university of manchester phd mathematics

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PhD Pure Mathematics / Entry requirements

Year of entry: 2024

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Academic entry qualification overview

The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree in a discipline directly relevant to the PhD (or international equivalent) OR any upper-second class (2:1) honours degree and a Master’s degree at merit in a discipline directly relevant to the PhD (or international equivalent).

Other combinations of qualifications and research or work experience may also be considered. Please contact the admissions team to check.

English language

All applicants will need to demonstrate competency in English language.  Applicants who do not already possess an acceptable English Language qualification will need to take a recognised test and attain a minimum IELTS 6.5 overall with a minimum of 6 in writing and listening, and 5.5. in all other sub-tests. TOEFL iBT:  At least 90 overall with no subtest below 20. We do not accept 'MyBestScore'.

Pearson Test of English (PTE): At least 70 overall with no subtest below 59. Further information on language requirements can be found on our website . 

Pre-sessional English: We also accept successful completion of a pre-sessional English course run by the University Language Centre to meet our English language requirements.  

Other international entry requirements

Manchester Mathematical Modelling in Science & Industry

Welcome to the homepage of MMMSI. This initiative is a community for PhD students in the Department of Mathematics at the University of Manchester whose project involves mathematical modelling and engagement with researchers based in another department or in industry.

The directors of MMMSI are Prof. Will Parnell and Dr Marcus Webb .

PhD studentships for September 2021

• You can also become a part of MMMSI through the normal PhD application channels in the Department of Mathematics, if the PhD project and supervisory team is appropriate for MMMSI.
• Provide PhD students with training opportunities via internal seminars, journal reading groups, and participation in events such as modelling challenges or industrial problem events

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PhD Mathematical Logic University of Manchester

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Qualification.

PhD/DPhil - Doctor of Philosophy

The University of Manchester

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Course summary

Programme description

The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.

Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.

Opportunities for PhD research are available in a wide range of topics in Mathematical Logic. For some of the available areas of possible PhD research see the current Mathematical Logic research areas . For more details about the specific projects, please contact the relevant individual members of staff or get in touch with the Logic Postgraduate Admissions Tutor .

Students may enter our graduate programme in Mathematical Logic by initially taking our taught M.Sc. course over 1 year. This will provide an introduction to mathematical logic and, subject to satisfactory progress, can lead to admission to the PhD programme.

Tuition fees

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£ 28,000 per year

Tuition fees shown are for indicative purposes and may vary. Please check with the institution for most up to date details.

University information

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The University of Manchester, Oxford Road, Manchester, Manchester, M13 9PL, England

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We have 7 The University of Manchester, Department of Mathematics PhD Projects, Programmes & Scholarships

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The University of Manchester, Department of Mathematics PhD Projects, Programmes & Scholarships

Simulation-based inference for financial econometrics models, phd research project.

PhD Research Projects are advertised opportunities to examine a pre-defined topic or answer a stated research question. Some projects may also provide scope for you to propose your own ideas and approaches.

Competition Funded PhD Project (Students Worldwide)

This project is in competition for funding with other projects. Usually the project which receives the best applicant will be successful. Unsuccessful projects may still go ahead as self-funded opportunities. Applications for the project are welcome from all suitably qualified candidates, but potential funding may be restricted to a limited set of nationalities. You should check the project and department details for more information.

Modelling and Computation for Radiation Chemistry at Interfaces

Lie algebra actions on noncommutative rings, data fusion of x-ray imaging with magnetic induction tomography, funded phd project (students worldwide).

This project has funding attached, subject to eligibility criteria. Applications for the project are welcome from all suitably qualified candidates, but its funding may be restricted to a limited set of nationalities. You should check the project and department details for more information.

Connections between Numerical Analysis of Differential Equations and Machine Learning

Funded phd project (uk students only).

This research project has funding attached. It is only available to UK citizens or those who have been resident in the UK for a period of 3 years or more. Some projects, which are funded by charities or by the universities themselves may have more stringent restrictions.

Algebraic groups with finitely many unipotent conjugacy classes

The university of manchester - department of mathematics, funded phd programme (students worldwide).

