Mathematical Sciences PhD

PhD Mathematical Sciences

Our research courses reflect the extensive research experience of staff in a wide range of areas.

Fact file

PhD Mathematical Sciences
Entry requirements
2:1 (or international equivalent) in mathematics or a closely related subject with substantial mathematical content
6.5 (no less than 6.0 in any element) If these grades are not met, English preparatory courses are available
Start date
Please contact the school
University Park
Other requirements




Major research themes

We are home to seven research groups within the school, together with Uncertainty Quantification as a particular cross-group focus.

Algebra and Analysis


The Algebra subgroup deals with relationships and operations and structures that underpin other areas such as geometry, mathematical physics and computer science.

Research includes:

  • algebraic cobordism
  • combinatorial and geometric group theory
  • commutative algebra and invariant theory
  • milnor K -theory
  • nonassociative algebras
  • quadratic forms and forms of higher degree
  • quadratic and hermitian lattices

The Analysis subgroup focuses on the mathematical description of natural phenomena.

Research includes:

  • banach algebras 
  • calculus of variations and PDE 
  • complex analysis  
  • functional analysis 
  • complex dynamics 
  • data and image analysis 

Industrial and Applied Mathematics

The Industrial and Applied Mathematics Group focuses on mathematical modelling of problems and challenges in engineering and industry.

Typical applications are related to formulating equations describing the behaviour of, for example, particles, solids and fluids as well as linear and non-linear wave problems. These equations are generally either nonlinear and/or involve many degrees of freedom and cannot be solved exactly.

We explore various techniques to obtain approximate solutions to these equations, including asymptotic methods, similarity methods, bifurcation theory, and numerical methods.

Our research is divided into three main areas:

Wave modelling
  • Wave and Quantum Chaos
  • Vibro-acoustics
  • High frequency asymptotic
  • Electromagnetics and Optics
  • Transfer Operator
  • Random Coupling Model and Random Matrix Theory
  • Electromagnetic Compatibility and Wireless Communication
  • Complex Ray Theory
  • Diffraction and Scattering
  • Transformation Optics

Group members also coordinate the Wave Modelling Research Group, together with colleagues in the Schools of Engineering and Physics. 

Fluid and particle dynamics
  • Nucleation/Coagulation-fragmentation processes
  • Fluid Mechanics
  • Molecular modelling
  • Polymers under flow
  • Prediction via Computational Chemistry
  • Porous Media
  • Multiscale modelling
  • Multiphase flows
  • Vortices in Rotating Fluids
Nonlinear dynamics
  • bifurcation theory
  • asymptotic methods
  • power electronics
  • uncertainty quantification

Students at the University of Nottingham can join our Student Chapter of the Society for Industrial and Applied Mathematics (SIAM) for free.


Mathematical Medicine and Biology

The Centre for Mathematical Medicine and Biology (CMMB) is led from within the School of Mathematical Sciences and comprises members of the University of Nottingham who use mathematical methods to provide insights into biological and biomedical phenomena.

We aim to promote the application of mathematical modelling to medicine and the biomedical sciences, and to stimulate multi-disciplinary research within the University and beyond.

The CMMB is an Institute Partner of the Mathematical Biosciences Institute (MBI) at Ohio State University.


Mathematical Physics

The Mathematical Physics Research Group is formed of three subgroups.

Quantum Chaos and Disorder

The Quantum Chaos and Disorder subgroup gathers leading experts to understand realistic quantum systems. 

  • anderson localization
  • multifractality
  • quantum chaos
  • quantum tunnelling
  • quantum mechanics in phase space
  • random Matrix Theory
  • short-wavelength approximations
  • topological insulators and superconductors
Quantum Gravity

The Quantum Gravity subgroup is one of the leading groups in the field of (non-string) quantum gravity.  

  • analogue models of quantum gravity 
  • modified gravity 
  • strong gravity
  • non-commutative geometry 
  • quantum black holes and cosmology
  • quantum Field Theory in curved spacetime 
  • quantum gauge theories 
  • higher categorical structures in physics 

The subgroup runs the Quantum Gravity Laboratory carrying out experiments demonstrating analogue quantum gravity effects in matter systems.

The subgroup's research has been supported by grants from the Royal Society, EPSRC, STFC and the ERC. Steffen Gielen, Alexander Schenkel and Silke Weinfurtner all hold current Royal Society University Research Fellowships. 

The subgroup also runs the MSc programme in Gravity, Particles and Fields jointly with the Particle Theory group in the School of Physics and Astronomy.

Quantum Information

The Quantum Information subgroup is part of a young and eclectic field that considers information and computation, probability and statistics, and control theory.  

