Originally, cluster algebras were introduced to axiomatize part of the `dual canonical basis' for the ring of functions on a Lie group, and similar patterns have been found in many other contexts. This naturally begs the question: do cluster algebras have a natural basis, extending the set of cluster monomials?
Recent work by Gross, Hacking, Keel, and Kontsevich has proposed a basis of `theta functions'. These theta functions are defined by counting certain tropical curves in a `scattering diagram', a task which appears to be prohibitively difficult in practice. In this talk, I will demonstrate how to use this counting problem to restrict the behavior of the theta functions; in particular, to constrain their monomial support. In simple cases (that is, rank 2), such constraints completely characterize the theta functions, and may be used to give alternate characterizations which are far more computable in practice. Parts of this talk will cover joint work with Man Wai Cheung, Mark Gross, Gregg Musiker, Dylan Rupel, Salvatore Stella, and Harold Williams