Towards a nonlinear Schwarz's list

TL;DR

This paper surveys the search for algebraic solutions to the sixth Painleve equation, exploring its geometric interpretation and connections to algebraic groups, braid groups, and cubic surfaces.

Contribution

It provides a comprehensive overview of algebraic solutions of Painleve VI and their geometric and algebraic structures, highlighting nonlinear analogues of classical hypergeometric equations.

Findings

Discussion of algebraic solutions and their classification.

Connection of Painleve VI to geometric structures like cubic surfaces.

Interpretation of Painleve VI as a nonlinear Gauss-Manin connection.

Abstract

This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin connection, i.e. as a nonlinear differential equation 'coming from geometry', much as Picard-Fuchs equations arise in the case of cohomology with abelian coefficients.