Manchester Geometry Seminar 2014/2015

Cluster algebras were introduced by Fomin and Zelevinsky in 2002 for studying total positivity of Grassmannians. Very soon after that it turned out that the notion is connected to numerous different fields in mathematics (such as Poisson geometry, representation theory, integrable systems, combinatorics of polytopes, Teihmuller theory, dilogorithm identities and many others). In this talk we will introduce cluster algebras and show some connections to Coxeter groups. More precisely, we will see how an experience in classification of Coxeter hyperbolic polytopes helped to solve some classification problem in cluster algebras. The work is joint with Pavel Tumarkin and Michael Shapiro.