Join our vibrant and supportive community of academics and postgraduates with expertise in operational research, statistics, pure and applied mathematics and theoretical physics.
In the 2021 Research Excellence Framework (REF 2021), 100% of our research impact and research environment was specifically rated as internationally excellent or of world-leading quality. We are ranked 17th in the Sunday Times Good University Guide 2021 and 12th for Mathematical sciences in the Russell Group.
You'll be supported by supervisors who are international experts in their field, spanning pure and applied mathematics, statistics and operational research. They'll provide guidance and also direct you to further in-depth training.
We offer a wide range of training opportunities. You'll have access to specialised in-house post-graduate courses and three National Postgraduate Training Networks in Applied and Pure Mathematics (MAGIC), Operational Research (NATCOR) and Statistics (APTS).
You’ll be encouraged to participate in research seminars and discussion and attend conferences. You'll have a personal computer, a desk in a shared office and a conference attendance allowance.
Our academics have direct contact with potential employers. Our postgraduate students are highly sought after by other universities, business and industry, NGOs and governments worldwide.
We offer 2 PhD routes.
This is a research degree of up to 4 years duration. When you apply, you'll choose one of the following specialisms:
Our iPhD is a doctoral programme in mathematical sciences, which includes a taught first year. The duration of the programme is up to 5 years.
We have one of the broadest communities of mathematicians in the UK. We collaborate not only with other mathematicians, but also with engineers, scientists, biologists and social scientists.
You can either apply for a structured studentship or propose your own PhD idea.
Taking a structured studentship will give you access to additional training, conferences and secondments.
This PhD project focuses on improving aircraft noise prediction for emerging technologies at early design stages. It involves developing whole-aircraft noise models, incorporating operational factors and fleet-level scenarios. The research supports Rolls-Royce’s noise prediction systems and informs certification standards and airport noise policies.
This PhD applies AI to inverse design, a method that works backwards from desired performance to generate efficient photonic circuits. You'll develop algorithms that intelligently explore vast design spaces, enabling compact, manufacturable light-based chips.
This exciting PhD project explores the use of Physics-Informed Neural Networks (PINNs) to model complex environmental flows. By integrating AI with fluid mechanics, it aims to enhance simulations of the 2D Shallow Water Equations for applications in flood prediction, infrastructure design, and sustainable water management.
This project aims to study and identify the beneficial effects of using bio-inspired geometry on aerofoils, enhancing aerodynamic efficiency and reducing the noise generated from non-equilibrium turbulent boundary layers. It is a computational study by using a highly optimised CFD code on the UK supercomputer ARCHER2.
This PhD project investigates noise from eVTOL aircraft during transitional flight phases, aiming to improve prediction methods and assess perceived noise using virtual reality. It supports design-performance trade-offs, informs policy, and contributes to quieter, more acceptable urban air mobility systems through semi-empirical modelling and integration with whole-aircraft noise tools.
The project develops intelligent nanorobots for autonomous control and adaptive behaviour in liquid nanoscale environments, exploring algorithms for simulation, control, and prediction with applications in drug delivery, diagnostics, and nano-manipulation.
Extreme bursts in fluid flows are hard to predict and control. This project develops modern optimisation and dynamical-systems tools to understand and limit these events using low-dimensional models.
In this project you will develop advanced machine learning tools to convert real X-ray measurement data into artefact free 3D images. The work draws on a range of ideas from artificial intelligence, optimization, X-ray physics, applied mathematics and computer science.
This PhD will develop AI-based predictive and control methods for nonlinear, regolith-resilient mechanisms in collaboration with ESA, combining structural dynamics, compliant design, and topology optimisation to create lightweight, intelligent space structures for next-generation exploration.
Do you want to shape the future of quieter, more sustainable aviation? This PhD develops efficient computational methods to simulate aeroengine noise, combining fluid dynamics, acoustics, and high-performance computing to create faster, more accurate tools that help reduce environmental impact.
This PhD project develops next-generation multicore fibre amplifiers for sustainable submarine networks. The research combines simulation and experiment to create energy-efficient, high-capacity amplification technologies that reduce power consumption, cost per bit, and enhance future global communication infrastructure.
Agentic AI is rapidly affecting diverse aspects of our life, including social, technological and financial interactions. They may seem like human agents, yet their "thinking" and "reasoning" is not the same. The aim of this project is to study, understand and design the strategic interaction between LLM agents.
