maths phd edinburgh

A world-class PhD training programme which equips students with the skills and confidence to lead their discipline .

TRAINING FOR LEADERSHIP

Skills in analysis and data-driven, computational modelling

Collaborative projects focus

Responsible research and innovation

Entrepreneurship training

About the mac-migs programme.

MAC-MIGS students have a world of opportunity for developing and applying mathematics in collaboration with scientists and engineers, interacting with over 30 industrial and governmental partners, and visiting international collaborators around the world.  MAC-MIGS is not just PhD instruction, it is world-class PhD training that equips students with the skills and confidence to lead their discipline.

We accept applications from students interested in a wide variety of areas of mathematics, including mathematical modelling, applied and pure ordinary and partial differential equations, calculus of variations, numerical analysis, pure and applied statistics, machine learning, inverse problems, fluid dynamics, stochastic analysis, and any interdisciplinary topic involving these branches of mathematics and their interplay with other fields such as biology, chemistry, engineering and physics.

WHAT TO EXPECT FROM THE PROGRAMME

MAC-MIGS is a joint PhD programme of the University of Edinburgh and Heriot-Watt University, leading to the award of a joint degree from both universities. During your first year as a student, you are based at the new Bayes Centre in central Edinburgh, where you take courses and carry out group and individual projects, often involving industrial or government partners. Towards the end of the year, you are matched with a PhD project proposed by MAC-MIGS supervisors.

There are opportunities for interdisciplinary internships and periods of time spent in industry or with one of the overseas academic partners in our global network, including Brown University; Duke University; Ecole des Ponts; Norwegian University of Science & Technology; University of Potsdam; University of Turin; Technology University of Berlin; Vienna University of Technology; Utrecht University and the Technical University of Denmark. 

As a MAC-MIGS student, you study topics such as mathematical modelling, computational mathematics, analysis of ordinary, partial and stochastic differential equations, optimal transport theory, statistical methods and applied probability, optimisation, calculus of variations, high performance computing, data analytics (e.g. machine learning), and uncertainty quantification.

MAC-MIGS staff are well aware of the need to address state of the art challenges in data-driven mathematical modelling; our colleagues in the Bayes Centre , EPCC and the Alan Turing Institute will assist us in giving you access to advanced skills to future-proof your education and prepare you for leadership in this rapidly evolving field.

Students in the programme are trained to work on interdisciplinary projects – analysing equations, identifying relevant model structures and testing computational methods in a real-world setting. Our training is outward facing and involves participation of researchers from the sciences and engineeering, as well as industry partners. The training also includes topics in support of research and future employment, presentation skills, entrepreneurship, and responsible innovation.

A PHD WITH INTEGRATED STUDY IN MATHEMATICAL MODELLING, ANALYSIS AND COMPUTATION

The PhD programme will provide a broad training that cuts across disciplinary boundaries to include mathematical analysis – pure, applied, numerical and stochastic – data-science and statistical techniques and the domain-specific advanced knowledge necessary for cutting-edge applications.

Students on this integrated degree will join the broader Maxwell Institute Graduate School in its Bayes Centre. They will benefit from dedicated academic training in subjects that include mathematical analysis, computational mathematics, multi-scale modelling, model reduction, Bayesian inference, uncertainty quantification, inverse problems and data assimilation, and machine learning; extensive experience of collaborative and interdisciplinary work through projects, modelling camps, industrial sandpits and internships; outstanding early-career training, with a strong focus on entrepreneurship; a dynamic and forward-looking community of mathematicians, scientists and engineers, sharing strong values of collaboration, respect, and social and scientific responsibility.

The students trained on this programme will have expertise in a broad array of mathematical modelling techniques and of their application in multidisciplinary contexts as well as experience of industrial collaboration. 

Their skills will be highly valuable for all sectors of business and government as is reflected by our wide network of industry and agency partners covering manufacturing, energy, finance, healthcare, digital technologies, and environmental protection. Immediate benefits to these sectors will be realised through the projects carried out by the students in collaboration with these partners; long-term benefits will be achieved throughout the students’ careers, as they take up leadership positions and influence the future of their sectors.

Industrial and agency partners will provide internships, development programmes and research projects, and help maximise the impact of the students’ work. Our collaborations with academic partners representing leading institutions in Europe and in the US, will provide further opportunities for collaborations and research visits. 

The students will integrate into a vibrant research environment, closely interacting with many academics drawn from the faculties of both Edinburgh and Heriot-Watt Universities.