Some or all of the PhD opportunities in this programme have funding attached. Applications for this programme are welcome from suitably qualified candidates worldwide. Funding may only be available to a limited set of nationalities and you should read the full programme details for further information.

Maths Research Programme

PhD Research Programmes describe the opportunities for postgraduate research within a University department. You may often be asked to submit your own research project proposal as part of your application, although predefined research projects may also be available.

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PhD Studentship: Advancing Joint Modelling Methodology for Clinical Prediction Models

The university of manchester - mathematics.

This 3.5 year PhD project is funded by the MADSIM Project, https://www.madsim.manchester.ac.uk/ . Funding is for Home students and EU students with settled status. Funding covers home tuition fees and provides a stipend at the UKRI rate (£19,237 for 2024/25). The start date is 1 st October 2024.

This is an interdisciplinary project between the Department of Mathematics (Christiana Charalambous, Timothy Waite) and the Centre for Health Informatics (David Jenkins). Clinical prediction models (CPMs) are algorithms that use information about a patient at a given time point to generate risk estimates for an outcome. These models are widely adopted throughout healthcare and can be used to inform clinical decisions, for example, if an individual should receive an intervention. Traditionally, CPMs have used data from a single time point and often consider a single outcome. However, the adoption of electronic health records and the increase in availability of data provides rich longitudinal (e.g. repeatedly measured biomarkers) and time-to-event (e.g. death or disease progression) data, which are often underutilised. Complex models that use multi-outcome (potentially correlated) data, such as joint models, are increasingly being adopted and evidence from the literature suggests this could improve predictive accuracy and in turn patient outcomes. Nonetheless, there are strong assumptions required and a lack of methodological development for healthcare usage. In addition, it is unclear when repeated longitudinal measurements should be recorded. Objectives and outcomes of the project:

• Review the existing literature on joint modelling for clinical prediction.
• Undertake methodological development for the formulation and validation of joint models, testing the developed methods throughout a range of scenarios in simulated and real-world health data.
• Extend time-dependent AUC methodology for joint model validation.
• Perform simulation studies to compare the proposed method to existing approaches for prediction and model validation.
• Extend current minimum sample size methodology for joint models.
• Undertake simulation studies to investigate the impact of longitudinal measurement frequency on joint model performance.
• Investigate techniques for Bayesian design of experiments in joint models.

The combination of simulation and real-world data will allow us to evaluate the methods under a range of scenarios and parameter combinations and assess the real-world impact of the methods. The collaboration with the Centre for Health Informatics will provide access to real-world health data, such as the Greater Manchester care record, UK Biobank and cardiovascular data from the Manchester University NHS Foundation Trust and Wythenshawe Hospital, that the centre regularly utilises. The project will also provide recommendations to determining when to monitor patients and guidance for developing and validating joint models for clinical prediction.

Applicant’s should have:

• Obtained or working towards a 1st class degree in Mathematics (BSc/MMath) or Distinction level Masters in (Bio)Statistics, Data Science or similar.
• Research experience, e.g. UG/MSc project, research internship or other.
• Background in some or all of the following: longitudinal data analysis, survival analysis, design of experiments, Bayesian statistics.
• Good programming skills in a language such as R or Python
• Good communication skills (oral and written)
• Openness to working across disciplines

Before you apply, please contact Dr Christiana Charalambous at [email protected] .

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MSc Pure Mathematics / Overview

Year of entry: 2024

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Full entry requirements

Apply online

As there is high demand for this course we operate a staged admissions process with selection deadlines throughout the year.

Due to the competition for places, we give preference to students with grades above our minimum entry requirements. If we make you an offer, you will have 3 weeks in which to accept. Any offers not accepted within 3 weeks will be withdrawn so that an offer can be made to another candidate. 

Please find more information in our Application and Selection section .

Course options

Course description

This one year taught postgraduate programme leads to the degree of MSc in Pure Mathematics. The programme is suitable not only for students who wish to improve their background knowledge prior to applying to undertake a PhD by research, but also for students who wish to enhance their knowledge of postgraduate-level abstract mathematics.  The MSc comprises of the taught component, running from the start of the academic year in September until the end of the second semester in late Spring, followed by the dissertation component running from June until September.

