Mathematics: Fully Funded PhD Scholarship in Toric Geometry and Multivariate Splines

The aim of this PhD project is to explore the properties of splines through the lens of applied algebraic geometry, with a particular focus on their relationship to toric geometry. Splines are mathematical representations built upon simpler pieces (usually defined by low-degree polynomials) glued together to form smooth curves, surfaces, or higher-dimensional objects. They are fundamental tools for approximation and computational design, crucial in finite element methods, computer vision, and geometric deep learning.

Splines can be viewed as functions defined over polyhedral complexes. These complexes, in turn, define toric varieties in algebraic geometry, and it has been proved that the set of splines on a polyhedral complex can be seen as the equivariant cohomology ring of the associated toric variety. This project will focus on the equivariant cohomology subrings and representations corresponding to splines with continuous derivatives of order at least 1, with the aim of making significant progress toward a deeper understanding of these structures.

The ideal candidate for this project should have a background in algebra, combinatorics, and/or geometry, and will have the opportunity to complement their knowledge in the remaining areas throughout the project, as well as develop an understanding of the various applications of spline methods. The project can be tailored to the applicant's interest in either the theoretical or computational aspects of the topic. Coding skills are desirable but not essential.

Informal scholarship enquiries from applicants to prospective supervisors – prior to the application deadline – are warmly encouraged.

Funding Details

Funding Comment

Each SURES scholarship includes tuition fees and a stipend at UKRI rate (currently £19,237 for 2024/25) – reviewed annually at the UKRI standard rate and a £1,000 annual training allowance to support immersive training experiences, conference attendance, engagement with industry and international collaborative  opportunities.

Each scholarship is offered for a maximum period of three years; SURES students are expected to submit their theses within this timescale.

Scholarship open to UK fee eligible students only, however applications can be considered from International students, if they wish to be considered for the SUIPRES Award to run alongside this Studentship.

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