The pentagram map, Poncelet polygons, and commuting difference operators
Series: Mathematical Physics and Probability Seminar
Presenter: Anton Izosimov, University of Arizona
The pentagram map, introduced by Richard Schwartz in 1992, is a discrete dynamical system on planar polygons. By definition, the image of a polygon P under the pentagram map is the polygon whose vertices are intersections of shortest diagonals of P (i.e. diagonals connecting second nearest vertices). The pentagram map is a completely integrable system which can be thought of as a lattice version of the Boussinesq model in hydrodynamics.
In this talk, we will discuss polygons which are projectively equivalent to their image under the pentagram map. One can think of such polygons as pentagram map "solitons". The result is that a real convex polygon P is projectively equivalent to its image under the pentagram map if and only if there is a conic inscribed in P and a conic circumscribed about P. I will explain the idea of the proof, which is based on the theory of commuting difference operators, elliptic curves, and theta functions.