During the taught component of the course, you will normally take six units together with a research skills and project unit.  You may choose exclusively pure topics, or a mixture of pure mathematics units and mathematical logic units. In the research skills and project unit you will first complete a mini-project to improve mathematical writing skills, and then undertake a project. The project is normally an expository account of a piece of mathematics and you will write this under the guidance of a supervisor. The taught component comprises of conventional lectures supported by examples classes, project work and independent learning via reading material.

After successfully completing the taught component, you will prepare a dissertation on an advanced topic in pure mathematics or mathematical logic, normally of current or recent research interest, chosen in consultation with your supervisor. 

You can also take the programme part-time, over a period of two years.  There is some flexibility in the precise arrangements for this programme, but you would normally attend two lecture courses each semester for three semesters before commencing work on your dissertation.

Within two weeks from the start of the academic year, students in this programme can opt to change to the Mathematical Logic Pathways leading to the degree of MSc in Pure Mathematics and Mathematical Logic.

For entry in the academic year beginning September 2024, the tuition fees are as follows:

• MSc (full-time) UK students (per annum): £14,000 International, including EU, students (per annum): £30,000

Further information for EU students can be found on our dedicated EU page.

Policy on additional costs

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

Scholarships/sponsorships

For more information, see the Department of Mathematics Fees and funding  page or visit the University of Manchester  funding for master's courses website for more information.

Contact details

Our internationally-renowned expertise across the School of Natural Sciences informs research led teaching with strong collaboration across disciplines, unlocking new and exciting fields and translating science into reality.  Our multidisciplinary learning and research activities advance the boundaries of science for the wider benefit of society, inspiring students to promote positive change through educating future leaders in the true fundamentals of science. Find out more about Science and Engineering at Manchester .

Courses in related subject areas

Use the links below to view lists of courses in related subject areas.

• Mathematics

Regulated by the Office for Students

The University of Manchester is regulated by the Office for Students (OfS). The OfS aims to help students succeed in Higher Education by ensuring they receive excellent information and guidance, get high quality education that prepares them for the future and by protecting their interests. More information can be found at the OfS website .

You can find regulations and policies relating to student life at The University of Manchester, including our Degree Regulations and Complaints Procedure, on our regulations website .

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Department of Mathematics

Funding for postgraduate research

There are a variety of funding options for postgraduate research programmes in the Department of Mathematics.

All PhD projects are eligible for funding via a variety of scholarships.

The scholarships listed are awarded competitively among the eligible applicants by the relevant postgraduate funding committees.

Please note that academics regularly apply for research grants and may therefore be able to offer funding for specific projects without requiring approval from these committees.

Explore how you can make the next steps towards your academic or industry career by exploring the postgraduate funding options below.

UK PhD students

The arrangements for supporting postgraduate study are completely different from those for undergraduates. Typically, finance for grants comes from the Government, via the Engineering and Physical Science Research Council (EPSRC) or via one of the other Research Councils. These grants are called studentships and are not means tested. Each research council studentship covers all course fees of £4,712 per annum (2023/24) and provides a stipend for maintenance of £18,622 per annum (2023/24). Studentships are awarded for three and a half years, subject to satisfactory progress.

The financing of EPSRC doctoral training studentships is now administered by the Departments. In order to be awarded such a studentship an applicant will typically need to have a first class honours degree or possibly a good upper second. It is not necessary to make a separate application for these studentships. All UK students who apply for admission to a PhD programme in the Department will automatically be considered for such funding. Applicants awarded EPSRC funds are required by EPSRC to satisfy certain formal UK/EU residence criteria.

We also have a number of Department scholarships available to UK and overseas students which cover course fees and, for UK students, a stipend of £18,622 per annum (2023/24) to cover living costs. These scholarships are awarded for three and a half years, subject to satisfactory progress.

