In the 18th and 19th centuries\, the classical special fu nctions such as Bessel\, Airy\, Legendre and hyper geometric functions\, were recognized and develope d in response to the problems of the day in electr omagnetism\, acoustics\, hydrodynamics\, elasticit y and many other areas.

Around the middle of the 20th century\, as science and engineering cont inued to expand in new directions\, a new class of functions\, the Painleve functions\, started to a ppear in applications. The list of problems now kn own to be described by the Painleve equations is l arge\, varied and expanding rapidly. The list incl udes\, at one end\, the scattering of neutrons off heavy nuclei\, and at the other\, the distributio n of the zeros of the Riemann-zeta function on the critical line Re(z) =1/2. Amongst many others\, t here is random matrix theory\, the asymptotic theo ry of orthogonal polynomials\, self-similar soluti ons of integrable equations\, combinatorial proble ms such as the longest increasing subsequence prob lem\, tiling problems\, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are compara ble and large\, and also random growth problems.

The Painleve equations possess a plethora of interesting properties including a Hamiltonian structure and associated isomonodromy problems\, w hich express the Painleve equations as the compati bility condition of two linear systems. Solutions of the Painleve equations have some interesting as ymptotics which are useful in applications. They p ossess hierarchies of rational solutions and one-p arameter families of solutions expressible in term s of the classical special functions\, for special values of the parameters. Further the Painleve eq uations admit symmetries under affine Weyl groups which are related to the associated Backlund trans formations.

In these lectures I shall fi rst review many of the remarkable properties which the Painleve equations possess. In particular I w ill discuss rational solutions of Painleve equatio ns. Although the general solutions of the six Pain leve equations are transcendental\, all except the first Painleve equation possess rational solution s for certain values of the parameters. These solu tions are usually expressed in terms of logarithmi c derivatives of special polynomials that are Wron skians\, often of classical orthogonal polynomials such as Hermite and Laguerre. It is also known th at the roots of these special polynomials are high ly symmetric in the complex plane. The polynomials arise in applications such as random matrix theor y\, vortex dynamics\, in supersymmetric quantum me chanics\, as coefficients of recurrence relations for semi-classical orthogonal polynomials and are examples of exceptional orthogonal polynomials. LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR