Cluster algebra and Poisson geometry

Michael Shapiro

In the minicourse we describe relations between cluster algebras and Poisson structures (dually, 2-forms). One of the example of this correspondence is the Weil-Petersson form on the (decorated) Teichmueller space. Using the correspondence we prove the "cluster determines seed" conjecture for cluster algebras of geometric type. We explain also cluster structure on planar oriented networks and give an interpretation of cluster transformation as Baecklund-Darboux transformation for open Toda lattices.