Doctoral loans

Doctoral loans  are available for those studying PhDs (and equivalent doctoral programmes). The amount you'll get isn't based on you or your family's income (Non-means tested) and is available for all types of doctoral study (eg PhD, as well as all professional doctorates such as DBA, EdD). The loan is paid directly to you in three instalments across each year of the course and can be used for course fees and living costs.

For more information, to check your eligibility and find out how to apply,  visit our postgraduate loans for doctoral students page , email  [email protected]  or visit the  GOV.UK  website.

EPSRC Doctoral Training Partnership (DTP) in Mathematical Sciences

Our EPSRC DTP Studentships enable exceptional postgraduate research candidates to typically undertake a 3.5-year research project in the field of mathematical sciences. Each year we have a number of studentships (fees plus standard UKRI stipend) available for application. Overseas applicants are eligible for these studentships. 

How to apply

All applicants who do not have their own funding are automatically considered for all funding schemes they are eligible for.

Our Graduate Development Training Programme is designed to provide research and transferable skills linked to academic progression, personal development and future employability.

The training programme includes discipline-focused training as well as essential workshops addressing areas such as research integrity, research conduct and intellectual property. We also offer a range of transferrable skills courses to support your development as a researcher. All training is provided on a needs-based model.

Early career researchers undertake an annual skills audit to identify gaps in their skills appropriate to their stage of study. They can then be directed to the appropriate training programme to address their needs.

Student support

Students benefit from a wide range of support initiatives ensuring that both their academic and pastoral needs are addressed. PGR Life helps you to make the most of your time as a postgraduate researcher, supporting your research and planning for your future.

For queries regarding the PhD application process, available projects and eligibility please contact:

• Email: [email protected]
• Tel: +44 (0)161 275 0178

EU PhD students

View the latest information for prospective students from the EU .

We have a number of Departmental scholarships available which EU students can apply for, these cover all course fees and provide a stipend of £18,622 per annum (23/24) for living costs. These scholarships are awarded for three and a half years, subject to satisfactory progress.

Heilbronn Doctoral Partnership (HDP)

The Heilbronn Doctoral Partnership (HDP) is a partnership of the Universities of Bristol, Manchester and Oxford with the Heilbronn Institute for Mathematical Research (HIMR) . It funds PhD studentships in Discrete Mathematics, interpreted in its broadest sense, which includes most areas of Pure Mathematics, Computational Statistics, Data Science, Probability and Quantum Information.

In addition to undertaking cutting-edge research in one of the partner universities, students on this programme will have the opportunity to spend nine weeks each summer (in years 1-3) at the HIMR, where they will contribute to the classified research activities of the Institute. However, working at HIMR is not a requirement of the HDP programme. Successful candidates who wish to work at HIMR must satisfy security vetting; UK nationals will normally be able to meet this condition.

Students from traditionally under-represented groups are strongly encouraged to apply.

To apply, please submit a standard PhD application in a relevant area, indicating that you wish to be considered for HDP funding.

International PhD students

Annually, we often can allocate a number of Overseas Department Scholarship Awards. These awards normally constitute tuition fees (£26,500 per annum for 2023/24) and are awarded competitively. These scholarships are awarded for three years or three and a half years, subject to satisfactory progress.

Chinese students may be eligible for the China Scholarship Council awards .

Faculty of Science and Engineering funding

The Faculty of Science and Engineering also offer a number of awards and scholarships to support postgraduate researchers:

President's Doctoral Scholar (PDS) Award

The President’s Doctoral Scholar (PDS) Award is the University’s flagship funding initiative, and is strongly supported by the University's President and Vice Chancellor, Professor Dame Nancy Rothwell.

The award is open to all nationalities and supports students of the highest calibre, who have the commitment and desire to work on challenging research projects in world-leading research environments.

For further information, please visit the  PDS website .

Dean's Doctoral Scholarship

The Dean's Doctoral Scholarship also supports researchers of the highest calibre from any country. It provides researchers with the chance to make a difference and become part of an innovative community whose work changes lives

For further information, please visit the Dean's Doctoral Scholarship page.

