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The School of Mathematics offers an exceptionally wide range of opportunities for postgraduate studies in mathematics.

The programme aims to offer knowledge and expertise for a career in the academic world, or to pursue a variety of other opportunities in which a strong mathematical background is important. Past students are on the staff of universities in the UK and abroad, and senior statisticians and managers in industry and business.

Available research areas

  • number theory
  • functional analysis and spectral theory of ordinary and partial differential operators
  • calculus of variations and partial differential equations
  • numerical analysis
  • theoretical and computational fluid dynamics
  • operator algebras and non commutative geometry
  • algebraic topology
  • algebraic geometry
  • combinatorics
  • quantum field theory and statistical mechanics
  • conformal field theory and vertex operator algebras
  • image processing
  • operational research
  • time series analysis
  • probability theory and statistics.

Distinctive features

  • Lively research environment, extensive seminar programme.
  • Conference and workshop support, including annual Welsh Mathematics Colloquium.
  • Optional participation in undergraduate teaching, including HEA accreditation.
  • Colloquia are held regularly and each research group has a weekly research seminar.
  • The School hosts and participates in frequent workshops organised by WIMSC research clusters, with which the research groups are associated.
  • The many collaborative projects with groups in other institutions in the UK and abroad, bring a steady stream of distinguished visitors to the School.

Key facts

Mode of study Full-time, part-time
Qualification PhD, MPhil
Full-time duration PhD 3.5 years; MPhil 1 year
Part-time duration PhD 5 years; MPhil 2 years
Start dates January, April, July, October

As a research student, in the first and second year you will follow an agreed programme of lectures and reading courses agreed with your supervisor to introduce you to research skills and methods, and to advance your knowledge in your chosen field. These include formal assessment.

This includes a broad choice of postgraduate courses provided through the national collaborative networks:

  • MAGIC (for pure/applied mathematics)
  • NATCOR (for operational research)
  • APTS (for statistics).

Regular seminars in all subject areas, as well as the Cardiff Mathematics Colloquia and the yearly Welsh Mathematics Colloquium, provide insight into cutting-edge research.

Moreover, research students benefit from a carefully developed graduate program including computational, presentational and subject-specific skills.

Students will also write a substantial end-of-year report and give a seminar in the department.

The PhD programme leads the student, through taught courses and individual project supervision, to the frontier of an area of mathematics, with the aim of creating a piece of original research, written up in the doctoral thesis. In the process, the student acquires essential research, problem-solving and communication skills. The MPhil programme is similar, but less ambitious due to its more limited scope; work on an MPhil can lead on to completing a PhD.

Skills developed

In addition to in-depth knowledge of a mathematical research area, PhD students gain general research and presentation skills, and (optionally) undergraduate teaching skills, HEA accreditation possible.

The School of Mathematics offers an exceptionally wide range of opportunities for postgraduate studies in mathematics. The academic staff are engaged in research projects in fields that are as varied as number theory, functional analysis and spectral theory in pure mathematics, image processing, operational research, statistics and quantum field theory.

Many research projects undertaken by members of staff in the School are being generously funded by the public and private sectors. The leader of the Operator Algebras and Non-Commutative Geometry group is the co-ordinator of the European Network on Non-Commutative Geometry funded by the EU TMR programme. Other sources of funding include EPSRC, Leverhulme Trust, NATO, INTAS, Ministry of Defence, Wellcome Trust, GlaxoSmithKline, Proctor and Gamble.

We have close links with several universities and institutions abroad. These include the Institute of Advanced Study in Princeton, the Australian National University and the Australian Road Research Board in Melbourne, Ecole Superieure d'Electricite in Gif-Sur Yvette, ETH in Zurich and universities in Canada, the Czech Republic, France, Germany, Italy, Ireland, Norway, Saudi Arabia, Japan, Malaysia and USA.

Research environment

The individual research groups run weekly seminars with guest speakers, and postgraduate research students regularly attend the seminar in their research area, and are encouraged to profit from talks in other areas, too. We organise about three mathematics colloquia with distinguished guest speakers per term.

Moreover, Cardiff is the venue for a number of international workshops and conferences organised by members of the School , thus giving postgraduate students a welcome opportunity to learn about world-leading current research. They are also encouraged to take part and present their results at major conferences in the UK and abroad.

At the annual Welsh Mathematics Colloquium at Gregynog, postgraduate students can showcase their work in the group of Welsh Mathematics departments.


Informal contact with potential supervisors in the School of Mathematics is recommended before applying.

Research areas

Applied Mathematics

Research in this area spans: Wave propagation in inhomogeneous media; Homogenisation; Fluid mechanics; Structural and solid mechanics; Numerical analysis and scientific computing; Applied mathematical modelling; Memory effects; Inverse problems; Integral transforms.

