Arbeitsgemeinschaft: Cluster Algebras

Abstract

Cluster algebras, invented by Sergey Fomin and Andrei Zelevinsky around the year 2000, are commutative algebras endowed with a rich combinatorial structure. Fomin-Zelevinsky's original motivations came from Lie theory but in the past two decades, cluster algebras have had strikingly fruitful interactions with a large array of other subjects including Poisson geometry, discrete dynamical systems, (higher) Teichmüller spaces, commutative and non-commutative algebraic geometry, representation theory,.... In this Arbeitsgemeinschaft, we have focused on 1) basic definitions and theorems, 2) cluster structures on algebraic varieties and 3) the recent connection between cluster algebras and symplectic topology, with its recent application to the construction of cluster structures on braid varieties.