[School of Natural Sciences PhD Scholarships] Differential algebra and the model theory of differential equations

About the project

About the Project Differential algebra studies differential equations using algebraic methods, treating differentiation as an operation on a field. Model theory adds tools for understanding the structure of solution sets and the relations between their elements. Together, these approaches reveal features of differential equations that are difficult to see from explicit solutions alone. This project will explore questions in differential algebra and the model theory of differential fields, with scope to study equations associated with classical functions. The broad objective is to understand how a differential equation controls algebraic dependence and the geometry of its solutions. Possible directions include studying algebraic relations among solutions of selected equations, investigating their definable subsets, or establishing criteria for particularly simple model-theoretic behaviour. One example is strong minimality: roughly, a solution set is strongly minimal if every definable subset is finite or has finite complement, including after extending the ambient field. Equations associated with the modular j-function provide an important source of examples linking this property with functional transcendence. A further possible direction concerns reducts of differential fields. Here one retains the field operations and selected relations defined by differential equations, while omitting the differentiation operation itself. The student could investigate how much information about differentiation can be recovered from those relations, or how the resulting structures can be described by axioms. Related questions concern existential closedness: which algebraic compatibility conditions ensure that systems involving a chosen differential relation have solutions? The exact focus will be agreed with the student, allowing the project to develop towards algebra, model theory or their applications. Training and research will draw on differential polynomial rings, differential field extensions and differentially closed fields. Depending on the chosen problem, further methods may include quantifier elimination, stability theory, algebraic geometry and Ax-Schanuel type inequalities. The initial phase will combine foundational reading with detailed study of examples and proofs, providing a basis for formulating a manageable research question. Potential outcomes include structural results for particular differential equations, criteria for algebraic independence or definability, and examples distinguishing different model-theoretic behaviours. Prior specialist knowledge of differential algebra or stability theory is not essential. 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 here . 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 1 st November 2026, you can expect a decision by 18 th December 2026. If your application is submitted by 15 th January 2027, you can expect a decision by 30 th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

Research areas