Pentagram map

Introduced by R. Schwartz about 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. I shall describe a new family of completely integrable discrete dynamical systems, including the pentagram map, and explain their connection to cluster algebras and weighted directed networks on surfaces (joint work with M. Gekhtman, S.Tabachnikov, and A. Vainstein).