Applications
Cluster algebra is, of course, interesting in its own right, but it can also be used to study the properties of other objects. Notably, cluster algebras of particular ranks can correspond to the behavior of triangulated oriented surfaces, where clusters and cluster variables have explicit connections to arcs and triangulations [4]. In other words, certain cluster algebras can provide information into the topological nature of surfaces that they relate to.
Additionally, cluster algebra was partly inspired by the idea of "total positivity" in matrices, as it arose from the study of such topics --- in fact, the construction of a cluster algebra inherently creates a relation to total positivity, as detailed in Section 2 of [5]. This means cluster algebra can be a useful tool in studying anything to do with totally positive matrices.