Operational Research

Research in this area spans: Modelling of traffic flow; Healthcare modelling; Modelling of the spread of infectious diseases; Queueing theory; Scheduling and timetabling problems; Metaheuristics; Discrete optimisation.

Probability and Statistics

Research in this area spans: Multivariate statistical analysis; Time series analysis; Statistical modelling in market research; Optimal experimental design; Stochastic global optimisation; Change point detection; Probabilistic methods in search and number theory; Fisheries; Medical statistics; Random fields; Mathematical finance.

Pure Mathematics

Research in this area spans: Ordinary and partial differential equations; Functional analysis; Analytical and computational spectral theory; Quantum mechanics; Number theory and its applications; Mathematical physics; Operator algebras; Algebraic geometry.


This project will develop knowledge and skills in several general areas of algebraic, geometric and enumerative combinatorics, including polytope theory, poset theory and symmetric function theory.

Game Theory

This project will aim to build on research by exploring novel reinforcement learning algorithms and/or other techniques from machine learning.

Integer Optimisation

The project aims to develop novel algebraic and geometric methods that can be successfully applied to study integer optimisation problems, with a special focus on sparsity of solutions in context of (linear or nonlinear) integer optimisation.

Spectral element mimetic least squares PGD method

The project will develop new numerical discretisations for solving PDEs that can subsequently be applied to problems in fluid mechanics and will build on current expertise in the School.

Metaheuristic Methods for Probabilistic Graphical Models

The aim of this project is to develop approaches for learning probabilistic graphical models that can be used for classification problems and to explore causal relationships between attributes.

Spectral approximation

This project will deal with problems of spectral approximation for operators and operator pencils and try to identify classes of operator for which efficient spectrally inclusive algorithms can be devised.

Dynamics of bubbles rising in viscoelastic fluids

This project will develop new numerical discretisations for solving PDEs that can subsequently be applied to problems in fluid mechanics.

Homogenisation Theory

The aim of this project is to understand the properties of composites by approximating them with “effective” (homogeneous) materials.

Mathematical analysis - Dynamical systems and spectral theory

This project will develop cutting edge techniques in the interface between pure mathematics and exciting applications.

Design of experiments in regression models with correlated observations

This project aims to develop the theoretical and methodological principles for the problem of optimal experimental design in the case of correlated observations.

Adaptive modelling and optimisation of hyperelastic cellular structures

This project will identify mechanical properties amenable to mathematical treatment and produce physically realistic mathematical models for natural cellular structures.

Development of a platform for healthcare operations management underpinning strategic, tactical and operational decisions

This project will develop a platform for healthcare operations management underpinning strategic, tactical and operational decisions.

Coordinated movement of random fish schooling

This project aims to develop a theoretical model for three-dimensional random movement in fish schools, with the absence of a centralised coordination.

Mathematical methods for scale-bridging: From interacting particle systems to differential equations

The focus of the project is on applying two new mathematical developments for the purpose of scale bridging.

Problems in microlocal analysis with an emphasis on spectral asymptotics and scattering theory

In this project, you will gain exposure to a variety of ideas from microlocal analysis and how they are used in problems initiated with the field of quantum mechanics, more specifically quantum chaos.

The Influence of Compressibility on the Dynamics of Encapsulated Microbubbles

The dynamics of EMBs in viscoelastic fluids forced by ultrasound forms the focus of this project.

Persistent sheaf cohomology

This project is about a fairly recent development: the application of sheaf theory to topological data analysis.

Vertex algebras and generalisations of Lie theory

This project focuses on symmetric functions and their implications in representation theory, finite dimensional semisimple Lie algebras and their representations, affine Kac-Moody algebras and infinite dimensional Lie algebras.

Optimisation of placement of charging stations for electrical vehicles in road networks

The aim of this PhD project is to develop novel modelling and algorithmic solutions for the charging limitations of electrical vehicles in road networks by drawing from existing research in the fields of optimisation and network theory.

Least Squares PGD Mimetic Spectral Element Methods for Systems of First-Order PDEs

The goal of this project is to develop a least squares PGD mimetic spectral element method for the Stokes problem.

A broad variety of careers in academia or industry.

UK government postgraduate doctoral loans

Candidates for the Professional Doctorate programme may be eligible to apply for a UK government postgraduate doctoral loan.

Find out more about UK government postgraduate doctoral loans


A small number of EPSRC-funded doctoral training grants are available for UK/EU applicants, awarded on a competitive basis, subject to research council residence requirements.

There are currently no funding opportunities available.

Tuition fees

UK and EU students

Get the latest information on postgraduate fees.

Students from outside the EU

Get the latest information on postgraduate fees.

Suitable for graduates in Mathematics (or a suitable related field). A 1st or upper 2nd class UK Honours degree, or equivalent is required.

English language requirements

Applicants whose first language is not English are normally expected to meet the minimum University requirements (e.g. 6.5 IELTS). Please see our English Language Requirements guidance for more details.


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