MAC-MIGS PROGRAMME STUDIES

All students are based in the Bayes Centre during their first year. Students take around 180 credits of study, divided as follows:

• A 15 credit module on Computational Methods for Data-Driven Modelling
• A 15 credit module on Mathematical Modelling and Applied Analysis
• 60-70 units of additional approved coursework (typically at Master’s level) from the Scottish Mathematical Sciences Training Centre, or the two universities, to be agreed with the Cohort Director
• A 15 credit Group Project
• A 15 credit Group Project on an industry or government-relevant theme
• A 60 credit Extended Individual Project

In addition, during the first year, students engage in a Modelling Camp, an Industrial Sandpit, training in and Presentation Skills. Near the end of their first year, all students participate in the 3-day Residential Camp, held at Millport on the Isle of Cumbrae or at The Burn, a large country house in the Scottish Highlands where they gain practice in presentation, hear science lectures and talks on responsible research and innovation, and learn about state of the art industrial challenges.

Years Two-Four

In the second year, students are based in the same building as their supervisor, at one of the campuses of Edinburgh and Heriot-Watt Universities. The work is focussed on the student’s agreed research project, which is typically supervised by a team including staff from both Edinburgh and Heriot-Watt Universities. Students are expected to take around 20 credits of academic coursework in each year. They are also required to participate in MAC-MIGS Skills and Citizenship courses which provide training in responsible research and innovation, equality, diversity and inclusion, project and time management, and other important topics. They also obtain specific instruction in Entrepreneurship.

Students return frequently to the Bayes Centre throughout their studies, for cohort activites, for seminars, for research group meetings and study groups, and to attend workshops at ICMS.

Taster Projects

First-year students work on two taster projects of their choice – one each term – in groups of three or four students. This is a way to get a flavour of the different research areas and to challenge themselves with more advanced material. 

Projects in 2021/22

• Machine Learning for new Numerical Methods in Viscoelastic Fluid Dynamics
• Optimal Infrastructure Planning for Large Scale  CO 2 removal from the Atmosphere
• What is a Quantum Annealer?
• Baysian Inference of the Double-Glazing Model (IBM Research)
• Comparison between DEM and NSDEM (EDEM)
• Understanding the Public Health Waiting Times Landscape in Scotland: Finding Key Drivers and Forecasting Demand (Public Health Scotland)
• Performance Validation using Reference Turbines (Ventient Energy)
• Model Selection by Simulation (Moody’s Analytics)
• Reconciling Robustness and Interpretability in Machine Learning (Moody’s Analytics)

Projects in 2020/21

• Reinforcement Learning via Relaxed Stochastic Control Approach
• Rare event Simulation
• Quantifying Uncertainty in Chaotic Systems using Multi-Level Monte Carlo Methods
• Computational Optimal Transport and Modern Methods of Optimisation
• Deep Learning in Computational Imaging
• Classification of Self-Assembled Structures using Machine Learning 
• Defining Good Neighbours: Modelling Root Traits for Beneficial Plant-Plant Interactions
• Validating Hidden Markov Models as Tools to Identify Seabird Foraging Areas
• Constructing Land Valuation Models to Find Profitable Investments
• Fluid Damping Model for Wave Energy Conversion

Projects in 2019/20

• Modelling Opinion Dynamics
• Control Variates for Path Tracing
• Dispersion in Random Flows: Homogenisation and beyond
• Modelling and Simulation of Multiscale Stochastic Systems in Living Cells
• Deep Learning in Imaging Inverse Problems and Application to Medicine
• Efficient Approximation of Pricing Functions
• Machine Learning for the Prediction of Battery Life
• Stochastic Plant Roots Growth in Granular Media

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Welcome to GlaMS PhD training Centre, a joint PhD training centre between the University of Glasgow , the University of Edinburgh and Heriot-Watt University .

Our 60+ supervisors cover the full range of Algebra Structures, across Algebra, Mathematical Physics, and Geometry & Topology, and we train within the remit of algebraic methods, interpreted broadly.  We offer innovative training , including courses and working seminars in the first year, group projects, and 3-month placements.  For more information, please consult About , and our FAQs .

We have now completed four intakes of students. We very strongly encourage applications from across the mathematical sciences, and across traditional boundaries.  For more details, see our Apply page.

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Undergraduate Study

Postgraduate taught programmes.

• Postgraduate Research Studies
• School of Mathematics

Studying Here

Find out about our undergraduate, postgraduate taught and postgraduate research programmes.

