Mathematics

The Mathematics Department seeks to provide a unique research environment, unconstrained by traditional disciplinary boundaries, in which our researchers can develop world-leading theory and methodology, and apply them successfully to cutting-edge problems in science, engineering and medicine. With over 60 permanent academic staff, we have internationally leading research groups in several areas of pure mathematics, applied mathematics and statistics: number theory, geometry and algebra; dynamical systems, control theory and analysis; fluid dynamics, astrophysical fluids and magnetic fields; statistical science and epidemiology; mathematical biology and medicine; and climate dynamics and renewable energy. Our reputation is reflected in the fact, for example, that we are one of only four UK universities to enjoy an academic partnership with the Met Office. The Department spans the University’s campuses at both Streatham, in Exeter, and Penryn, near Falmouth. We run several regular series of seminars and colloquia, and host visits from internationally leading scholars, providing opportunities for students and staff to meet and discuss the latest mathematical research.

Applying for a Studentship

This webpage lists research themes based in Mathematics. When you submit an application you will be expected to submit a research proposal aligned to a theme listed below. Examples of research projects previously available are included for reference. 

For full details of eligibility criteria and the application process, please click here.

Two common themes of the research in the Dynamical Systems, Control Theory and Analysis group are how simple rules can lead to complex behaviour, and the role of feedback as a moderator of dynamical behaviour. A PhD position within our group will give you a range of opportunities, from exciting pure mathematical research in ergodic theory, to stability and robustness analysis, through to applications in healthcare and climate research. The following are a few of our current topics.

The potential supervisors can give more details. See also http://emps.exeter.ac.uk/mathematics/research/csdc/postgraduate/ and personal homepages such as http://empslocal.ex.ac.uk/people/staff/vnb262/PhD-topics for further information. 

Possible PhD topics are listed below. For further details of individual projects, please contact the supervisor, and for other enquiries about research in Dynamical systems, control theory and analysis, please contact Prof. Jan Sieber (J.Sieber@exeter.ac.uk). 

“Synchronization and chimera states” (Peter Ashwin) 

“Dynamics and number theoretic aspects of piecewise isometries” (Peter Ashwin) 

"Networks of dynamical systems" (Peter Ashwin)

“Causodynamics of waves and patterns in reaction-diffusion systems” (Vadim Biktashev) 

“Hybrid asymptotic-numerical methods for cardiac excitation models” (Vadim Biktashev) 

“Coevolution of bacteria and phages” (Vadim Biktashev) 

“Phase-locking of meandering spiral waves” (Vadim Biktashev) 

“Iterative methods of calculation of response functions of spiral waves” (Vadim Biktashev)

 “Statistical properties of strange non-chaotic attractors” (Mark Holland) 

“Noisy intermittent dynamical systems” (Mark Holland) 

“Extreme Value Theory and Risk Analysis” (Mark Holland) 

“Combinatorial dynamics and topological entropy” (Ana Rodrigues) 

“Measure theory and piecewise isometries” (Ana Rodrigues) 

“Theory of dynamical systems with delay” (Jan Sieber) 

“Tracking bifurcations in experiments” (Jan Sieber)


"Data-driven prediction of critical transitions in dynamical systems" (Frank Kwasniok)

"Characterising rare trajectories in dynamical systems using large-deviation theory" (Frank Kwasniok)

 

 

The Fluid Dynamics Research Group employs a combination of theory, computation and observation to explore the physical properties of the sun, stars and the planets, including our own planet Earth. We use the equations governing fluid flow to examine processes deep inside Jupiter12, important for the planet’s overall structure. We consider how flows in the interior of the sun and stars generate the magnetic fields seen in sunspots and solar storms2 4 7 12. We examine in detail how solar storms develop and how they impact the near-Earth environment3 5. We apply fluid mechanics to create numerical models for weather and climate prediction1 8 9 10 11. We also develop basic theory in turbulence4 9 10 11, vortex dynamics4 10, flight? and the geometry and topology of magnetic fields2 7. We apply geometric mechanics and Hamiltonian dynamics to develop mathematical theories8 11 and structure-preserving numerical methods for dynamics of water waves, rigid bodies with fluid-filled cavities, fluid sloshing in moving containers and wave-structure interactions.

