Aarhus University Seal / Aarhus Universitets segl

Seminar by Mikhael Shapiro (Michigan State University)

Title: Poisson geometry of directed networks and integrable lattices

2013.11.25 | Christine Dilling

Date Fri 13 Nov
Time 16:30 17:30
Location Aud. D3


Recently, Postnikov used weighted directed planar graphs in a disk to parametrize cells in Grassmannians. We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson in the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as an appropriate Poisson homogeneous space. Next, we generalize Postnikov's construction by defining a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We use a special kind of directed networks in an annulus to study a cluster algebra structure on a certain space of rational functions and show that sequences of cluster transformations connecting two distinguished clusters are closely associated with Backlund-Darboux transformations between Coxeter-Toda flows in GL(n). This is a joint work with M.Gekhtman and A.Vainshtein