Research Degrees (PhD)

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• Postgraduate study

Postgraduate research opportunities A-Z

• Mathematics
• Staff research interests search

Postgraduate research  

Mathematics PhD/iPhD/MPhil/MSc (Research)

The School of Mathematics & Statistics combines world-leading research expertise in pure and applied mathematics and statistics in areas such as mathematical biology, fluid dynamics, geometry and topology and environmental statistics.

Research projects

• Mathematics thesis topics

With a sizeable complement of academic staff and postgraduate students, the School of Mathematics and Statistics is one of the largest in the UK. 

Our research interests cover several areas of Pure and Applied Mathematics. These areas are not mutually exclusive and there are considerable benefits from interactions between the different areas that enhance the research environment.

All our research areas are highly rated internationally and most members of the School have ongoing collaborations with mathematicians overseas and elsewhere in the UK. Many overseas mathematicians spend periods in Glasgow working with members of the School.

There are several series of research seminars with invited speakers from the UK and overseas. There is also a regular postgraduate seminar where our PhD students can develop their presentational skills. All this helps to generate a lively and mutually supportive research environment, which has led to the award of coveted prizes and competitive fellowships for several of our younger staff.

Research areas

• Continuum Mechanics
• Geometry and Topology
• Geophysical & Astrophysical Fluid Dynamics
• Integrable Systems and Mathematical Physics
• Mathematical Biology
• Number Theory

Study options

Integrated phd programme (4 years).

Our PhD with Integrated Study in Mathematical Sciences   is a four-year PhD programme in the School of Mathematics and Statistics.

Completion of taught Masters level courses in the first nine months will provide you with a valuable introduction to academic topics and research methods, whilst providing key training in the critical evaluation of research data.

Upon successful completion of the taught component, you will progress to your research degree. You will submit a thesis to be examined by the end of your fourth year.

Entry requirements

Phd programmes.

2.1 Honours degree or equivalent

Required documentation

Applicants should submit:

• Transcripts/degree certificate 
• Two references
• Name of potential Supervisor

Integrated PhD programmes

2.1 Honours degree or international equivalent in a relevant subject area

English language requirements

For applicants whose first language is not English, the University sets a minimum English Language proficiency level.

International English Language Testing System (IELTS) Academic module (not General Training)

• 6.5 with no subtests under 6.0
• Tests must have been taken within 2 years 5 months of start date. Applicants must meet the overall and subtest requirements using a single test.

Common equivalent English language qualifications accepted for entry to this programme:

Toefl (ibt, my best or athome).

• 79; with Reading 13; Listening 12; Speaking 18;Writing 21
• Tests must have been taken within 2 years 5 months of start date. Applicants must meet the overall and subtest requirements , this includes TOEFL mybest.

Pearsons PTE Academic

• 59 with minimum 59 in all subtests

Cambridge Proficiency in English (CPE) and Cambridge Advanced English (CAE)

• 176 overall, no subtest less than 169

Oxford English Test

• Oxford ELLT 7
• R&L: OIDI level no less than 6 with Reading: 21-24 Listening: 15-17
• W&S: OIDI level no less than 6

Trinity College Tests

Integrated Skills in English II & III & IV: ISEII Distinction with Distinction in all sub-tests.

University of Glasgow Pre-sessional courses

Tests are accepted for 2 years following date of successful completion.

Alternatives to English Language qualification

• students must have studied for a minimum of 2 years at Undergraduate level, or 9 months at Master's level, and must have complete their degree in that majority-English speaking country  and  within the last 6 years
• students must have completed their final two years study in that majority-English speaking country  and  within the last 6 years

For international students, the Home Office has confirmed that the University can choose to use these tests to make its own assessment of English language ability for visa applications to degree level programmes. The University is also able to accept UKVI approved Secure English Language Tests (SELT) but we do not require a specific UKVI SELT for degree level programmes. We therefore still accept any of the English tests listed for admission to this programme.

Pre-sessional courses

The University of Glasgow accepts evidence of the required language level from the English for Academic Study Unit Pre-sessional courses. We also consider other BALEAP accredited pre-sessional courses:

• School of Modern Languages and Cultures: English for Academic Study
• BALEAP guide to accredited courses

Fees and funding

• UK: £4,786
• International & EU: £30,240

Prices are based on the annual fee for full-time study. Fees for part-time study are half the full-time fee.

Irish nationals who are living in the Common Travel Area of the UK, EU nationals with settled or pre-settled status, and Internationals with Indefinite Leave to remain status can also qualify for home fee status.

