BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Peter A. Clarkson (University of Kent\, UK) DTSTART:20211116T140000Z DTEND:20211116T150000Z DTSTAMP:20260422T201809Z UID:CAvid/56 DESCRIPTION:Title: S pecial polynomials associated with the Painlevá equations\nby Peter A . Clarkson (University of Kent\, UK) as part of CAvid: Complex Analysis vi deo seminar\n\nLecture held in N/A.\n\nAbstract\nThe six Painlevé equatio ns\, whose solutions are called the Painlevé transcendents\, were derived by Painlevé and his colleagues in the late 19th and early 20th centuries in a classification of second order ordinary differential equations whose solutions have no movable critical points. In the 18th and 19th centuries \, the classical special functions such as Bessel\, Airy\, Legendre and hy pergeometric functions\, were recognized and developed in response to the problems of the day in electromagnetism\, acoustics\, hydrodynamics\, elas ticity and many other areas. \n\nAround the middle of the 20th century\, a s science and engineering continued to expand in new directions\, a new cl ass of functions\, the Painlevé functions\, started to appear in applicat ions. The list of problems now known to be described by the Painlevé equa tions is large\, varied and expanding rapidly. The list includes\, at one end\, the scattering of neutrons off heavy nuclei\, and at the other\, the distribution of the zeros of the Riemann-zeta function on the critical li ne Re(z) =1/2. Amongst many others\, there is random matrix theory\, the a symptotic theory of orthogonal polynomials\, self-similar solutions of int egrable equations\, combinatorial problems such as the longest increasing subsequence problem\, tiling problems\, multivariate statistics in the imp ortant asymptotic regime where the number of variables and the number of s amples are comparable and large\, and also random growth problems.\n\nThe Painlevé equations possess a plethora of interesting properties including a Hamiltonian structure and associated isomonodromy problems\, which expr ess the Painlevé equations as the compatibility condition of two linear s ystems. Solutions of the Painlevé equations have some interesting asympto tics which are useful in applications. They possess hierarchies of rationa l solutions and one-parameter families of solutions expressible in terms o f the classical special functions\, for special values of the parameters. Further the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations. \n\nIn this talk I shall discuss special polynomials associated with rational solutio ns of Painlevé equations. Although the general solutions of the six Painl evé equations are transcendental\, all except the first Painlevé equatio n possess rational solutions for certain values of the parameters. These s olutions are expressed in terms of special polynomials. The roots of these special polynomials are highly symmetric in the complex plane and specula ted to be of interest to number theorists. The polynomials arise in applic ations such as random matrix theory\, vortex dynamics\, in the description of rogue wave patterns\, in supersymmetric quantum mechanics\, as coeffic ients of recurrence relations for semi-classical orthogonal polynomials an d are examples of exceptional orthogonal polynomials.\n LOCATION:https://researchseminars.org/talk/CAvid/56/ END:VEVENT END:VCALENDAR