Applications of multivariate orthogonal polynomials (Ref: MA/BW-SF1/2026)

About the Project

Random Matrix Theory (RMT) arose in the 1950s out of a need for a model to predict the spectral behaviour of systems that were too complex to describe precisely. Investigations of large matrices whose entries were random variables were able to reproduce features of complex quantum systems that were universal---that is to say, do not depend on the precise details of the system under consideration, but more general features such as whether or not time-reversal symmetry is present.

Although RMT was introduced to study complex quantum systems, for example excitation levels of large atomic nuclei, it was observed that spectral statistics of a single particle could be predicted using RMT, provided that the classical dynamics are chaotic. This leads to the Random Matrix Conjecture, still not fully resolved to this day, that the spectral statistics of a chaotic system with time-reversal symmetry, correspond exactly to the large matrix size limit of Gaussian symmetric matrix statistics; if time-reversal symmetry is absent then the spectral statistics instead are Gaussian Hermitian.

RMT has also been observed in the distribution of zeros of the famous{Riemann zeta function. This correspondence was given theoretical evidence by Montgomery and established beyond reasonable doubt by a virtuoso effort of numerical computations performed by Odlyzko.

The objective of this project will be to develop new tools and methods for RMT, based on the theory of Jack polynomials. These are multivariate homogeneous symmetric polynomials that are eigenfunctions of certain partial differential operators. Their importance in RMT comes from an integration formula due to Kadell for a Jack polynomial against measures naturally occurring in RMT that can be explicitly evaluated. For other ensembles of random matrices the multivariate orthogonal polynomials are expected to appear in analogous ways. These multivariate polynomials are expanded in the basis of Jack polynomials. The utility of Jack polynomials and multivariate orthogonal polynomials has only recently started to be realised, and there are numerous open problems that are ripe for a Ph.D. thesis.

This project will aim to solve the following problems about multivariate orthogonal polynomials:

1. The value of multivariate Hermite polynomials at (0,...,0). Explicit, tractable formulae are known for multivariate Laguerre and Jacobi polynomials at the origin, but only partial and unsatisfying results for Hermite polynomials. This part of the project will aim to find usable closed formulae for the values of Hermite polynomials at the origin.
2. Generating functions for Jacobi polynomials. For Hermite and Laguerre multivariate orthogonal polynomials, closed-form generating functions are known which are analogues of the generating functions in the 1-variable case. For multivariable Jacobi polynomial, no such formula is known. In this part of the project you will investigate this unsatisfying situation and look to find a possible closed formula for their generating function.
3. Integration formula relating Jacobi and Laguerre polynomials. In the one-variable case, there is a known integration formula that transforms a Laguerre polynomial into a Jacobi one. It is not known whether-or-not this formula can be generalised to n-dimensions. This part of the project will be to investigate whether that is possible, or whether it is not.

There are potentially many useful applications to knowing answers to these problems. For example a good solution to problem 1) would lead directly to better evaluations of certain integrals used in RMT.

Name of primary supervisor/CDT lead:

Brian Winn b.winn@lboro.ac.uk

https://www.lboro.ac.uk/departments/maths/staff/brian-winn/

Entry requirements:

A first-class or good 2:1 degree in Mathematics.

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 October 2026

Start date: April 2026, July 2026, October 2026, January 2027

Full-time/part-time availability: Full-time 3 years, Part-time 6 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/BW-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 selffunded 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:

computational mathematics, mathematics, probability, pure mathematics, random matrix theory, orthogonal polynomials