[School of Natural Sciences PhD Scholarships] Geometry of numbers beyond lattices

About the project

About the Project The geometry of numbers studies the remarkable interplay between arithmetic and geometry through lattices. But how much of this theory really depends on lattices being groups? This project will explore that question for cut-and-project sets: highly ordered but typically aperiodic point sets which arise naturally in number theory, dynamical systems and the mathematical theory of quasicrystals. They include well-known examples such as the Fibonacci chain, Ammann–Beenker sets and Penrose-type point sets. There are several possible directions. The student might investigate how efficiently cut-and-project sets can pack or cover space, seek analogues of classical theorems of Minkowski, or study the lower-dimensional patterns produced by taking projections and intersections. These questions combine concrete geometric examples with broader ideas from number theory and dynamics, and there is flexibility for the project to develop according to the student’s interests. 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 Pure Mathematics or similar subject OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Pure Mathematics or similar subject. Applicants should have a strong background in number theory and dynamical systems. Familiarity with geometry of numbers, Diophantine approximation or related areas would be helpful, but prior knowledge of cut-and-project sets or aperiodic order is not required. 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. FSESoNS

Research areas