Potential project supervisors:  1 Bob Beare, 2 Mitchell Berger, 3 Claire Foullon, 4 Andrew Gilbert, 5 Andrew Hillier, 6 Frank Kwasniok, 7 Joanne Mason, 8 Jemma Shipton, 9 John Thuburn, 10 Geoffrey Vallis, 11 Beth Wingate, 12 Keke Zhang

Possible PhD topics are listed below. For further details of individual projects, please contact the supervisor, and for other enquiries about research in Fluid dynamics, astrophysical fluids and magnetic fields, please contact Prof. Mitchell Berger (m.berger@exeter.ac.uk). 

 “Mathematical modelling of pollution dispersion” (Supervisor: Bob Beare)

“Understanding scale interactions in weather and climate models” (Supervisor: Bob Beare)

 “Multi-spacecraft investigations of solar and heliospheric plasmas” (Supervisor: Claire Foullon)

 "Vortex dynamics and instabilities" (Supervisor: Andrew Gilbert)

"Magnetic field generation in conducting fluid flows" (Supervisor: Andrew Gilbert)

"Simulation of magnetically driven low Reynolds number swimmers" (Supervisor:Andrew Gilbert) 

 "Predicting material failure using an elasto-plastic fracture mechanics approach to composite materials" (supervisor: Layal Hakim)

 "Instabilities, turbulence and magnetohydrodynamic mixing" (supervisor: Andrew Hillier)

"Instability and nonlinearities of nonlinear MHD waves" (supervisor: Andrew Hillier) 

 "Nonlinear resonance in the formation of coherent structures in rotating, stratified, and magnetic fluids" (supervisor: Beth Wingate)

 "How to take a long time step? Frequency-averaging in numerical methods for geophysical and astrophysical fluid dynamics" (supervisor: Beth Wingate)

"Stochastic subgrid modelling in geophysical fluid systems" (Frank Kwasniok)

 "Characterising predictability in geophysical fluids with Lyapunov exponents and vectors" (Frank Kwasniok)

 

Researchers in the mathematical biology and medicine group develop and use mathematical methods to understand biological systems. The interests of our researchers span multiple levels of biological organisation, from subcellular mechanisms to cells, organs, individuals and ecosystems. Many of our researchers are based in the Living Systems Institute (LSI) or the Environment and Sustainability Institute (ESI), where the University has invested significantly in developing interdisciplinary approaches to understand living systems and the environment. In the LSI, a main focus is the development of predictive models to better understand, diagnose and treat chronic health conditions that present significant challenges to society, such as diabetes and dementia, and the EPSRC-funded Centre for Predictive Modelling in Healthcare brings together a world-leading team to tackle these problems. These activities are driven by close collaborations with clinicians, including neurologists, cardiologists, endocrinologists and dermatologists, as well as experimental biologists and neuroscientists. In the ESI, we apply dynamical systems and control theory methodology to issues in natural and human population demography, resource management, conservation ecology, biodiversity and environmental growth, and cyber-physical systems in the context of health and well-being.

Our research includes the development and analysis of mathematical models of biological systems at different scales of description, the quantification of clinical and experimental data, and the development and application of methods for linking models and data. These activities draw upon diverse fields including dynamical systems theory, complexity science and networks, uncertainty quantification, evolutionary computing and machine learning.

Possible PhD topics are listed below. For further details of individual projects, please contact the supervisor, and for other enquiries about research in Mathematical biology and medicine, please contact Dr. Marc Goodfellow (m.goodfellow@exeter.ac.uk). 


