The pentagram map

The pentagram map was introduced by Schwartz as a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of real tori. We will explain how the real, complex, and finite field dynamics of the pentagram map are all related by the following generalization: the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties (except possibly in characteristic 2). The second hour will get into the details of the proof, especially the definition of the Lax representation and the spectral curve.