About the Project
This project will involve working on unlikely intersection problems (an active area in Diophantine geometry) using ideas from o-minimality (part of model theory).
On the Diophantine side, it is likely that the problems considered will involve elliptic curves and perhaps families of elliptic curves, and related objects. From o-minimality, we would use recent versions of the Pila-Wilkie counting theorem.
Unlikely intersection results usually involve the finiteness of a certain set, for example a set of algebraic numbers. In this project, the goal would sometimes be to show more, and give effectively computable bounds on some measure of complexity of the objects in the set. For example, when the set consists of algebraic numbers, we would try to give bounds on the degrees and absolute values of the coefficients of the minimal polynomials of the algebraic numbers.
The particular problems to be considered are flexible, and could be adapted to the skills and interests of the student
This project is expected to start in September 2027.
Before you apply: We strongly recommend that you contact the supervisors for this project before you apply.
How to apply: To be considered for this project you must complete a formal application through our online application portal.
When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications. Please note late or incomplete applications will not be considered.
Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status.
We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder.
Eligibility: The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Mathematics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Mathematics.
This project will remain open until filled.
If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026.
If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027.
Self or externally funded students can also be considered for this project.
FSE_SoNS
Funding Notes
Funding covers tuition fees, an annual stipend at the UKRI minimum rate (currently £21,805 per year), and a research training support grant of up to £2,500 per year for the full 3.5-year programme.