Noncommutative bispectral Darboux transformations

Milen Yakimov (Louisiana State University)

Monday 5th October, 2015 15:00-16:00 Maths 522


The bispectral problem of Duistermaat and Grunbaum asks for finding all functions in 2 variables that are eigenfunctions in each variable. It was first posed for the purposes of applied mathematics (tomography and time-band limiting) but was soon related to various structures in pure mathematics: KdV and Calogero-Moser integrable systems, ideal structure of the first Weyl algebra, quiver varieties and others. We will describe a general result that proves bispectrality of Darboux transformations of such functions with values in arbitrary finite dimensional algebras. It captures all previous examples of bispectral Darboux transformations and has many new applications. We will also describe a classification of these transformations in the rank 1 case. In the special case when the target algebra equals the base field, it reproduces the classical Calgero-Moser spaces. This is a joint work with Joel Geiger (MIT) and Emil Horozov (Sofia Univ).

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