Cluster algebras, defined by Fomin and Zelevinsky, are a type of commutative algebra whose generators (called cluster variables) appear in fixed-size distinguished subsets (called clusters). Two hallmark features of cluster algebras are the Laurent phenomenon and positivity, which together say that every cluster algebra element can be written as a Laurent polynomial with positive integer coefficients in terms of any choice of cluster. These Laurent polynomials are called cluster expansions. A particularly approachable type of cluster algebra is that of surface type, which can be modeled using triangulated surfaces. In this talk, we survey three different cluster expansion formulas for cluster algebras of surface type, each of which produces the integral coefficients by counting a concrete combinatorial object. Through concrete examples, we will understand each cluster expansion formula and explore the relationships between the combinatorial objects that they use. (This talk will assume no prior knowledge of cluster algebras.)
