The pentagram map, Poncelet polygons and commuting difference operators

The pentagram map is a discrete integrable system on the space of projective equivalence classes of planar polygons. By definition, the image of a polygon P under the pentagram map is the polygon P' whose vertices are the intersection points of consecutive shortest diagonals of P, i.e. diagonals connecting second nearest vertices. In the talk, I will discuss the problem of describing the fixed points of the pentagram map. In other words, the question is: which polygons P are projectively equivalent to their "diagonal" polygons P'? It is a classical result of Clebsch that all pentagons have this property. Furthermore, in 2005 R.Schwartz proved that this property is also enjoyed by all Poncelet polygons, i.e. polygons that are inscribed in a conic section and circumscribed about another conic section. In the talk I will argue that in the convex case the converse is also true: if P is convex and projectively equivalent to its diagonal polygon P', then P is a Poncelet polygon. The proof is based on properties of commuting difference operators, real elliptic curves, and theta functions.