Integrability, polynomials and partitions

Integrable systems covers a broad area of mathematical problems that can be solved exactly in some sense.   An early example includes the Kepler problem describing planetary motion, solved by Newton despite the differential equations being nonlinear.  Such problems would typically have a number of conserved quantities and corresponding symmetries.  More recent examples in classical integrable systems includes the Korteweg-de Vries equation describing shallow water waves, which has soliton solutions, and in quantum integrable systems the XXZ spin chain, which is a quantum model of interacting spins in one dimension. <br /> <br /> Wronskian polynomials such as the Wronskian of a finite  set of Hermite polynomials have been studied in recent years in classical and quantum integrable settings.  Curiously if we use a partition of an integer to label the set of Hermite polynomials in the Wronskian, then some properties of the polynomials can be described in terms of properties of the chosen partition.  Specifically, the coefficients of Wronskian Hermite polynomials depend on the hook lengths of the partition and those of the partitions in the Young lattice found by removing 2-hooks in all possible ways.<br /> <br /> This project seeks to understand some of several open problems that have arisen especially those related to Wronskian polynomials that arise in the Painleve hierarchy generalising the six Painleve equations.

<p><meta charset="UTF-8"></p> <p><strong>Bibliography</strong><br /> <br /> The ODE/IM Correspondence,Patrick Dorey, Clare Dunning and Roberto Tateo,<br /> Encyclopedia of Mathematical Physics 2nd edition. Editors-in-Chief: Richard Szabo, Martin Bojowald. (2025)<br /> Pages 145-161<br /> <br /> Harmonic locus and Calogero-Moser spaces, Giovanni Felder, Alexander P. Veselov, arXiv:2404.18471</p> <p><br /> Rational Solutions of the Fifth Painleveé Equation. Generalised Laguerre Polynomials, Peter A. Clarkson, Clare Dunning, Stud. Appl. Math, 152 (2024) 453-507<br /> <br /> Counting monster potentials. R Conti, J High Energy Physics, JHEP02 (2021), 059. </p> <p>Coefficients of Wronskian Hermite polynomials, Niels Bonneux, Clare Dunning, Marco Stevens, Studies in Applied Mathematics 144 (2020)<br /> <br /> Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials, David Gomez-Ullate, Yves Grandati, Robert Milson, Stud. Appl. Math, 141 (2018) 596-625<br /> <br /> Studying for a PhD is different to studying for a BSc, MMath or MSc. It requires curiosity, tenaciousness, creativity, ability to collaborate and to work independently, resilience and to be self-motivated.  Every doctoral candidate is different and there isn’t a typical maths PhD student.<br /> <br /> You will develop your skills throughout your study through a tailored development programme that prepares you for your PhD and a career afterwards, whether that is in research or otherwise.   You will also have the opportunity to take advanced technical courses, attend conferences and meet other researchers and present your research.  You will be part of a large community of researchers in the School of Mathematics with several relevant seminar programmes as well as being part of an international research community.<br /> <br /> <br />  </p>

<p>To apply for this project you will need to make a formal application for research degree study through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University website</a>. You will need to create a login ID with a username and PIN. </p> <p>•    For <strong>Application type</strong> please select <strong>Research Degrees – Research Postgraduate</strong>. <br /> •    The admission year for this project is <strong>2027/28</strong> Academic Year. <br /> •    You will need to select your <strong>Planned Course of Study</strong> from a drop-down menu. For this project, scroll down and select <strong>PHP Applied Mathematics Full Time.</strong> <br /> •    The project start date for this project is <strong>1 October 2027</strong>, please use this as your <strong>Proposed Start Date of Research</strong>. <br /> •    Please state clearly in the<strong> Research Information Section</strong> that the research degree you wish to be considered for is <strong>Integrability, polynomials and partitions</strong><em><strong> </strong></em>as well as <a href="https://eps.leeds.ac.uk/maths/staff/16704/professor-clare-dunning">Professor Clare Dunning</a> as your proposed supervisor.</p> <p><strong>Please state clearly in the Finance section</strong>, <strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong>.</p> <p>More information on how to apply is available on our website <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">here</a>. You will be required to provide a personal statement which outlines your interest in the project you are applying for, why you have chosen it and how your skills map onto the requirements of the project.</p> <p>We will assess applications continuously as we receive them. We welcome and strongly encourage any potential applicants to contact the supervisor(s) for an informal discussion, prior to applying, and recommend submitting your application early.</p> <p><strong>If you are applying with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<p><meta charset="UTF-8" /></p> <p>This is a non funded PhD project. Applicants are expected to be self funded or to secure external funding.</p> <p>You can find out more about our Funding Opportunities on our <a href="https://phd.leeds.ac.uk/">Postgraduate Research Opportunities</a> website.</p> <p><strong>Important: </strong>Please note that all costs associated with your arrival at Leeds (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source.</p>

<p>For further information about this project, please contact Professor Clare Dunning by email to <a href="mailto:t.c.dunning@leeds.ac.uk">t.c.dunning@leeds.ac.uk</a> or by calling +44 (0)113 343 7257.</p> <p>For questions related to the application process and eligibility, please contact PGR Admissions by email to <a href="mailto:epspgradmissions@leeds.ac.uk">epspgradmissions@leeds.ac.uk</a></p>