  • continuous variable quantum information
  • entanglement and quantum correlations
  • quantum thermodynamics
  • open quantum systems and decoherence
  • quantum optics
  • quantum metrology and engineering;
  • quantum probability, statistics and control
  • relativistic quantum information and metrology

The subgroup hosts the Physics branch of the Penrose Institute to investigate high-precision quantum sensors of gravitational effects.

The subgroup also forms part of the Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems (CQNE).


Number Theory and Geometry

Number theory studies the deepest properties of numbers using methods from all areas of mathematics, and it is the most applicable part of pure mathematics through its use in coding, cryptography and computer sciences.

Research areas
  • analytic number theory
  • arithmetic, algebraic and anabelian geometry
  • computational number theory
  • geometric and categorical theories and correspondences
  • higher class field theories, higher adelic analysis and geometry, higher automorphic forms 
  • local number theory, Iwasawa theory
  • representation theory and quantum field theory
  • zeta and L functions

Our current awards include Symmetries and correspondences intra-disciplinary developments and applications, a Nottingham-Oxford EPSRC programme grant (2015-2021).


Scientific Computation

Scientific Computation is a vibrant research group with extensive expertise in numerical analysis, scientific computing, and mathematical modelling.

Scientific computation and numerical analysis are concerned with the design and mathematical analysis of computational algorithms used within a variety of application areas, including engineering, physics, biology, chemistry, and finance. 

Research areas include:

  • computational PDEs (high-order finite element and discontinuous Galerkin methods, adaptivity, a posteriori error analysis)
  • multiscale modelling and computation
  • computational fluid mechanics and electromagnetics
  • free and moving boundary problems
  • computational biology and medicine
  • bayesian inverse problems
  • uncertainty quantification
  • stochastic numerics and stochastic modelling

Students at the University of Nottingham can join our Student Chapter of the  Society for Industrial and Applied Mathematics (SIAM) for free.


Statistics and Probability

The Statistics and Probability Research Group is formed of four subgroups.

Applied and Theoretical Probability

The Applied and Theoretical Probability subgroup has interests ranging from the theoretical foundations of random objects in pure mathematics through to modern practical applications such as developments in computing.

  • diffusion processes on manifolds
  • limit theorems on Riemannian manifolds with or without singularities
  • mathematical finance
  • resource allocation, through restless bandits
  • stochastic geometry
  • stochastic optimal control
  • stochastic numerics and modelling
  • Theory of Markov renewal processes and semi-Markov processes 
Computational and Theoretical Statistics

This subgroup is primarily concerned with the science of reasoning under uncertainty.

We collaborate with the Centre for Plant Integrative Biology, on analysing large “-omics” data sets, for example for gene network inference.

  • analysis of computer experiments and uncertainty quantification
  • analysis of large high-dimensional data sets
  • directional statistics
  • bayesian computational statistics
  • bootstrap and empirical likelihood methods
  • extreme value analysis
  • inference for dynamical models defined by stochastic and ordinary differential equations
  • spatial and spatio-temporal modelling
  • statistical archaeology
  • statistics for directional data and other non-Euclidean data types
  • structural bioinformatic
Epidemic Modelling

Epidemic Modelling uses mathematics to study the mechanisms underlying the spread of infectious diseases such as influenza, foot-and-mouth disease, AIDS and Ebola. The subgroup has particular expertise in stochastic models.  

  • modelling epidemics with a structured underlying population (such as households, random networks, hospital wards and farms)
  • rigorous analysis of stochastic epidemic models
  • analysis of disease outbreak data
  • bayesian computational methods for infectious disease data
  • assessing if future disease outbreaks can be prevented
  • analysis of the effect of intervention strategies such as vaccination

The subgroup has strong collaborative links with institutions such as Public Health England, leading UK Hospitals such as Guy's and St Thomas' Hospital in London, and the University's School of Veterinary Medicine and Science

Shape and Object Data Analysis

Shape analysis is a thriving research field driven by wide applications in areas such as bioinformatics, medicine, biology and engineering.

An even broader research area is Object Data Analysis which includes the analysis of data objects' such as images, shapes, trees, dynamical systems and functional data.

The impact of our current Shape and Object Data Analysis research includes improving the recognition of disease from medical images.


Uncertainty Quantification

Mathematical models are imperfect because they are based on our flawed understanding of the world and they often rely on unknown or immeasurable information.

Uncertainty Quantification is concerned with understanding and calculating the uncertainties inherent in such scientific problems. For example, we might use mathematical models that allow us to predict climate change, but the models used can only approximate the real world and so the results from the model will be uncertain.

Researchers at the University of Nottingham are focusing on Uncertainty Quantification to improve our predictions of climate change and its impacts, and assessing the safety of geological disposal of radioactive wastes and carbon capture/storage schemes.

