The pentagram map, Poncelet polygons, and commuting difference operators
The pentagram map takes a planar polygon P to a polygon P' whose vertices are the intersection points of consecutive shortest diagonals of P. This map is known to interact nicely with Poncelet polygons. A polygon is called Poncelet if it is inscribed in a conic section and circumscribed about another conic section. A theorem of R.Schwartz says that if P is a Poncelet polygon, then the image of P under the pentagram map is projectively equivalent to P. In this talk, I will argue that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its image under the pentagram map, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.