Cluster algebras and their associated quivers were introduced in 2002 by Fomin and Zelevinsky to provide a framework to study total positivity in matrix groups. Since then, cluster algebras have been successfully linked to a wide range of subjects including Poisson geometry, integrable systems, higher Teichmuller spaces, commutative and non commutative algebraic geometry.
The Laurent phenomenon exhibited by some particular recurrences (Gale-Robinson, Octahedron, Somos) has first been proved using this framework.
In this seminar I will explain how to obtain a recurrence from a mutation-periodic cluster algebra and present some results concerning reduction of the associated dynamical system to lower dimension by using pre-symplectic geometry.

This is joint work with Esmeralda Sousa-Dias (IST/CAMGSD).