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Bäcklund transformations, discrete Painlevé equations, and Sakai’s geometric classification scheme

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Date and time:Ìý

Tuesday, February 17, 2015 - 4:30pm

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ECOT 226

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The goal of this talk is to give a brief introduction into the geometric approach of H. Sakai to the theory of discrete (primarily, difference) Painlevé equations. We begin by showing how discrete Painlevé equation arise from Bäcklund transformations of the usual differential Painlevé equations using P_II and alt. d-P_I as an example. We then show how to construct, from the resulting difference Painlevé equation, a certain rational algebraic surface that is called the space of the initial conditions (or the Okamoto surface) of the equation. In the process of constructing this surface we will introduce all of the main ingredients that are needed to explain the general classification scheme of H.Sakai for discrete Painlevé equations. In this approach each discrete Painlevé equation corresponds to a particular translation in the Picard lattice of its Okamoto surface. We will conclude the talk by indicating how to obtain the equation starting from the given translation vector.