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.[-]

The pentagram map and its analogs act on interesting and complicated spaces. The simplest of them is the classical moduli space $M_{0,n}$ of rational curves of genus $0$. These moduli spaces have a rich combinatorial structure related to the notion of "Coxeter frieze pattern" and can be understood as a "cluster manifolds". In this talk, I will explain how to describe the action of the pentagram map (and its analogs) in terms of friezes. The main goal is to understand how does this action fit with the cluster algebra structure, in particular, with the canonical (pre)symplectic form.[-]