We live in a rapidly changing world. It is crucial that we understand such a complex system and build reliable predictive tools in time. The computational fluid dynamics (CFD) model includes buildings and abrupt terrains in a city-scale environment. It aims to understand across-scale physical processes with a focus on metre to kilometre scales and develop a fast CFD tool.
Hollow core anti-resonant fibres (ARFs) enable strong light-matter interaction through functional material deposition. This PhD project advances composite material ARF (CM-ARF) technology using 2D materials and chalcogenides for photonic applications, combining cleanroom fabrication, device characterization, and simulations—ideal for candidates with physics, materials, or engineering backgrounds.
Acoustic security is rapidly emerging at the intersection of cybersecurity, privacy, cyber physical systems, and acoustical physics. While machine learning has produced notable results, this project goes further—advancing both attacks and defenses through new mathematical approaches and deeper insights from acoustics.
This project will develop a multi-scale surrogate modeling framework to optimize passive surface textures (like dimples) for maximum fluid drag reduction. By enabling efficient shape optimization and identifying critical flow parameters, this research seeks to resolve conflicting results and advance the theoretical understanding and practical application of cost-effective flow control in transportation.
Soil is one of the most complex and important self-assembling organo-mineral composites in the world. All human food supply, ecosystem and infrastructure services depend on soil, yet we have very little understanding of what happens in soil. Developing an understanding of soil composition is important, as climate change is likely to alter soil function. We need to find new robust engineering practices and modify existing ones to enable effective soil resource management.
This PhD project will explore gust–wing interactions in vertical axis wind turbines (VAWTs). Integrating computational fluid dynamics, data-driven modelling, and experiments on a custom-built VAWT rig, you will examine how gusts influence blade aerodynamics in curvilinear flows and develop predictive tools to mitigate their adverse effects.
Spin-based quantum sensing converts tiny quantum signals into detectable responses by aligning microscopic spins, for example in diamond nitrogen-vacancy centres. Can this alignment be exploited to amplify responses in other systems? This project addresses that question—theoretically and experimentally—via novel transfer protocols utilising periodic control fields and Floquet-engineering methods.
Join our dynamic research team to explore cutting-edge microscale optical resonator designs for quantum technologies. This PhD will combine photonics, quantum physics, and computational modelling to design devices that enhance the interaction between matter and light on the quantum level to unlock new capabilities in quantum computing, communication, and sensing.
Dive into the mysterious world of polarization in antiresonant hollow core fibres, where conventional wisdom is turned on its head, and unexpected phenomena emerge every day. Through your insights and innovation, you will shape the future of this cutting-edge technology from data centres, to high-power lasers, to space systems.
This project explores how quantum computing can transform energy system planning for a net-zero Europe. By integrating quantum and classical optimisation methods, it will address uncertainty in renewable generation and develop scalable algorithms for large-scale stochastic models, advancing both optimisation theory and practical tools for the energy transition.
This project explores the emerging field of Quantum Computational Fluid Dynamics (QCFD), combining quantum computing and CFD to simulate nonlinear systems such as turbulence and shockwaves. You will be working and implementing quantum variational algorithms in quantum computers that bridge fundamental physics with quantum algorithmic innovation for next-generation fluid simulation.
The main challenge in the adoption of quantum computing is the gap between algorithmic requirements and current quantum hardware. In this project, you will codevelop novel qubit efficient quantum approaches and techniques that can be used to solve optimization problems and apply them to logistics, pharma, transport, or manufacturing industries.
When gas flows from a coaxial nozzle surrounded by a liquid, it forms reproducible liquid shells useful in industries needing hollow spheres, such as pharmaceuticals and 3D printing. This project aims to understand and optimize production rates, reproducibility based on nozzle design and operating parameters.
The aim of this project is to develop formal models of accountability and liability using logic, game theory, and agent-based simulation. You will explore responsibility under uncertainty, delegation, and trust, with applications in autonomous systems, digital governance, and ethical AI.
This project combines cutting-edge data assimilation with experimental and computational fluid dynamics to uncover hidden flow properties in porous materials. By tuning RANS model parameters to experimental data, we infer permeability from geometry-based porosity alone—advancing our ability to predict, model, and understand complex flows through porous structures.
This PhD project focuses on developing silicon photonic sensors that can detect early biomarkers of sepsis in children - quickly, accurately, and at the point of care.
This project aims to pioneer advancements in the energy-efficient Generative AI models (GenAI), focusing on achieving faster inference times and reduced model sizes without compromising performance and increasing carbon emissions.
