The Painleve equations - nonlinear special functions Peter A. Clarkson Institute of Mathematics and Statistics University of Kent, Canterbury, CT2 7NF, UK. To be held on Wednesday, July 17, 2002 at 2:00pm, Western Science Centre, Room 156. Coffee and Refreshments available. ABSTRACT The six Painleve equations (PI-PVI) were first derived around the beginning of the twentieth century by investigation by Painleve, Gambier and their colleagues in a study of nonlinear second-order ordinary differential equations. There has been considerable interest in Painleve equations over the last few years primarily due to the fact that they arise as reductions of soliton equations solvable by inverse scattering. Further, the Painleve equations are regarded as completely integrable equations and possess solutions which can be expressed in terms of the solutions linear integral equations. Although first discovered from strictly mathematical considerations, the Painleve equations have appeared in various of several important applications including statistical mechanics, random matrices, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. The Painleve equations may also be thought of as nonlinear analogues of the classical special functions such as Bessel functions. Their general solutions are transcendental in the sense that they cannot be expressed in terms of previously known functions. However, for special values of the parameters, PII-PVI possess rational solutions and solutions expressible in terms of special functions. For example, there exist special solutions of PII-PVI that are expressed in terms of Airy, Bessel, parabolic cylinder, Whittaker and hypergeometric functions, respectively. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations. In this talk I shall give an overview of some of plethora of remarkable properties which the Painleve equations possess (including connection formulae, Backlund transformations, associated discrete equations and hierarchies of exact solutions) and some of their applications.