and growth-rate predictions. Staquet and Sommeria (2002) review how such internal-wave instabilities fit into the broader geophysical picture, emphasising them as a key pathway from coherent wave motions to wave breaking, turbulence, and mixing in stratified environments. Sutherland’s textbook provides the foundational linear internal-wave theory - dispersion and polarisation relations, energetics, and standard modelling assumptions - that will be used to set notation and derive the governing equations needed to reproduce Drazin’s results.

Proposed project

The aim of this project is to reproduce and validate the main instability results in Drazin (1977), using modern numerical tools while keeping the mathematical structure of the original analysis explicit. The student will first work through the derivation of the internal-wave dispersion relation and the form of the primary monochromatic wave in a uniformly stratified Boussinesq fluid, using Sutherland as the main background source. You will then implement the perturbation problem studied by Drazin-linearised equations for disturbances evolving on the time-periodic wave field - and compute growth rates as functions of the relevant nondimensional parameters (such as wave amplitude and frequency ratios) using an eigenvalue/Floquet approach or an initial-value simulation with exponential fitting. Finally, you will reproduce one or more of Drazin’s stability diagrams (stable/unstable bands and representative growth rates), and interpret the physical mechanisms and geophysical significance with guidance from Staquet and Sommeria (2002), including how these instabilities connect to wave–wave interactions and routes to mixing.

References

Drazin, P. G. (1977). On the instability of an internal gravity wave. Proceedings of the Royal Society of London. Series A, 356, 411–432.
Staquet, C., & Sommeria, J. (2002). Internal gravity waves: From instabilities to turbulence. Annual Review of Fluid Mechanics, 34, 559–593.
Sutherland, B. R. (2010). Internal Gravity Waves. Cambridge University Press.

Supervisor: Dr Debasish Das

Supervisor: Dr Debasish Das

Supervisor: Dr Laura Miller

Biological tissues often behave like porous materials where fluid moves through collagen networks, proteoglycan gels, or tissue microvascular. This project introduces you to the idea that the “effective” permeability of a tissue emerges from its microscale structure. You will build simple microscale geometries representing biological tissue (such as collagen networks, aligned fibres, random pores), simulate Stokes or Darcy–Brinkman flow through the microstructure, compute the homogenised permeability tensor and compare their results to published values for real tissues.

Supervisor: Dr Elizabeth Dombi

Game theory provides a mathematical framework to model and analyse situations in which two or more parties make competitive or collaborative strategic decisions. These situations are abundant in everyday life and therefore game theory became one of the most important areas of applied mathematics having applications in economics, politics, social sciences, psychology or biology to name but a few.

Games can be classified in several ways. For example, characteristics can include the number of players, whether participants have perfect or imperfect information, whether players have conflicting interests or if there is scope for negotiations. John von Neumann and Oskar von Morgenstern, the founding fathers of game theory, first considered the mathematical description and analysis of two-person zero-sum games in The Theory of Games and Ecomonic Behaviour. 

This  project could explore key concepts in game theory and the mathematical formulation of two-person zero-sum games,  von Neumann's Minimax theorem and explore the computation of optimal strategies

Supervisor: Dr Elizabeth Dombi

Cryptographic systems are divided into two main families: symmetric-key and asymmetric key (public-key) systems. Symmetric-key cryptography uses the same key for encryption and decryption and one of its main challenges is for the parties to exchange the secret key. Public-key cryptography uses two keys, an encryption key which is made public and a decryption key which is kept secret. RSA is the most widely used public-key cryptographic system that was developed in 1977 by Rivest, Shamir and Adleman. 

This project could investigate the number theoretical background needed to understand public-key algorithms, describe how RSA works and consider possible attacks against RSA

Supervisor: Dr Tunde Csoban

This project will investigate the challenges and solutions related to missing data in real life datasets, which often occur due to factors such as patient availability, equipment and staff constraints, or participants not feeling comfortable with disclosing sensitive information. You will explore some traditional and/or machine learning imputation techniques, apply these methods to a dataset, and assess how they impact correlation between variables, and simple regression models.  

Supervisor: Dr Laura Miller

Many engineering materials are made from fibres embedded in a matrix. When fluids flow through these materials (such as during manufacturing or cooling), the permeability becomes direction dependent. This project explores how fibre arrangement controls anisotropic flow. You will create microscale geometries containing aligned fibres, simulate flow through the microstructure, compute the anisotropic permeability tensor, compare results to classical analytical models. and explore how fibre volume fraction and orientation affect flow.

Supervisor: Dr Laura Miller

Bacterial biofilms are communities of microorganisms embedded in a self produced polymeric matrix. They behave mechanically like soft, heterogeneous porous materials, and fluid transport through them controls nutrient delivery, waste removal, and importantly, how antibiotics penetrate and act. This project introduces you to the idea that a biofilm’s effective permeability and transport properties emerge from its microscale structure. You will explore how the arrangement of bacteria and extracellular substances influences fluid flow at the tissue scale. You will build a microscale model of a biofilm, simulate fluid flow through the microstructure, compute homogenised transport properties, and apply this to real-world problems such as how permeability changes during biofilm growth, or how antibiotic penetration depends on effective diffusivity.

