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  Dirac operators and global symmetries

   School of Natural and Computing Sciences

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  Dr Markus Upmeier, Prof Jarek Kedra  No more applications being accepted  Funded PhD Project (UK Students Only)

About the Project

Dirac operators are first order differential operators that are ‘square roots’ of Laplace operators. They are of fundamental importance in both Pure Mathematics and Theoretical Physics, and their study is the subject of Index Theory and combines techniques from Analysis, Topology, and Geometry.

Global symmetries are possibly non-invertible symmetries governed by a category rather than a group. Recently, many powerful new ideas have emerged in this rapidly expanding area, and the project aims to investigate the applications of these new categorical techniques. Possible research directions include constructing spin structures on moduli spaces (for example, in the context of Seiberg-Witten theory), investigating applications to the theory of loop groups, and using global symmetries to study elliptic boundary value problems on Riemann surfaces with corners.

Candidate Background: 


  • Candidates should have a minimum of a UK Honours degree at 2.1 or above (or equivalent) in Pure Mathematics along with some background in algebraic topology, category theory, differential geometry and functional analysis.

We encourage applications from all backgrounds and communities, and are committed to having a diverse, inclusive team.

Informal enquiries are welcome. To apply, please send a copy of your undergraduate transcript to Dr Upmeier ([Email Address Removed]) with a brief description of why you are interested in this project.



  • Formal applications can be completed online:
  • You should apply for Degree of Doctor of Philosophy in Mathematics to ensure your application is passed to the correct team for processing.
  • Please clearly note the name of the lead supervisor and project title on the application form. If you do not include these details, it may not be considered for the studentship.
  • Your application must include: A personal statement, an up-to-date copy of your academic CV, and clear copies of your educational certificates and transcripts.
  • Please note: you DO NOT need to provide a research proposal with this application
  • If you require any additional assistance in submitting your application or have any queries about the application process, please don't hesitate to contact us at [Email Address Removed].
Mathematics (25)

Funding Notes

This is a three year, fully funded project. Funding covers tuition fees at the UK/Home rate (this includes EU nationals that hold UK settled or pre-settled status), research costs, and a doctoral stipend for living costs (£18,622 for the 2023/2024 Academic year).
Due to funding criteria we cannot accept applications from International students.
The start date of the project is October 2024


H.B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton Math. Series 38, PUP, Princeton, NJ, 1989.
D. Joyce and M. Upmeier, On spin structures and orientations for gauge-theoretic moduli spaces, Adv. Math. 381, 2021.
A. Pressley and G. Segal, Loop groups, Oxford University Press, New York, 1986.

Where will I study?

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