Next Seminar
11 February 2026 (Wed) 17:30～(JST)/9:30～(CEST/UTC+１)
Peter Clarkson (the University of Kent)
Classical solutions of the fifth Painlevé equation
Abstract :
The general solutions of the Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions. However, it is well known that all Painlevé equations except the first equation possess rational solutions, algebraic solutions and solutions expressed in terms of the classical special functions for special values of the parameters. These solutions of the Painlevé equations are often called ``classical solutions" and frequently can be expressed in the form of determinants.
In the generic case of the fifth Painlevé equation, special function solutions are expressed in terms of Wronskians of Kummer functions, with special cases in terms of modified Bessel functions. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms
of the generalised Umemura polynomials. Both these polynomials can be expressed as Wronskians of Laguerre polynomials.
In this talk I shall discuss properties of the classical solutions of the fifth Painlevé equation, including giving non-unique rational solutions,
associated discrete Painlevé equations which are derived from B\"{a}cklund transformations, including a new discrete equation which has ternary symmetry, as well as applications to random matrices and quantum minimal surfaces.
Joint work with Clare Dunning (University of Leeds, UK) and Ben Mitchell (University of Kent, UK).
Painlevé Seminar Topics
Recently, we have encountered many interesting links between differential
equations of Painlevé types and other fields of mathematics such
as algebraic geometry, non-abelian Hodge theory and topological recursion
and so on.
In this web seminar, we will organize research talks on Painlevé
equations and related topics on web (zoom), basically about once every
two or three weeks.
From September 2022, Regular time for seminar talk will be on Wednesday from 17:30--18:30 + discussion time
in JST(Japan) or 10:30--11:30 in UTC+1. We are looking forward to your participation in
the seminar. In order to get zoom access, please register through the form.
List of Talks
You can find the titles and abstracts of all seminars. ⇒ List of Talks
Registration
Online. Please register at the form, we will send you about web seminar address.
Organizers
Arata Komyo (Hyogo), Frank Loray (Rennes 1),
Ryo Ohkawa (Osaka Metropolitan), Masa-Hiko Saito (Kobe Gakuin), Takafumi Matsumoto (RIMS, Kyoto)
Scientific Committee (other than the Organizing Committee)
Supported by
French ANR-16-CE40-0008 project "Foliage" (Frank Loray)
JSPS Grant-in-aid (A) 22H00094 (PI: Masa-Hiko Saito)
Osaka Central Advanced Mathematical Institute:
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.
Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi-Yau category and its associated Donaldson-Thomas invariants, and have also appeared in the context of isomonodromic systems associated with Painlevé equations. Under certain non-degeneracy conditions, they encode a complex hyperkähler structure on the tangent bundle of the base of the Joyce structure. Taking inspiration from this work, I'll give a description of (real) hyperkahler metrics compatible with an integrable system structure that is very similar to the description of Joyce structures. In particular, I will give examples of such hyperkähler metrics where there is a precise analog of the Plebański function appearing in the context of Joyce structures. This is work in progress, which is a follow-up to https://arxiv.org/abs/2403.00548.