Mathematics > Classical Analysis and ODEs
[Submitted on 23 Jul 2007 (v1), last revised 31 Jul 2008 (this version, v2)]
Title:Towards a nonlinear Schwarz's list
View PDFAbstract: 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.
Submission history
From: Philip Boalch [view email][v1] Mon, 23 Jul 2007 12:51:13 UTC (635 KB)
[v2] Thu, 31 Jul 2008 01:07:08 UTC (634 KB)
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