“Computer-in-the-loop control of cellular population dynamics” (Wedgwood)

"Pattern formation via long-range cell-to-cell contact: revisiting Turing's morphogenesis hypothesis" (Wedgwood)

“Developing a whole brain model to generate scalp level spatio-temporal brain rhythms” (Creaser)

“Mathematical modelling and analysis of brain dynamics in PTSD” (Creaser)

“Mathematical modelling and analysis of antibiotics uptake in gram negative bacteria and implications for antimicrobial resistance” (Tsaneva)

“Mathematical modelling for precision medicine” (Tsaneva)

 “Machine learning and optimisation for combined antibiotic treatments” (Akman)

 “Data science approaches to optimising synthetic minimal media” (Akman)

 “The nonlinear neural dynamics of synchronising to complex rhythms” (Rankin)

"The impact of hearing loss on language networks" (Rankin)

 “A mathematical and computational framework for neurobiological modelling: Behaviour-driven optimisation of neural connectivity” (Borisyuk)

 “What does that neuron do: A study of the neural circuits that produce swimming in the tadpole” (Borisyuk)

“Dynamics of locomotion and phagocytosis in shape-changing cells: theory and experiments” (Wan)

 “A novel opto-hydrodynamical platform for studying microorganism movement in three-dimensions” (Wan)

 “Modelling convergent cell signalling pathways mediating the neurophysiological stress response” (Walker)

 “Modelling the relationship between ion channel expression and electrical activity in stress-sensitive cells” (Walker)

“Linking models of large-scale brain networks with data to understand neurological disorders” (Goodfellow)

 “Personalised brain models for the management of epilepsy” (Goodfellow)

 

The Number Theory, Algebra and Geometry research group comprises seven permanent members of staff and nine current PhD students. There is a weekly research seminar with speakers from around the UK and rest of Europe. Research interests include anabelian geometry, arithmetic geometry, p-adic Hodge theory, Hopf-Galois theory, Galois module structure, Iwasawa theory, analytic number theory in function fields, homogeneous dynamics and Diophantine approximation. These topics are all connected to Number Theory in some way, which, broadly speaking, can be described as the study of the integers. This is a very active research area, the biggest breakthrough in the last 25 years perhaps being Andrew Wiles's proof of Fermat's Last Theorem. 

Possible PhD topics are listed below. For further details of individual projects, please contact the supervisor, and for other enquiries about research in Number Theory, Geometry and Algebra, please contact Prof. Nigel Byott (N.P.Byott@ex.ac.uk). 

 

“Moments of L-functions in function fields" (Supervisor: Julio Andrade)

“Comparing integral Galois module properties in different Hopf-Galois structures” (Supervisor: Nigel Byott) 

“Minimal Hopf-Galois structures of elementary abelian type” (Supervisor: Nigel Byott) 

“Applications of integral representation theory to problems in algebraic number theory” (Supervisor: Henri Johnston) 

“On special cases of the leading term conjectures” (Supervisor: Henri Johnston) 

"Number fields and their Galois groups" (Supervisor: Mohamed Saidi)

"Reconstruction of number fields from m-step solvable Galois groups" (Supervisor: Mohamed Saidi)

The Statistics and Data Science Research Group uses statistics to address major environmental and health challenges. As well as fundamental problems on statistics and data science areas of expertise include weather forecasting, climate prediction, natural hazards (especially tropical and extratropical cyclones), renewable energy, air pollution, human physical and mental health, and infectious diseases in humans and animals. Long-term collaborations with the insurance industry and organisations such as the Met Office, the World Health Organisation and the Alan Turing Institute reflect the Group’s influence. 

 These challenging applications often demand new statistical and data science methodology. A common feature of the Group’s research is the combination of data with mathematical models of complex systems, such as numerical weather prediction models and dynamical models of infectious diseases, and much of the Group’s research focuses on how to do this effectively. Specific topics of methodological interest include uncertainty quantification, hierarchical space-time modelling, forecast evaluation, extreme-value theory, epidemiology, clinical trials, accelerometry for healthcare, operations research and Bayesian theory and computation

Possible PhD topics are listed below. For further details of individual projects, please contact the supervisor, and for other enquiries about research in Statistical science and epidemiology, please contact Prof. Peter Challenor  (p.g.challenor@exeter.ac.uk). 

