Program Overview

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Program Details

Introduction

The program in question is a PhD opportunity at Loughborough University, focusing on the Eigenvalues of functional difference operators associated to mirror curves.

Qualification(s) Available

• PhD

Entry Requirements

Applicants should have, or expect to achieve, at least a 2:1 Honours degree (or equivalent) in mathematics or physics. A relevant Master’s degree and/or a strong background in analysis are of advantage.

Fees for Entry

• UK: £5,006 (Full-time degree per annum)
• International: £22,360 (Full-time degree per annum)

Duration

• Full-time: 3 years

Start Date

• January 2026

Application Deadline

• 1 October 2025

Project Reference

• MA/LS - SF2/2025

Location

• Loughborough

Subject Area(s)

• Mathematical Sciences

Project Details

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 functional-difference 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.

Supervisors

• Primary supervisor: Lukas Schimmer

English Language Requirements

Applicants must meet the minimum English language requirements. Further details are available on the International website.

Fees and Funding

Tuition fees cover the cost of your teaching, assessment and operating University facilities such as the library, IT equipment and other support services. University fees and charges can be paid in advance and there are several methods of payment, including online payments and payment by instalment. Fees are reviewed annually and are likely to increase to take into account inflationary pressures.

How to Apply

All applications should be made online. Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MA/LS – SF2/2025 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.