Various Aspects on the Painlevé Equations

November 26-30, 2012

RIMS, Kyoto University, Japan


The Painlevé equations were found in 1898 by Paul Painlevé but were once forgotten after Painlevé died. But the Painlevé equations are now applied in many fields in mathematical sciences, such as soliton equations, quantum field theory, the Random matrices, traffic flow analysis and so on. They are related to many fields of mathematics such as differential geometry, foliations, probability theory, representation theory, algebraic geometry, number theory, etc.

In the recent study of the Painlevé equations, we need many mathematical tools: complex analysis, moduli of connections, differential Galois theory, projective differential geometry and so on. The Painlevé equations are a crossing point of many fields of mathematics. The Painlevé equations are generalized to many different directions. The Garnier systems are among their earliest generalizations, which is given by isomonodromic deformations of a general Fuchsian linear differential equation of the second order.

Around 1990, difference Painlevé equations were introduced in the study of quantum gravity. Sakai's theory of initial value spaces explains degeneration of whole of difference-differential Painlevé equations, and he found the elliptic Painlevé equation as a top equation of the degeneration diagram. The Painlevé hierarchy is related to the solition equations. Now higher order Painlevé-type equations are systematically studied by reductions from Drinfeld-Sokolov hierarchies or the UC hierarchies. Higher order Painlevé-type equations are also related to the Deligne-Simpson problem. Recently quantum extensions of the Painlevé equations are developping.

Monodromy preserving deformations are a powerful tool to study Painlevé equations. Recently asymptotic expansions of the Painlevé transcendents have been studied by many researchers. One hundred years have passed since Rene Garnier derived the Garnier systems from monodromy preserving deformations for general Fuchsian equations of the second order at 1912. And Pierre Boutroux obtained asymptotic expansions of the first Painlevé transcendent in 1913. In modern research on asymptotics of the Painlevé equations, the complex WKB methods are used to explain nonlinear Stokes phenomenon for the Painlevé transcendents.

The objective of the workshop "Various Aspects on the Painlevé Equations" is to show recent development of the study on the Painlevé equations. We study not only the Painlevé equations themselves, but also their applications to many mathematical fields and related topics.