Mathematics: Fully Funded PhD Scholarship in Toric Geometry and Multivariate Splines SURES (RS856)

Funding provider: Swansea University Research Excellence Scholarships (SURES)  

 Subject area: Mathematics   

Project start date: 1 October 2025 (Enrolment open from mid-September)  

Supervisors:  

• Primary Supervisor: Dr Nelly Villamizar                         
• Secondary Supervisor: Dr Gary Tam.  

Aligned programme of study:  PhD in Mathematics  

 Mode of study: Full-time   

 Project description:   

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.  

UK Fee Eligible Students Only 

Due to funding restrictions, this scholarship is open to applicants eligible to pay tuition fees at the UK rate only, as defined by UKCISA - Full list of categories for HE in Wales  

Candidates must have attained, or must be expected to attain, a first-class UK undergraduate degree (or equivalent international qualification), or an upper second-class UK undergraduate degree and a Distinction in a UK master’s degree (or international equivalent)  

Where applicants have multiple master’s degrees, a distinction must be held in the degree that is most relevant to the intended PhD study.  

If you are currently studying for a master’s level qualification with an expected award date that is later than 01/10/2025, you should hold a minimum of an upper-second-class (2:1) honours degree.    

You should be able to demonstrate a pass with a minimum grade average of at least 70% for your part-one master’s degree modules (the taught aspect of your master’s course rather than a research-focused dissertation) and submit your dissertation by no later than 30/09/2025.   

NB: If you hold a non-UK degree, please see the Swansea University degree comparisons to find out if you meet the eligibility criteria.   

Applicants must be able to begin their course of study in October 2025.  As a cohort-based programme, deferral to an alternative enrolment window within the academic year or to another academic year is not permissible.  

English Language requirements: If applicable – IELTS 6.5 overall (with a score of at least 6.5 in each individual component) or equivalent that is recognised by Swansea University.  Details on the Swansea University English Language entry policy can be found here.  

Please note that both the degree and language-proficiency entry requirements for SURES are higher than the baseline standard for entry that is stipulated for most of the PhD programmes at Swansea University.  

If you have any questions regarding your academic or fee eligibility based on the above, please email pgrscholarships@swansea.ac.uk with the web-link to the scholarship(s) you are interested in. 

Please note that the programme requires some applicants to hold ATAS clearance, further details on ATAS scheme eligibility are available on the UK Government website.  

ATAS clearance IS NOT required to be held as part of the scholarship application process. Successful award winners (as appropriate) are provided with details as to how to apply for ATAS clearance in tandem with a scholarship course offer.  

This scholarship covers the full cost of tuition fees and an annual stipend at UKRI rate (currently £20,780 for 2025/26). 

Additional research expenses of up to £1,000 per year will also be available. 

SURES
Swansea University is pleased to offer fully-funded Swansea University Research Excellence Scholarships (SURES) for full-time doctoral students to its seventh cohort, commencing in October 2025.  

 Each SURES scholarship includes tuition fees and a stipend 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.   

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

To apply, please complete the entire application form:

In order to be considered for this scholarship award the following steps are also required. 

1) In section ‘Programme Related Information’ please input the relevant RS Code for the scholarship award i.e. RS856 

2) In section ‘Research’ you will see ‘Proposed project title/studentship title’* (Mandatory) 

• In ‘Proposed project title/studentship title’ please input either:  

• the RS Code RS856
• the scholarship title

• Please leave Proposed Supervisor field blank 
• Please leave Research Project (if applicable) blank 
• In ‘Do you have a proposal to upload?*’(Mandatory) Please select Yes 
• Then upload copy of advert (you can save the advert by clicking print, and then print to pdf) 

3) In section ‘Funding information’ please choose the option ‘Scholarship Funding’ only. Please ensure no other options are selected.  

*It is the responsibility of the applicant to list the above information accurately when applying, please note that applications received without the above information listed will not be considered for the scholarship award. 

One application is required per individual Swansea University led research scholarship award; applications cannot be considered listing multiple Swansea University led research scholarship awards. 

If you’ve previously applied for this programme, the system will display an “Application Submitted” warning and block a new submission. In this case:

1. Apply for the same course with the next available start date (e.g., select January if October is unavailable).
2. Email pgrscholarships@swansea.ac.uk with your student number and the relevant scholarship RS code, requesting the start date be amended to match the advert.
3. Admissions staff will then update your application accordingly.

NOTE: Applicants for PhD/EngD/ProfD/EdD - to support our commitment to providing an environment free of discrimination and celebrating diversity at Swansea University you are required to complete an Equality, Diversity and Inclusion (EDI) Monitoring Form in addition to your programme application form.    

Please note that completion of the EDI Monitoring Form is mandatory; your application may not progress if this information is not submitted. 

As part of your online application, you MUST upload the following documents (please do not send these via email): 

• CV 

• Degree certificates and transcripts (if you are currently studying for a degree, screenshots of your grades to date are sufficient) 

• A cover letter including a ‘Supplementary Personal Statement’ to explain why the position particularly matches your skills and experience and how you choose to develop the project. 

• One reference (academic or previous employer) on headed paper or using the Swansea University reference form. Please note that we are not able to accept references received citing private email accounts, e.g. Hotmail. Referees should cite their employment email address for verification of reference. 

• Evidence of meeting English Language requirement (if applicable). 

• Copy of UK resident visa (if applicable) 

• Confirmation of EDI form submission 

Informal enquiries are welcome; please contact Dr Nelly Villamizar  N.Y.Villamizar@Swansea.ac.uk  

*External Partner Application Data Sharing – Please note that as part of the scholarship application selection process, application data sharing may occur with external partners outside of the University, when joint/co- funding of a scholarship project is applicable.