 

“A comparison of Bayesian neural networks and Gaussian processes for emulating complex numerical models” (Peter Challenor)

“Uncertainty quantification of complex numerical models using the ensemble method” (Peter Challenor)

“Bayesian deep learning for real-time calibration of infectious disease models” (Danny Williamson)

“Bayesian decision support systems for UK tree planting policies” (Danny Williamson)

“New approaches for understanding changes in frequency and severity of weather extremes due to climate change” (David Stephenson)

“New approaches for estimating changes in modes of climate variability due to climate change” (David Stephenson)

 “Statistical modelling of extreme weather events generated by climate models” (Ben Youngman)

“A statistical framework for extreme weather risk based on climate model projections” (Ben Youngman)

“Statistical methods for analysing the performance of multivariate, space-time forecasts” (Chris Ferro)

“Statistical approaches to synthesizing data from various sources for robust estimation and prediction of extreme values in weather and climate” (Theo Economou)

“Statistical approaches for modelling big environmental data with applications from the UK Met Office” (Theo Economou)

“Statistical correction of flawed epidemiological data (e.g. Dengue fever, Covid-19) in collaboration with the Fiocruz Public Health institute in Brazil” (Oliver Stoner)

“Accelerometry data analysis: what are the best methodologies to support the analysis of accelerometry data for population health” (Mark Kelson)

“Time series data imputation for missing data in accelerometry analysis” (Mark Kelson)

“Advanced optimisation techniques for manufacturing and supply chain management” (Stephen Maher)

“From theory to software: Novel approaches for improving the computational performance of Benders' decomposition” (Stephen Maher)

“Physical-statistical modelling for precipitation radar nowcasting” (Stefan Siegert)

“Bayesian methods for quantifying and reducing uncertainty in life cycle assessment of renewable energy” (Stefan Siegert)

"Statistical post-processing of ensemble forecasts of compound weather risk" (Frank Kwasniok)

"Quantifying weather and climate risk using large-deviation theory" (Frank Kwasniok)


 

The Mathematics and the Environment group at the Penryn campus works on mathematical, statistical and computational modelling, fluid dynamics, systems and control theory, optimization, big data analytics and machine learning. We are involved in numerous diverse inter-disciplinary applications and work closely with the Centre for Ecology and Conservation, Camborne School of Mines, Renewable Energy and Energy Policy Group at the Penryn Campus and the European Centre for Environment & Human Health at the Truro Campus. There are also strong links to local industries, small and medium-size business and other stakeholders. Many of the researchers in the group are affiliated to the Environment and Sustainability Institute (ESI) and/or the Institute of Data Science and Artificial Intelligence (IDSAI), where the University has invested significantly in developing interdisciplinary approaches to understand the environment and interdisciplinary use of data science. In the ESI, we apply dynamical systems and control theory to issues in natural and human population demography, resource management, conservation ecology, biodiversity and environmental growth, and cyber-physical systems in the context of health and well-being. The IDSAI affiliates in the centre are an interdisciplinary team of researchers with diverse and complementary research interests in real-world data-driven fundamental research in machine learning, computational statistics and artificial intelligence methods to solve challenges arising in control theory and optimisation, dynamical systems, signal and image processing, large-scale and computing intensive numerical modelling.

Possible PhD topics are listed below. For further details of individual projects, please contact the supervisor, and for other enquiries about research in Mathematics and the Environment, please contact Prof. Stuart Townley (S.B.Townley@exeter.ac.uk). 

“Structure-preserving numerical discretisation for Hamiltonian partial differential equations” (Hamid Alemi Ardakani)

“Mathematical modelling of the coupled fluid-body dynamics using the Hamiltonian particle mesh theory” (Hamid Alemi Ardakani)

“Multi-agent reinforcement learning control for renewable energy integration in smart grids with economic load dispatch” (Saptarshi Das)

“Joint state/parameter estimation, uncertainty modelling and Bayesian model selection in smart grid signal processing” (Saptarshi Das)

“Exploiting physical certainties in robust control” (Tim Hughes)

“Control theoretic paradigms in an equation-based modelling framework” (Tim Hughes)

"Applications of generalised trigonometric and special functions in signal processing" (Houry Melkonian)

"Understanding image processing through the use of generalised Fourier-type analysis" (Houry Melkonian)

“Robust control designs for integrated renewable energy systems” (Markus Mueller)

“Adaptive power management strategies for arrays of oscillating water column wave energy converters” (Markus Mueller)

“Feedback loops, selection pressures and anti-microbial resistance and/or pesticide resistance" (Stuart Townley)

“The spread and control of information and power in dynamic social networks” (Stuart Townley)