Cluster algebras from surfaces

Dylan Thurston

Triangulations of surfaces with at least one puncture provide many examples of cluster algebras which are mutationally finite: there are only a finite number of different combinatorial types of clusters. We establish basic properties of the cluster algebras associated with oriented, bordered surfaces with marked points. We further show how to introduce coefficients into these cluster algebras and relate them to the Teichmüller space of the corresponding hyperbolic surface, which we will introduce.

This is joint work with Sergey Fomin and Michael Shapiro.