Titles and Abstracts

Speaker: Peter A. Clarkson
Title: The Painlev´e Equations — Nonlinear Special Functions
Abstract: The six Painlev´e equations (PI–PVI) were first discovered around the beginning of the twentieth century by Painlev´e, Gambier and their colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently there has been considerable interest in the Painlev´e equations primarily due to the fact that they arise as reductions of the soliton equations which solvable by inverse scattering. Although first discovered from strictly mathematical considerations, the Painlev´e equations have arisen in a variety of important physical applications including statistical mechanics, random matrices, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. Further the Painlev´e equations may be thought of a nonlinear analogues of the classical special functions. In this lecture I will give an introduction to the Painlev´e equations. In particular I shall discuss many of the remarkable properties which the Painlev´e equations possess including connection formulae, B¨acklund transformations associated discrete equations, and hierarchies of exact solutions.

Speaker: Andy Magid
Title: The Complete Paicrd--Vessiot Closure of the Constants
Abstract: The compositum of all the Picard--Vessiot extensions of a given base differential field, unlike the algebraic closure of the field, may itself have proper Picard--Vessiot extensions. Iterating this, in general countably many times, produces a differential field that has no proper Picard--Vessiot extensions, and is minimal over the base with this property. This field is called the complete Picard--Vessiot closure. Its group of differential automorphisms over the base controls the differential subfield structure, even though the group is not (pro)algebraic and the correspondence is not a full Galois connection. We will focus on the natural special case when the base field is the (algebraically closed, characteristic zero) field of constants.

Speaker: Marius van der Put
Title: Solving linear differential equations.
Abstract: We concentrate on linear differential equations (or differential modules) over the differential field C(z). The theme, probably introduced by L. Fuchs, is to solve a differential equation in terms of equations of lower order. This problem has led to the highly interesting paper of G. Fano (1900). The work of M.F. Singer opened a new perspective on this theme. We continue this direction and apply the powerful theory of representations of semi-simple Lie algebras in order to obtain a systematic way for solving the problem. This involves differential Galois theory, Tannaka theory, simple algebraic groups and it leads to algorithms.