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Boole Fellowship 2026-2027 Project Proposals
Please see below the list of proposed projects and the staff members that will supervise them.
Supervisor |
Discipline |
Project Abstract |
|---|---|---|
| Dr Claus Koestler | Pure Mathematics |
A geometric approach to noncommutative independence
This project adopts a geometric perspective on noncommutative notions of stochastic independence by investigating the structure of commuting squares in operator algebras and their probabilistic interpretation. The research will begin with concrete examples of commuting squares and analyze how these configurations reflect independence phenomena in noncommutative probability spaces.
The project is well suited to a motivated student interested in the interface between functional analysis and modern probability theory.
|
| Prof Sebastian Wieczorek | Applied Mathematics |
Non-autonomous Instabilities in Natural and Human Systems
Non-autonomous dynamical systems are mathematical models (ODEs or PDEs) that describe the time evolution of real systems with time-varying external inputs. There is no established bifurcation theory for such dynamical systems, but there are interesting true non-autonomous instabilities that can be explored using ideas from geometric singular perturbation theory and optimal control theory. This project will analyse these instabilities in order to identify new nonlinear phenomena in mathematical models from one of the following areas of application:
(i) Tipping points in the climate system or ecosystems [1].
(ii) Cancer development and evolutionary adaptive treatment strategies [2].
[1] https://en.wikipedia.org/wiki/Tipping_points_in_the_climate_system
[2] https://royalsocietypublishing.org/doi/full/10.1098/rsif.2024.0844
Potential co-supervisors: Dr. Hassan Alkhayuon, Dr. Serhiy Yanchuk
|
| Dr Ben Taylor | Statistics or Actuarial Sciences | Stochastic Geometry, Point Processes and HPC Algorithms This project offers an intensive, self-directed research opportunity in Stochastic Geometry, the statistical and probabilistic field used to analyse and model random spatial patterns. This discipline has high-impact applications in diverse areas - modelling data at any scale from microscopic structures, through wireless communication networks, to the cosmic web. The project is designed to be student-led, allowing you to focus on either the rigorous theoretical foundations or practical aspects of computational modelling. Successfully engaging with the theoretical track requires significant mathematical maturity and independent study, specifically in the areas of probability, measure theory, topology, and multivariate calculus. Computationally, there is scope to investigate and develop high performance algorithms through programming in CUDA (on GPUs, TPUs or quantum computing via CUDA-Q). |
| Dr Andreas Amann | Applied Mathematics | AI Techniques for Mathematical Modelling Despite the impressive recent progress of AI, most high-dimensional modelling problems which appear for example in physics or engineering are still tackled using traditional methods, for example the classical finite element method. The recent development around "Physics-informed neural networks" (PINs) has given rise to the hope that AI modelling might become competitive in the near future, but as of now practical applicability is limited. This project therefore seeks to modify and improve current PIN methods with the aim of making them suitable for current modelling problems for example in the area of electrodynamics. |
| Dr Edward Gunning | Statistics or Actuarial Sciences | Functional Data Analysis of Spatiotemporal Biomechanics Data Biomechanics research in sport performance and injury rehabilitation typically analyses one-dimensional kinematic or kinetic variables such as knee angles or vertical ground reaction forces. These signals are recorded at high frequency and are naturally represented as smooth functions denoted by x(t), which makes functional data analysis (FDA; Ramsay and Silverman, 2005) an effective modelling framework. Recent developments in pressure-sensing technology now allow the measurement of full underfoot pressure maps across hundreds of spatial locations and through time. For walking, running, or sport-related movements, these data form spatiotemporal functional objects x(t, s_1, s_2), where s_1 and s_2 index spatial coordinates on the sensing array. Each observation is therefore a smooth function on a three-dimensional domain consisting of time and a two-dimensional spatial surface. In principle, standard one-dimensional FDA methods can be extended to these data, although the extremely high dimensionality of modern pressure-sensing systems, such as the StepUP-P150 dataset (see https://arxiv.org/pdf/2502.17244), creates significant computational and statistical challenges. Efficient representation and dimension reduction are essential if we wish to model entire time-varying pressure maps rather than rely on traditional summary measures. Project aims Develop and compare representation methods for spatiotemporal biomechanics data. The student will investigate tensor-product basis expansions, low-rank functional representations, principal component approaches, and deep learning encoders such as convolutional autoencoders. Composite approaches that combine linear and nonlinear elements will also be explored. Integrate these representations into statistical modelling workflows. Once an appropriate representation has been identified, we will embed it within established statistical methods, for example, using the derived features as predictors or responses in regression models, mixed effects models, or clustering procedures. Apply the workflow to a high-resolution open-source biomechanics dataset. The project will use the UNB StepUP-P150 dataset, which contains high-resolution plantar pressure measurements collected across multiple footwear conditions and walking speeds. The student will also develop a small R or Python package with functions for data import, representation, and visualisation. Therefore, this project will suit a student interested in working on real data applications, with some prior coding experience in R, Python, MATLAB, C++, or similar preferred. References Ramsay, J. O., & Silverman, B. W. (2005). Functional data analysis. New York, NY: Springer New York. Larracy, R., Phinyomark, A., Salehi, A., MacDonald, E., Kazemi, S., Bashar, S. S., ... & Scheme, E. (2025). UNB StepUP: A footStep database for gait analysis and recognition using Underfoot Pressure. arXiv preprint arXiv:2502.17244. |
