Gloria Mari-Beffa (Wisconsin)

Discrete Geometry of polygons and soliton equations

The relation between the discrete geometry of surfaces and completely integrable discrete systems has been well established in the last few decades, through work of Bobenko, Suris and many others. The recent introduction of discrete moving frames by Liz Mansfield, Gloria Mari-Beffa and Jing-Ping Wang, and the study of the pentagram map by Richard Schwartz and many others, has produced a flurry of work connecting the discrete geometry of polygons to completely integrable PDEs (both discrete and continuous) in any dimension, including connections to Combinatorics and to bi-Hamiltonian structures that often describe Liouville integrability. In this talk I will first review definitions and background on Liouville integrable systems and discrete moving frames on polygons. I will then describe a recent proof of the integrability of a discretization of Adler-Gelfand-Dikii systems (or generalized KdV systems), that used projective geometric vector fields and their direct connection to the moduli space of projective polygons.

We will describe in detail how introducing the projective group allowed us to describe the generalized KdV discretizations as completely integrable bi-Hamiltonian systems, a task that seemed out of reach through more direct classical methods. This is joint work with Jing-Ping Wang and Annalisa Calini.