Fecha y lugar: Lunes, Abr 12, 16:30 hrs CST.

Por: Anton Izosimov, Univ of Arizona, Tucson, USA.

Título: The pentagram map and Poncelet polygons

Abstract: Let P be a planar pentagon, and let P' be the pentagon whose vertices are the intersection points of diagonals of P. Then a classical result due to Clebsch says that P and P' are projectively equivalent (i.e. there is a projective transformation taking P to P').

In 2007, R. Schwartz generalized this result to Poncelet polygons, i.e. polygons which are simultaneously inscribed in a conic and circumscribed about another conic (clearly, any pentagon has this property, so Poncelet polygons can be regarded as "generalized pentagons"). Namely, Schwartz proved that if P is a Poncelet polygon with odd number of vertices, and P' is the polygon whose vertices are the intersection points of shortest diagonals of P (i.e. diagonals connecting second nearest vertices), then P and P' are projectively equivalent.

In the talk, I will argue that in the convex case this property characterizes Poncelet polygons. In other words, if P is a convex odd-gon which is projectively equivalent to its “diagonal” polygon P', then P (and hence P') is a Poncelet polygon. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.