Marco Bertola (Concordia)
Title:听Pad茅 approximants on Riemann surfaces and KP tau functions.
Abstract:听The talk has two relatively distinct but connected goals; the first is to define the notion of Pad茅 approximation of Weyl-Stieltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Pad茅鈥搇ike approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann鈥揌ilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. 听
The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a 鈥渢au鈥� function. We show how this tau function satisfies the Kadomtsev鈥揚etviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2鈥揟oda hierarchy when considering the w
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