[School of Natural Sciences PhD Scholarships] Existential Closedness for exponential and automorphic functions

About the project

About the Project Polynomial equations over the complex numbers are central to algebraic geometry. What happens when equations also involve transcendental functions, such as the exponential function or the modular j-function? The Existential Closedness conjectures predict that suitable systems have solutions whenever certain natural algebraic and geometric obstructions are absent. This project will investigate these predictions and the structure of the resulting solution sets, at the intersection of model theory, complex analysis and algebraic geometry. A useful geometric viewpoint is to ask when an algebraic variety meets the graph of a transcendental function. The objective will be to establish existence or density results for selected classes of varieties, or to clarify the conditions under which such results should hold. Possible directions include systems of exponential equations, equations involving the modular j-function and its derivatives, or related questions for other automorphic functions. The project could focus on extending known results beyond particular projection or dimension assumptions, understanding exceptional cases, or studying the distribution of solutions in families of examples. The methods will depend on the student's interests and the chosen setting. Complex analytic approaches may use local inverse functions, asymptotic behaviour and periodicity to construct solutions. Algebraic geometry provides tools for analysing projections, dimension and Zariski density. Model theory offers a language for studying generic solutions and the algebraic relations they satisfy, while functional transcendence results help identify possible obstructions. Another direction is to investigate connections with unlikely intersections and the Zilber-Pink conjecture, particularly the passage from existence of solutions to stronger statements about their genericity. The project will begin with guided study of foundational examples and existing existence theorems, followed by the selection of a tractable problem. Potential outcomes include new cases of Existential Closedness, improved criteria for solvability, density results, or a better understanding of the relationship between analytic and model-theoretic formulations. The research programme will remain flexible, allowing the student to develop an emphasis on complex analysis, geometry or logic as their interests evolve. Specialist knowledge of the conjectures is not expected at the outset; the necessary background will be developed during the project. 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

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