Consider a major global environmental issue. Perhaps you thought about ocean plastic pollution, air pollution, or sea level rise leading to coastal erosion. In this project, you will contribute to this goal through experimental, theoretical, numerical, or combined approaches, depending on your skills and interests.
Deep learning has been a driving force behind the rapid progress of AI for more than a decade, culminating recently in the success of large language models powered by transformer architectures—a class of deep neural networks. In parallel, bilevel optimization has emerged as a powerful framework for modeling complex machine learning tasks, giving rise to what we refer to as deep bilevel learning. This PhD project will investigate the mathematical foundations of deep bilevel learning, with the goal of uncovering structural properties that can be exploited to design more efficient, robust, and explainable learning algorithms. The results have the potential to influence the next generation of AI systems and advance theory at the intersection of optimization and deep learning.
This PhD will help build the next revolution, the “self-driving” laser, a new class of intelligent light sources that think for themselves. These lasers will sense their environment, learn optimal parameters in real time, and autonomously deliver perfect results across diverse materials and geometries.
We offer a wide range of fully funded research projects.
These projects:
We also offer studentships in all our current research areas.
Visit our research groups and project pages to contact members of staff listed in your area of interest to enquire about possible PhD projects.
Doctoral training centres offer fully funded studentships which include:
We administer the Engineering and Physical Sciences Research Council (EPSRC) DTP funding for research projects in mathematical sciences. Funding decisions are handled directly by the centre. The decision to award funds is independent of, and separate from, the decision on to offer you a place on a PhD degree.
Find out more about EPSRC DTP funding.
We offer scholarships and teaching bursaries ourselves. Your potential supervisor can guide you on what is available.
If you’re an international student you may be able to apply for a scholarship from your country.
Find out more about scholarships
Once you've found a supervisor, they can help you with potential funding sources. We offer match funding in some cases.
You'll need to state how you intend to pay for your tuition fees when you submit your application.
Find out more about funding your PhD
You may be able to fund your postgraduate research with funding from your current employer or from industry.
You can borrow up to £30,301 for a PhD starting on or after 1 August 2025. Doctoral loans are not means tested and you can decide how much you want to borrow.
Find out about PhD loans on GOV.UK
You may be able to win funding from one or more charities to help fund your PhD.
We charge tuition fees for every year of study. If you're applying for a fully funded project, your fees will be paid for you.
2023 to 2024 entry:
| Subject | UK fees | International fees |
|---|---|---|
| Mathematical sciences full time | tbc | £18,600 |
| Mathematical sciences part time | tbc | £9,300 |
2024 to 2025 entry:
| Subject | UK fees | International fees |
|---|---|---|
| Mathematical sciences full time | £4,786 | £19,200 |
| Mathematical sciences part time | £2,393 | £9,600 |
2025 to 2026 entry:
| Subject | UK fees | International fees |
|---|---|---|
| Mathematical sciences full time | tbc | £19,620 |
| Mathematical sciences part time | tbc | £9,810 |
You're eligible for a 10% alumni discount on a self-funded PhD if you're a current student or graduate from the University of Southampton. This will not apply for programmes that are externally funded. Please check the fees and funding section.
Mathematical sciences hosts a number of specialist research and teaching centres. In some cases we collaborate with other areas, such as business and social and political science.
We offer several research programmes including a general mathematical sciences iPhD:
Decide whether to apply to an advertised research project or create your own proposal.
It's a good idea to email potential supervisors working within your field of interest to discuss potential PhD projects. It's best to do this well ahead of the application deadline.
You’ll find supervisors’ contact details listed with the advertised project, or you can search for supervisors in the staff directory.
The application process is the same whether you're applying for a funded project, or have created a research proposal.
For both the PhD and the iPhD you should have a 2:1 honours undergraduate degree or equivalent qualification.
If you have completed a master’s degree, you should have achieved a merit or above including 60%, for the dissertation.
You will also need a satisfactory performance at interview.
If English is not your first language, you'll need an IELTS minimum level of 6.5 with a 6.0 in writing, reading, speaking and listening.
Your awarded certificate needs to be dated within the last 2 years.
If you need further English language tuition before starting your degree, you can apply for one of our pre-sessional English language courses.
Check the specific entry requirements listed on the project you’re interested in before you apply.
Research degrees have a minimum and maximum duration, known as the candidature. Your candidature ends when you submit your thesis.
Most candidatures are longer than the minimum period.
| Degree type | Duration (full time) | Duration (part time) |
| Mathematical sciences PhD | 2 to 4 years | 4 to 7 years |
| Mathematical sciences iPhD | 4 to 5 years | Up to 8 years |