Postgraduate Research Teaching Associate (PGRTA) Scholarship

The PGRTA Scholarship is open to UK researchers and allows students to tackle areas of key research, whilst developing teaching experience and earning a combined teaching salary and stipend.

For further information, please visit the  PGRTA page.

In addition to these three awards, there is other postgraduate research funding available at faculty level which can be found on the Faculty of Science and Engineering funding page .

Alternative sources of funding

Students who are not offered EPSRC or Department funding but who still wish to study here may choose to support themselves. Many students choose to do this and support may be available from a variety of sources, for example by taking out a career development loan .

Alternative funding sources are also available. The University has a subscription to The Alternative Guide to PG Funding . This published guide and online resource provides a comprehensive overview of how to find and apply to alternative sources of funding, particularly charities, that make awards to current and prospective postgraduate students.

Funded projects in mathematical modelling

Join a community of postgraduate researchers focusing on mathematical modelling, AI and big data as well as engagement with researchers based in other departments or in industry.

Research projects

Explore the research projects available to postgraduate students in our Department.

Explore our projects

Faculty funding

Discover the funding options offered by the wider Faculty of Science and Engineering.

President's Doctoral Scholar Award

The University’s flagship postgraduate funding scheme, supported by Dame Nancy Rothwell.

Office of the Registrar

Mathematics (Option II Graduate School Preparation) - Bachelor of Science 2024-25 Degree Map

• General Education Reqs
• Degree Reqs
• University Writing Reqs
• Race and Ethnic Studies Reqs
• Credit Reqs & Limits

Degree Maps & Advising

Degree maps show just one possible course plan for how to earn a degree in four years. To graduate in four years, you need an average of 45 credits per year to reach 180 total credits, including 62 upper division (300/400 level courses) credits. You may need to take electives (courses that are not used to meet specific requirements) to reach those totals. Most courses at PSU are 4 credits each. Talk with your advisor regularly to develop a personalized plan that works best for you, especially if you have any earned transferrable academic credit.

Bulletin or Catalog Year

This guide outlines the requirements in the 2024-25 Bulletin . Bulletin or Catalog year defines the specific set of requirements you need to complete a degree. Requirements occasionally change, so it may benefit you to follow a different catalog. Typically, your catalog year is the year you began at PSU, but if you are a transfer student you may be able to use an earlier catalog. Review PSU's course catalog policy and talk with your advisor about which catalog requirements make sense for you. 

DARS System  

Students are highly encouraged to run their own Degree Audit Reporting System (DARS) report in myPSU to track their progress through their program. You'll need to know which catalog year you'd like to use for your audit request. Students are also encouraged to contact an advisor if they have any questions about their DARS, catalog/bulletin year, or degree requirements in general. DARS audits are advising tools and are not an official record of earning a degree, major, minor, or certificate. You must apply to have a degree or certificate awarded.

First Year Milestones : Complete MTH 252 and MTH 261(by end of Spring term). 

Second Year

Second Year Milestones : Complete MTH 256(by Spring term).

Third Year Milestones : Complete MTH 311(by Spring term). 

Fourth Year

Fourth Year Milestones : Complete 400 level sequence(by Spring term). 

Additional Map Notes:

• Race and Ethnic Studies Requirement : Two courses required, each minimum 3 credits: one course must focus on the U.S. experience; the second course must center the experiences of groups under European and U.S. colonialism and imperialism beyond the United States. One must be within the School of Gender, Race and Nations or be cross-listed with one of the SGRN units: Black Studies (BST), Chicanx/Latinx Studies (CHLA), Indigenous Nations Studies (NAS), or Women, Gender and Sexuality Studies (WS). If you choose a RESR course that can be applied towards another requirement, take an elective here instead.
• Sophomore Inquiry or Honors : For University Studies, two Sophomore Inquiry courses are required, but for Honors, three Sophomore courses are required.
• MTH/STAT Upper Division Elective : MTH 300 recommended. See department for approved electives list
• MTH 4XX Sequence : See department for  approved sequences list.
• MTH 4XX Elective : See department for  approved electives list.
• You may need to add additional general elective credits to your plan to reach 180 total credits.

Resources for Mathematics