Cluster algebras and triangulated surfaces. Part I: Cluster complexes

Cluster algebras are a class of commutative rings endowed with an additional combinatorial structure, which involves a set of distinguished generators (cluster variables) grouped into overlapping subsets (clusters) of the same cardinality. The original motivation for cluster algebras came from representation theory, specifically the study of dual canonical bases and total positivity phenomena in semi-simple Lie groups. In subsequent years, constructions involving cluster algebras were discovered in various mathematical contexts, including a topological/geometric one that is the main focus of this paper.