| Dr Martin Kilian | Pure Mathematics | Finite Gap Curves In the space of all closed curves there is an interesting dense subset of so-called finite gap curves. These posess an algebro-geometric correspondence, and make appearances in various branches of non-linear structures. For example they appear as profiles of soliton solutions in wave equations, pulses in non-linear optics, and in energy transmission in microbiology. The aim of this project is to absorb some of the current literature and to study the moduli space of closed finite gap curves using the isoperiodic deformation. |
| Dr Hassan Alkhayuon (And Dr Serhiy Yanchuk as a co-supervisor) |
Applied Mathematics | Singular basins in multiscale systems Real-world systems often evolve on different timescales and possess multiple coexisting stable states. The difference in timescales can be represented by a small parameter $\epsilon$. Such systems are often characterised by singular basins of attraction that are geometrically different to the basin in the case of $\epsilon = 0$. In this project, we will examine the properties of singular basins, illustrate these properties with examples, and look at potential ramifications of singular basins, such as tunnelling between stable states. This project is well suited for a student with a strong background in nonlinear dynamics, singular perturbation, and numerical methods. |
| Dr David Henry | Discipline fluid | Mathematical Aspects of Geophysical Flows The study of geophysical fluid dynamics (GFD) in the ocean, and atmosphere, presents many interesting, and challenging, open mathematical questions. This is particularly the case when studying large-amplitude wave solutions: the nonlinear PDEs which represent the governing equations then form part of an (unknown) free-boundary problem. Nevertheless, rigorous mathematical analysis often leads to results which offer deep insight into the structure of a given problem, and a rich understanding of the underlying processes. The aim of this project is to apply mathematical analysis - employing techniques from topological degree theory - to study the issue of the well-posedness of solutions to the GFD governing equations. |
| Dr Luke Kelly | Statistics or Actuarial Sciences | Towards maximal couplings of Markov chain Monte Carlo algorithms in Bayesian phylogenetic inference Description: Phylogenies are graphs depicting evolutionary relationships between species descended from a common ancestor. Reconstructing phylogenies from data is a hard statistical problem. Markov chain Monte Carlo (MCMC) is the standard approach for fitting Bayesian phylogenetic models. However, the theoretical properties of MCMC algorithms in phylogenetic problems are not well understood. Recent work by Kelly, Ryder & Clarté (Ann. Appl. Stat., 2023) has established the feasibility of sampling for diagnosing convergence of MCMC algorithms in phylogenetic problems. The goal of this project is to develop improved couplings of MCMC proposals on the space of tree topologies in order to estimate tighter bounds on convergence and allow this approach to scale to larger trees and models. |
| Dr Supratik Roy | Statistics or Actuarial Sciences | The statistics of activation functions Activation functions play a crucial role in machine learning and artificial intelligence. Most current implementations use one of two variants, the sigmoid and the smooth approximations to the rectifiable linear unit. While more complex activations perform well, they also increase computational load which in turn have wider impact. Together, they show a rich variety of behaviours and performances as well as problems. While their performance in a particular type of machine learning task is often the primary consideration, statistical exploration of their behaviour is not available in general. This project will look at statistical generalisations of existing models, with a view towards optimising performance and computational load. The focus area is in statistics and machine learning. |
| Dr Ben McKay | Pure Mathematics | In 1890, Poincaré used geodesic motion on surfaces as a test case for understanding nonlinear dynamics more generally. In 1900, Zoll studied the surfaces all of whose geodesic motions are periodic. In 2002, Lebrun and Mason found a twistor transform of any smooth Zoll surface, producing a totally real surface in a certain complex surface. An example of a singular Zoll surface is the Tannery pear. The goal of this project is to study the Lebrun and Mason construction to see how that particular singularity expresses itself in terms of complex geometry. Moreover, we will ask the following natural questions: What can Zoll singularities look like more generally, and how they are reflected in singularities in the complex surface and the totally real surface? Which objects in the complex surface can generate singular but compact Zoll surfaces in the inverse twistor transform? |