University of Cambridge
Department of Applied Mathematics and Theoretical Physics
Supervisors: Mr Matthew Colbrook; Assoc Prof George Stepaniants
Application deadline: Friday, December 04, 2026
Competition Funded PhD Project (UK Students Only)
About the Project
AI_CDT_DecisionMaking
Details
This project will develop learning-guided, computer-assisted proofs that use AI for mathematical discovery while maintaining rigorous verification. It supports the CDT’s themes of uncertainty-aware exploration, human–AI collaboration and reliable decision-making, with spectral geometry providing a key application.
Starting from existing examples, the student will investigate new solution families as symmetry and boundary data vary, and develop validated continuation and geometric bounds. A complementary benchmark will concern certified eigenvalue enclosures for parameter-dependent differential operators. Specific targets will be selected with the student to balance mathematical significance, available validation theory and computational feasibility.
The methodological contribution will be AI that learns from the verification process itself. Probabilistic surrogate models and cost-aware active learning will select candidate geometries and spectral parameters, and decide when to refine a discretisation, increase arithmetic precision or attempt a full certificate. Training data will include successful certificates and diagnostic information from unsuccessful attempts. The objective is not simply a small numerical residual: it is to obtain mathematically meaningful, rigorously certified results within a computational budget. A failed verification attempt will be treated as inconclusive, not as evidence that the mathematical claim is false.
Discovery and certification will remain separate. Candidate results will be checked through proved a posteriori validation theorems, interval arithmetic and explicit control of discretisation errors and infinite-dimensional tails. For geometric constructions, the certificate must also establish the required properties of the domain. Learned confidence will guide the search but will never substitute for these mathematical checks. Building on AI-guided program search, an optional extension will investigate generated search heuristics or candidate analytic bounds, with human review and a fixed, independently maintained verification stage.
Evaluation will compare the learned methods with conventional optimisation, numerical continuation and non-learning adaptive strategies under matched total computational budgets. Tests on held-out problem families will assess transfer, certificate success rates, verification cost and the value of human mathematical guidance. The student will investigate which assumptions support reliable exploration and how to recognise when additional analysis, rather than more computation, is needed.
The project will produce new methods for verification-guided learning, reusable software and benchmark problems, and computer-assisted mathematical results. Reproducing existing non-AI certificates and improving validated bounds provide early milestones, while more ambitious discoveries provide longer-term targets. Success will not depend on resolving a single named conjecture.
Desirable Student Background:
The project would suit a student with a strong background in mathematics, applied mathematics, theoretical computer science or a closely related discipline. Essential preparation includes linear algebra, mathematical analysis, experience reading and writing proofs, and an interest in programming and computational experimentation. Familiarity with numerical analysis, differential equations, spectral theory, probability or machine learning would be valuable, but expertise in all these areas is not expected. Experience with scientific programming in Python, Julia, MATLAB or C++ would be helpful. The student should be enthusiastic about moving between abstract mathematics, algorithm development and reproducible computation. Specialist training will be developed through the CDT and supervision.
By submitting an application for a University of Cambridge project, you consent to your documents being shared with the relevant supervisory team at the partner institution. If you have questions about this process aidecisionscdt@manchester.ac.uk before applying.
How to apply:
Please apply under the University of Manchester application portal and select 'PhD in Artificial Intelligence' (OAA Application form)
When applying, you’ll need to specify the full name of this project, the name of your supervisor, if you already having funding or if you wish to be considered for available funding through the university, details of your previous study, and names and contact details of two referees.
Your application will not be processed without all of the required documents submitted at the time of application, and we cannot accept responsibility for late or missed deadlines. Incomplete applications will not be considered.
You will be asked to upload the following supporting documents:
- Final Transcript and certificates of all awarded university level qualifications
- Interim Transcript of any university level qualifications in progress
- CV
- Supporting statement: A one or two page statement outlining your motivation to pursue postgraduate research and why you want to undertake postgraduate research at Manchester, any relevant research or work experience, the key findings of your previous research experience, and techniques and skills you’ve developed. (This is mandatory for all applicants and the application will be put on hold without it).
- Contact details for two referees (Please use an official university or workplace email for referees, as references may be verified..)
- English Language Certificate (if applicable)
We actively encourage applicants from diverse career paths and backgrounds and from all sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation and transgender status.
We also support applications from those returning from a career break or other roles. We are dedicated to supporting work-life balance and offer flexible working arrangements to accommodate individual needs. Our selection process is free from bias, and we are committed to ensuring fair and equal opportunities for all applicants.
We are dedicated to supporting work-life balance and offer flexible working arrangements to accommodate individual needs, including a part-time study option.
Join us and help grow the UK’s national capability in artificial intelligence, part of the UK’s modern industrial strategy. Everyone is welcome if you are passionate about uncertainty quantification, decision making with humans in the loop and decision making for machine learning systems and their applications in Science and Engineering, and looking to develop your research and innovation skills at the doctoral level to further your career in the AI sector.
Equality, diversity and inclusion is fundamental to the success of The University of Manchester, and is at the heart of all of our activities. We know that diversity strengthens our research community, leading to enhanced research creativity, productivity and quality, and societal and economic impact. (Equality, diversity and inclusion | The University of Manchester)
We actively encourage applicants from diverse career paths and backgrounds and from all sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation and transgender status.
Funding Notes
Fully funded AI UKRI CDT 4 year program; The Project will be based in University of Manchester.
The UKRI AI CDT in Decision Making for Complex Systems is part of the TechExpert pilot offering stipends of £30,000 for students eligible for home fee status. The start date is September 2027.
The Project will be based in University of Cambridge.