Coordinate rings of some natural geometrical objects (for example, Grassmannians) possess distinguished sets
of generators (Plucker coordinates). Cluster algebras are generalizations of such coordinate rings.
Sets of generators of a cluster algebra are organized in a tree-like structure with simple transformation rules
between neighboring sets of generators.
Fomin and Zelevinsky obtained a complete classification for cluster algebras of finite type (i.e., with finite number of generators). This classification coincides with the Cartan-Killing classification of simple Lie algebras.
We try to describe cluster algebras with finitely many transformation rules, so called cluster algebras of finite mutation type.
We prove that a special class of cluster algebras originating from triangulation of Riemann surfaces is of finite mutation type.
This is joint work with Sergey Fomin.