Ptolemy theorem, cluster algebras from surfaces and a conjecture on friezes (Ana García Elsener). Ptolemy’s theorem is deeply tied with the recent theory of cluster algebras. We can define cluster algebras from unpunctured surfaces just by understanding an exchange relation similar to Ptolemy’s theorem. These cluster algebras are generated by algebraic expressions obtained from curves.

In this context a frieze can be interpreted either as a ring homomorphism from the cluster algebra to the integers, or as a function assigning a certain (modified hyperbolic) length to the set of curves. A frieze is unitary if there exists a set of curves forming a triangulation of the surface with all the arcs having length equal to one. We will explain these facts by examples and, hopefully, mention some recent theorems and conjectures.