BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Painleve Equations - Nonlinear Special Functions I - Peter Clarks on (University of Kent) DTSTART:20190910T080000Z DTEND:20190910T090000Z UID:TALK129313@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:The six Painleve equations\, whose solutions are called the Pa inleve transcendents\, were derived by Painleve and his colleagues in the late 19th and early 20th centuries in a classification of second order ord inary differential equations whose solutions have no movable critical poin ts.
In the 18th and 19th centuries\, the classical special functions such as Bessel\, Airy\, Legendre and hypergeometric functions\, were recog nized and developed in response to the problems of the day in electromagne tism\, acoustics\, hydrodynamics\, elasticity and many other areas.
A round the middle of the 20th century\, as science and engineering continue d to expand in new directions\, a new class of functions\, the Painleve fu nctions\, started to appear in applications. The list of problems now know n to be described by the Painleve equations is large\, varied and expandin g 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 Ri emann-zeta function on the critical line Re(z) =1/2. Amongst many others\, there is random matrix theory\, the asymptotic theory of orthogonal polyn omials\, self-similar solutions of integrable equations\, combinatorial pr oblems such as the longest increasing subsequence problem\, tiling problem s\, multivariate statistics in the important asymptotic regime where the n umber of variables and the number of samples are comparable and large\, an d also random growth problems.

The Painleve equations possess a p lethora of interesting properties including a Hamiltonian structure and as sociated isomonodromy problems\, which express the Painleve equations as t he compatibility condition of two linear systems. Solutions of the Painlev e equations have some interesting asymptotics which are useful in applicat ions. They possess hierarchies of rational solutions and one-parameter fam ilies of solutions expressible in terms of the classical special functions \, for special values of the parameters. Further the Painleve equations ad mit symmetries under affine Weyl groups which are related to the associate d Backlund transformations.

In these lectures I shall first revi ew many of the remarkable properties which the Painleve equations possess. In particular I will discuss rational solutions of Painleve equations. Al though the general solutions of the six Painleve equations are transcenden tal\, all except the first Painleve equation possess rational solutions fo r certain values of the parameters. These solutions are usually expressed in terms of logarithmic derivatives of special polynomials that are Wronsk ians\, often of classical orthogonal polynomials such as Hermite and Lague rre. It is also known that the roots of these special polynomials are high ly symmetric in the complex plane. The polynomials arise in applications s uch as random matrix theory\, vortex dynamics\, in supersymmetric quantum mechanics\, as coefficients of recurrence relations for semi-classical ort hogonal polynomials and are examples of exceptional orthogonal polynomials . LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR