Eigenvalues of functional difference operators associated to mirror curves (Ref: MA/LS-SF1/2026)

About the Project

Toric Calabi–Yau manifolds play an important role in mathematical physics. Their mirror manifolds can be described by algebraic curves and recently it was observed that quantisation of these curves leads to functionaldifference operators. The eigenvalues of these operators are conjectured to relate to geometric properties of the associated Calabi–Yau manifolds.

Formally, the operators are differential operators of infinite order. Studying properties of their eigenvalues is thus of twofold importance. Firstly, it sheds more light on the conjectured connection to Calabi–Yau manifolds. Secondly, any results obtained can be expected to form limit cases in known spectral results for finite-order differential operators.

This project will build on recent progress in establishing eigenvalue bounds for some of these operators. Specifically, it will investigate the asymptotic behaviour of the eigenvalues. Due to the peculiar nature of these operators, several concepts that are well known for finite-order differential operators cannot be directly applied and will need to be modified.

The successful candidate will be part of the Analysis and PDE group at Loughborough University, benefitting from a stimulating environment that includes weekly research seminars, diverse expertise in spectral theory and mathematical physics, as well as links with research groups across the UK and EU.

Name of primary supervisor/CDT lead:

Dr Lukas Schimmer l.schimmer@lboro.ac.uk

Entry requirements:

Applicants should have, or expect to achieve, at least a 2:1 Honours degree (or equivalent) in a mathematics-related degree. The project requires a good understanding of real, complex and functional analysis. A relevant Master’s degree is of advantage.

English language requirements:

Applicants must meet the minimum English language requirements. Further details are available on the International website (http://www.lboro.ac.uk/international/applicants/english/).

Bench fees required: No

Closing date of advert: 31st July 2026

Start date: October 2026

Full-time/part-time availability: Full-time 3 years

Fee band: 2025/26 Band RA (UK £5,006, International £22,360)

How to apply:

All applications should be made online. Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MA/LS-SF1/2026 in your application.

To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents.

The following selection criteria will be used by academic schools to help them make a decision on your application. Please note that this criteria is used for both funded and self-funded projects.

Please note, applications for this project are considered on an ongoing basis once submitted and the project may be withdrawn prior to the application deadline, if a suitable candidate is chosen for the project.

Project search terms:

pure mathematics, quantum mechanics Other search terms not listed: spectral theory; analysis; mathematical physics

Email Address Sci:

sci-pgr@lboro.ac.uk