Positivity in Cluster Algebras

Cluster algebras are commutative rings introduced by Fomin and Zelevinsky in 2001 as an algebraic/combinatorial framework for the study of total positivity and dual canonical basis in semisimple algebraic groups. Some cluster algebras can be realized as coordinate rings of algebraic varieties, like for example the Grassmannians, and they hence inherit a notion of total positivity. Inspired by this notion, one can introduce a notion of positivity in an abstract cluster algebra. The aim of this talk is to present such notion and to provide techniques to study and explicitly describe the positive elements.