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


Skills in analysis and data-driven, computational modelling

Collaborative projects focus

Responsible research and innovation

Entrepreneurship training


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.


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.




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.




Year One

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  CO2 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