Supervisor: Dr Vladimir Krivtsov

This summer project will investigate how pollutants move through an urban pond catchment, using Blackford Pond in Edinburgh to explore how blue green infrastructure will simultaneously reduce respiratory health risks and affect water quality. It will use existing monitoring data and published literature to quantify pollutant inputs from surface runoff and airborne deposition, examine processes such as leaf litter decomposition and pollutant wash off from vegetation, and contribute to the development of a conceptual simulation model of nutrient and pollutant cycling. You will benefit from interactions with an associated MSc project, as well as with local community groups and citizen scientists, and will contribute to the assessment of future ecosystem services and health benefits, and generate initial recommendations.

Relevant references

• Krivtsov, V. (2022) ‘Ecosystem services provided by urban ponds and green spaces: a case study of Blackford Pond, Edinburgh’, Blue-Green Systems, 4(1), pp. 1–24.
• Veerkamp, C.J. et al. (2021) ‘A review of studies assessing ecosystem services provided by urban green and blue infrastructure’, Ecosystem Services, 49, 101267.
• Pinto, L.V. et al. (2023) ‘Green and blue infrastructure and urban nature-based solutions: benefits for human and ecological well-being’, One Earth, 6(7), pp. 1–15.
  Perrelet, K. et al. (2024) ‘Engineering blue-green infrastructure for and with nature’, npj Urban Sustainability, 4, 18."

Supervisor: Professor Sergey Kitaev

In the Game of Life, invented by John Conway, a University of Cambridge mathematician, in 1967, "pieces" (specified cells in a regular grid) are placed in the starting position and certain rules determine everything that happens later. Thus, this is not a game in the conventional sense. Nevertheless, in most cases, it is impossible to look at a starting position (or pattern) and see what will happen in the future.
 
The goal of the project is to study behaviour of various patterns under natural variations of birth/survival rules. The main question here is whether infinite life exists under given conditions.  
 
No preliminary knowledge is assumed.

Supervisor: Professor Sergey Kitaev

A digital door lock code is normally four digits between 0 and 9 that must be pressed in the right order. However, the lock has bad memory and thus it does not matter if you start with pressing wrong digits: as soon as one presses the 4 digits in the right order, the door will be opened. If one forgot the code, how many presses must (s)he do in the worst case in order to open the door?

Answering the question above brings one to the theory of universal sequences, particular cases of which are de Bruijn sequences that found applications in genomics.

The goal of the project is not only in understanding all steps required to answer the question above via so-called de Bruijn graphs, but also in getting yourself familiar with some other examples of universal sequences.
 
No preliminary knowledge is assumed.

Supervisor: Professor Sergey Kitaev

A universal word for a finite alphabet A and some integer n>0 is a word over A such that every word of length n appears exactly once as a subword. The idea of universal partial words is to use non-deterministic symbols in order to shorten universal sequences and thus to make their storage less expensive while keeping the same amount of information. Universal partial words should eventually find applications in DNA sequence assembly, for example, in shortgun sequencing, where many gigabytes of storage are required for recording reads for a single genome. 
This project is about getting familiar with the theory of universal partial words.
 
No preliminary knowledge is assumed

Supervisor: Professor Sergey Kitaev

The study of patterns in permutations has a long history. For example, the study of descents or consecutive occurrences of the pattern 21 goes back to Leonhard Euler’s work in 1749. However, the origin of the modern day study of patterns in permutations can be traced back to papers by Rotem, Rogers, and Knuth in the 1970s and early 1980s. The first systematic study of permutation patterns was not undertaken until the paper by Simion and Schmidt, which appeared in 1985. Today the study of permutation patterns is a very active field with the book Patterns in Permutations and Words serving as a comprehensive overview over existing trends.
 
The goal of the project is to get you familiar with basic definitions and results in the theory of permutation patterns.
 
No preliminary knowledge is assumed.

Supervisor: Professor Sergey Kitaev

A partially ordered set (poset) is an interval order if it is isomorphic to some set of intervals on the real line ordered by left-to-right precedence. Interval orders are equivalent to so-called (2+2)-free posets. Interval orders are in 1-to-1 correspondences with several structures many of which are presented in the book On naturally labelled posets and permutations avoiding 12-34.

The goal of the project is to summarise our knowledge on a hierarchy of structures related to interval orders. 
 
No preliminary knowledge is assumed.

Supervisor: Dr Jiazhu Pan

Statistical techniques and concepts of time series analysis are applied to empirical analysis of financial markets. The software R is used as a vehicle for presenting practical implementations from financial modelling. The application includes calculation of value-at risk and expected shortfall.

Supervisor: Dr Alexander Wray

Ocean circulation is driven by a complex interplay between waves (such as Kelvin and Rossby waves), rotation, stratification and coherent vortices. This project introduces a mathematical model of these phenomena and uses it to understand how key length and time scales emerge in ocean contexts.

You will derive simple reduced models, compute and interpret dispersion relations, and use numerical solution techniques to visualise and interpret wave propagation and vortex evolution. Where possible, we will connect to existing real-world ocean datasets.

Supervisor: Dr. David Pritchard 

This project will numerically implement the model of erosion/deposition of polydisperse sediment in turbulent flow, originally developed by Dorrell et al. (J. Geophys. Res. Earth Surf., 118, 1939–1955, 2013; doi:10.1002/jgrf.20129). Following implementation, the student will carry out numerical experiments to explore the effect of the distribution of suspended sediment on the suspension capacity in a waning flow. In contrast to Dorrell et al. (2013), this project will consider non-normal grain size distributions. Students will require familiarity with numerical methods or packages for solving ODEs; some knowledge of fluid dynamics and/or environmental sediment transport would also be useful.