|XXXIV Workshop on Geometric Methods in Physics||28.06-4.07.2015|
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The Painleve VI equations, Poncelet polygons, and the Schlesinger equations
In 1990's Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation associated to the Poncelet polygons, inscribed in a conic and circumscribed about another conic. We will show that Hitchin's construction is the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation and it can be formulated in an invariant way, in terms of an Abelian differential of the third kind on the associated elliptic curve. The last observation allows to obtain solutions to the corresponding Schlesinger system in terms of this differential as well. The solution of the Schlesinger system admits a natural generalization to higher genera, and it is related to higher-dimensional Poncelet polygons. This is a joint work with V. Shramchenko. The research is supported by NSF grant no. 1444147.