About the Project
When can a system of polynomial equations have more special solutions than geometry predicts? This question lies at the heart of unlikely intersections, an area connecting number theory, algebraic geometry and model theory. The Zilber-Pink conjecture provides a broad framework for understanding such phenomena, encompassing celebrated problems such as the André-Oort and Manin-Mumford conjectures. This project will investigate aspects of this framework, with particular focus on the semi-abelian…
and modular settings.
The central objective is to understand how the geometry of an algebraic variety constrains its intersections with special subvarieties. An intersection is called atypical when its dimension exceeds the value predicted by a straightforward dimension count. In the modular setting, special subvarieties are described using modular relations and special values of the modular j-function. In the semi-ableian setting, special subvariaties are described using algebraic subgroups and torsion points (special values of the exponential function). Possible directions include studying atypical intersections for particular classes of varieties, exploring the role of genericity assumptions, or obtaining effective results.
The project will draw on algebraic geometry and model theory, with the balance determined by the questions chosen. Relevant methods include o-minimality, functional transcendence, and differential algebra. In particular, Ax-Schanuel theorems give lower bounds on algebraic independence unless prescribed functional relations occur, making them powerful tools for controlling atypical intersections. Depending on the direction taken, the student may also learn techniques from Diophantine and arithmetic geometry. Initial work will involve understanding established results and explicit examples before identifying a focused research problem.
Potential outcomes include new special cases of unlikely intersection statements, refinements of existing results, or a clearer account of the hypotheses needed for finiteness or non-density conclusions. The precise aims will be developed jointly with the student and adjusted as the research progresses, with an emphasis on questions of a suitable scale for a PhD. Prior knowledge of the Zilber-Pink conjecture is not required; the initial programme will be tailored to the student's mathematical background.
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. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships. If you don't already have an applicant account, please follow the instructions to create one.
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. Undergraduate and Masters courses covered must include: Algebraic Number Theory, Galois Theory, Elliptic Curves, Modular Forms as well as other advances courses in Algebra and Number Theory.
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.