Example projects include:

  • ABC methods for calibrating stochastic simulators
  • Carbon Capture and Storage
  • diagnosing errors for dynamical systems
  • engineered barrier for radioactive waste repository
  • gaussian process emulators for groundwater flow problems
  • GPEs applied to spreading of CO2 plumes in aquifers
  • Multilevel Monte Carlo for groundwater flow and radionuclide transport
  • numerical methods for SDEs applied to UQ
  • paleo climate reconstruction
  • parameter estimation for ODEs and SDEs
  • UQ applied to groundwater flow
  • UQ for performance of air riding seals and bearings
  • UQ in manufacturing composites

How to apply

Please note that you do not need to submit a detailed research proposal. A short overview indicating your areas of interest will suffice. Alternatively, you could cite the projects from our research groups that you are interested in. 



Conferences and seminars

Postgraduate students are encouraged to participate in the many school research-related activities, and attend appropriate national and international conferences. We also offer a diverse range of research seminars, which you are welcome to attend.

Maths research students also have access to specialised mathematical training courses such as those provided by the Academy for PhD Training in Statistics (APTS) or the Mathematics Access Grid Conferencing (MAGIC) group who run courses using videoconferencing technology.

Furnished offices

Each maths research student has a share of a furnished office with their own desk and personal computer or laptop and may make full use of general school facilities.

IT facilities

The University provides excellent computing facilities with access to specialist mathematical software.

The school also has dedicated postgraduate workrooms for both masters and research maths students that are also accessible 24-hours a day. The computers have electronic links to a high performance computing facility and specialist mathematical software.


Postgraduate students have access to the University's libraries including the £18m redeveloped George Green Library which supports the Faculty of Science.

Postgraduate maths student


Research support

A number of University support services exist to assist you during your time at Nottingham and beyond.

Students' Union

The Students' Union are a particularly important source of support.

Centre for English Language Education

Accredited by the British Council for the teaching of English in the UK, our Centre for English Language Education provides high-quality preparation and English language support, as well as a social programme for its students.


Find a supervisor

We encourage you to get in touch with a member of academic staff about your interests.

You do not need to submit a detailed proposall, a short overview indicating your areas of interest will suffice. Alternatively you could cite the projects from our research section that you are interested in.

See a list of projects on our school site. 

Details of research supervisors at the University can be found on our research A to Z.


Fees and funding

UK/EU students

Externally-funded studentships currently available include those from Engineering and Physical Sciences Research Council (EPSRC), and Biotechnology and Biological Sciences Research Council (BBSRC).

Details of eligibility for funding, and which research topics the funding can be used to support, can be found by following the links above. In addition, we have funding available direct from the school and the University of Nottingham

The University Graduate School operates two schemes of its own to help support current postgraduate research: 

The Graduate School holds a list of other sources of fundingStudentship opportunities are also available.

Government loans for doctoral study

The Government recently introduced doctoral student loans of up to £25,000 for PhDs and equivalent research programmes. Applicants must ordinarily live in England.

Doctoral training programmes

Linked to research councils, doctoral training programmes offer funding opportunities connected to our research priorities.

International and EU students

Research scholarships are available for outstanding international and EU students. You must already have an offer to study at Nottingham to apply. Please note closing dates to ensure your course application is submitted in good time.

Information and advice on funding your degree, living costs and working while you study is available on our website, as well as country-specific resources.



Studying a masters or PhD in mathematics can set you apart from the competition when it comes to finding employment on your chosen career path.

MSc graduates often pursue roles in

  • industry, business, commerce
  • medical research, pharmaceuticals
  • PhD study

PhD graduates are also in great demand for careers in university teaching or working for banks and financial institutions.

Average starting salary and career progression

In 2017, 100% of postgraduates in the school who were available for employment had secured work or further study within six months of graduation. The average starting salary was £30,800 with the highest being £60,000.*

* Known destinations of full-time home postgraduates 2016/17. Salaries are calculated based on the median of those in full-time paid employment within the UK.

Career prospects and employability

The University of Nottingham is consistently named as one of the most targeted universities by Britain’s leading graduate employers* and can offer you a head-start when it comes to your career.

Our Careers and Employability Service offers a range of services including advice sessions, employer events, recruitment fairs and skills workshops – and once you have graduated, you will have access to the service for life.

* The Graduate Market 2013–2017, High Fliers Research.


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This online prospectus has been drafted in advance of the academic year to which it applies. Every effort has been made to ensure that the information is accurate at the time of publishing, but changes (for example to course content) are likely to occur given the interval between publishing and commencement of the course. It is therefore very important to check this website for any updates before you apply for the course where there has been an interval between you reading this website and applying.

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Professor Kewei Zhang
School of Mathematical Sciences
University of Nottingham
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