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>

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<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 further information about your application, please contact PGR Admissions by email to <a href="mailto:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a></p>