Discrete Monodromy, Pentagrams, and the Method of Condensation

TL;DR

This paper explores the pentagram map's invariants, connecting it to monodromy of differential equations and Dodgson's condensation, revealing deep geometric and algebraic structures.

Contribution

It identifies conjectural invariants of the pentagram map and links the construction to monodromy and determinant condensation methods.

Findings

Identified conjectural invariants of the pentagram map

Connected the pentagram map to monodromy of differential equations

Related the construction to Dodgson's condensation method

Abstract

This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the invariants for this map, and along the way relates the construction to the monodromy of 3rd order differential equations, and also to Dodgson's method of condensation for computing determinants.