Abstract : The goal of this short course is to explain the concept of “triality”, which is an isomorphism between a large class of of (generalized) tame frieze patterns, certain spaces of linear difference equations, and the moduli space of configurations of points in the projective space. This approach will be used in several directions, in particular:
• to define “good” coordinates on moduli spaces related to cluster algebras and symplectic geometry
• to find simple proofs of some properties of friezes, such as periodicity
• to connect the subject to dynamical systems
• to create new types of friezes
• to count friezes of certain types.
The presentation is based on several joint papers with Sophie Morier-Genoud, Sergei Tabachnikov, and also Charles Conley, and Richard Schwartz. Coxeter friezes and geometry of the projective line. I will start with the classical Coxeter's frieze patterns and connect them to configurations of point in the 1-dimensional projective space P1. As a consequence, a (pre)symplectic structure on the space of Coxeter's friezes will be described. The basic notions of projective geometry, such as the cross-ratio and Schwarzian derivative will be recalled/explained and used.