Colloque des sciences mathématiques du Québec

29 novembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/HNE) Sur place

Higher Pentagram Maps via Cluster Mutations and Networks on Surfaces

Colloque par Michael Gekhtman (Université of Notre-Dame)

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras, a new and rapidly developing area with many exciting connections to diverse fields of mathematics. In this talk I will explain that one possible family of higher-dimensional generalizations of the pentagram map is a family of discrete integrable systems intrinsic to a certain class of cluster algebras that are related to weighted directed networks on a torus and a cylinder. After presenting necessary background information on Poisson geometry of cluster algebras, I will show how all ingredients necessary for integrability - Poisson brackets, integrals of motion - can be recovered from combinatorics of a network. The talk is based on a joint project with M. Shapiro, S. Tabachnikov and A. Vainshtein.


Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214