Introduction to cluster algebras

Cluster algebras were introduced by Fomin and Zelevinsky in 2002 while studying total positivity and canonical bases in Lie theory. The definition is elementary: begin with a finite set of variables and a directed graph, then repeatedly apply a combinatorial rule called mutation to produce new variables. Two surprises follow almost immediately: every variable produced this way is a Laurent polynomial in the original variables, and in many cases the resulting algebras turn out to be coordinate rings of familiar objects such as Grassmannians, double Bruhat cells, and decorated Teichmüller spaces of surfaces. We will start from the definition, work through small examples by hand, and then look at how these algebras arise in representation theory, algebraic and hyperbolic geometry, and number theory. No prior exposure to cluster algebras will be assumed.