On The Geometric View Of Pentagram Integrals Of Polygons Inscribed In Non-Degenerate Conics.

Abstract

The Pentagram map is a well notable integrable system that is de_ned on the moduli space of polygons. In 2005, Richard Evan Schwartz introduced certain polynomials called pentagram integrals (Monodromy invariants) of the pentagram map and de_ned certain associated integrals, the analogous _rst integrals. Schwartz further studied in 2011 with S. Tabachnikov on how these integrals behave on inscribed polygons. They discovered that the integrals are equal for every given weight of polygons inscribed in non-degenerate conics. However, the proof of their outcome was combinatorial which appeared to be more involving hence there was a need for quite a simple proof. Anton Izosimov in 2016 gave quite a simple conceptual geometric proof of these invariants of polygons inscribed in non-degenerate conics. In this thesis, we seek to analyse the geometry of these invariants by reviewing Anton's work. Our core analyses is that for any polygon inscribed in a non-degenerate conic, the analogous monodromy should satisfy a certain self-duality relation.