• Fee status and policies

Alumni discount

We offer a 20% discount to our alumni on all Postgraduate Research and full Postgraduate Taught Masters programmes. This includes University of Glasgow graduates and those who have completed Junior Year Abroad, Exchange programme or International Summer School with us. The discount is applied at registration for students who are not in receipt of another discount or scholarship funded by the University. No additional application is required.

Possible additional fees

• Re-submission by a research student £540
• Submission for a higher degree by published work £1,355
• Submission of thesis after deadline lapsed £350
• Submission by staff in receipt of staff scholarship £790

Depending on the nature of the research project, some students will be expected to pay a bench fee (also known as research support costs) to cover additional costs. The exact amount will be provided in the offer letter.

• View a full list of our  current scholarships

Our postgraduate students join a community of academic experts across a wide range of pure and applied mathematics and statistics and develop a mature understanding of fundamental theories and analytical skills applicable to many solutions.

There is a lively seminar program with members of the Schools involved in a number of networks including the North British Functional Analysis Seminar, The North British Differential Equations Seminar and the LMS Network on Classical and Quantum Integrability.

The School also welcomes many guest speakers from the UK and overseas. We hold regular postgraduate seminars, where our PhD students can develop their presentation skills, and international conferences and workshops.

Students can also attend our hugely popular week-long training course each year, which provides practically motivated training in key statistical analysis and modelling skills.

You will be part of a Graduate School which provides the highest level of support to its students.

The overall aim of our Graduate School is to provide a world-leading environment for students which is intellectually stimulating, encourages them to contribute to culture, society and the economy and enables them to become leaders in a global environment.

We have a diverse community of over 750 students from more than 50 countries who work in innovative and transformative disciplinary and interdisciplinary fields. An important part of our work is to bring our students together and to ensure they consider themselves an important part of the University’s academic community.

Being part of our Graduate School community will be of huge advantage to you in your studies and beyond and we offer students a number of benefits in addition to exceptional teaching and supervision, including:

• A wide-ranging and responsive research student training programme which enables you to enhance your skills and successfully complete your studies.
• Mobility scholarships of up to £4000 to enable you to undertake work in collaboration with an international partner.
• A diverse programme of activities which will ensure you feel part of the wider-research community (including our biannual science slam event).
• A residential trip for all new research students.
• The opportunity to engage with industry-partners through training, placements and events.
• Professionally accredited programmes.
• Unique Masters programmes run in collaboration with other organisations.
• State-of-the-art facilities including the  James Watt Nanofabrication Centre  and the  Kelvin Nanocharacterisation Centre .
• Highly-rated support for international students.

Over the last five years, we have helped over 600 students to complete their research studies and our students have gone on to take up prestigious posts in industries across the world.

Email:  [email protected]

How to apply

Identify potential supervisors.

All Postgraduate Research Students are allocated a supervisor who will act as the main source of academic support and research mentoring. You may want to identify a potential supervisor and contact them to discuss your research proposal before you apply. Please note, even if you have spoken to an academic staff member about your proposal you still need to submit an online application form.

You can find relevant academic staff members with our staff research interests search .

Gather your documents

Before applying please make sure you gather the following supporting documentation:

• Final or current degree transcripts including grades (and an official translation, if needed) – scanned copy in colour of the original document.
• Degree certificates (and an official translation, if needed): scanned copy in colour of the original document.
• Two references on headed paper and signed by the referee. One must be academic, the other can be academic or professional. References may be uploaded   as part of the application form or you may enter your referees contact details on the application form. We will then email your referee and notify you when we receive the reference.  We can also accept confidential references direct to  [email protected] , from the referee’s university or business email account.
• Research proposal, CV, samples of written work as per requirements for each subject area.
• If you have any questions about your application  before  you apply: email  [email protected]  
• If you have any questions  after  you have submitted your application:  contact our Admissions team
• Any  references  may be submitted by email to:  [email protected]

International Students

• Advice on visa, immigrations and the Academic Technology Approval Scheme (ATAS) can be found at Applying for a student visa outside the UK

More information

• External collaborations

• Schools & departments

Depending on the field of study you are interested in, you may be able to apply for a pre-defined PhD project, or you may need to develop your own research idea.

Research in the fields of medicine, science and engineering tend to require you to:

• apply for a project that already has specific, pre-defined aims
• compete for a place on a partially or fully funded programme (often with competitive application processes).

Research in the fields of arts, humanities and social sciences tend to require you to:

• develop your own research idea (be prepared to adapt your original idea)
• source your own funding, although some funded projects may be available.

PhD research projects

Doctoral training centres and partnerships, developing your own research idea, professional doctorates.