Eigenvalues of functional difference operators associated to mirror curves
Loughborough UniversityLoughborough, United Kingdom
Course overview
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. 94% of Loughborough’s research impact is rated world-leading or internationally excellent.
REF 2021
Key highlights
- Research on eigenvalues of functional-difference operators from quantised mirror curves
- Explores links between operator spectra and Calabi–Yau geometry
- Investigates asymptotic behaviour of eigenvalues of infinite-order differential operators
- Membership of the Analysis and PDE research group
- Weekly research seminars and expertise in spectral theory and mathematical physics
- Links with research groups across the UK and EU
Entry requirements
- Degree: 2:1 — 2:1
Scholarships
Alumni Bursary
10% of tuition fees which will be credited to the student’s tuition fee account.
Up to 10% of feesAlumni Discount
UK students: £1,500 tuition fee discount for one academic year. International students: £3,000 tuition fee discount for one academic year.
Care Leaver's Scholarship
Excellence Scholarships
UK students: £3,000 off tuition fees; International students: £6,000 off tuition fees which will be credited to the student’s tuition fee account.
International Sports Student Postgraduate Bursary
Our International Sports Student Postgraduate Bursary offers a £12,000 tuition fee discount (set against the tuition fees of your chosen postgraduate taught programme ONLY).
Up to £12,000London 10th Anniversary Award
£5,000 off tuition fees.
Up to £5,000Loughborough Business School Scholarship
£1,500 (UK students) and £3,000 (international students) off tuition fees which will be credited to the student’s tuition fee account.
MBA Progress with Purpose Scholarship
50% of tuition fees which will be credited to the student’s tuition fee account.
Up to 50% of feesQatari Ghanim Al-Muftah Excellence Scholarship
Careers this course leads to
- Postdoctoral research associate in mathematics
- University lecturer in mathematics
- Mathematical physicist
- Quantitative analyst
- Data scientist
Cost of study
UK/Home: £5,238 | International: £23,100
Admission
Applications are made through AHZ at no cost. Share your transcripts, English test scores and personal statement, and your AHZ Counsellor will review your profile, submit your application to Loughborough University, and guide you through interview, offer, CAS and visa — right up to enrolment.
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