Marco Bertola (Concordia)

Title: Padé approximants on Riemann surfaces and KP tau functions.

Abstract: The talk has two relatively distinct but connected goals; the first is to define the notion of Padé 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 Padé–like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert 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.  

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 “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2–Toda hierarchy when considering the w

Web -Please fill in this form: https://forms.gle/S1NcNQ8BxkzfAXcj9