Title: The pentagram map and algebraic integrability

Abstract: The pentagram map was introduced by Schwartz as a dynamical system on 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, after changing coordinates by a diffeomorphism, it is a translation 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 is birational to a translation on a family of Jacobian varieties. Soloviev proved this over the complex numbers in 2013, and we extend the result to algebraically closed fields of any characteristic not equal to 2.