The dimer model is the study of the set of dimer configurations (or perfect matchings) of a graph. In this talk, I will begin with an overview of the combinatorics of the dimer model, highlighting surprising connections between the dimer model and other areas of math such as algebraic geometry.
I will then present joint work with Ben Young and Gautam Webb which uses the dimer model and the less well-studied double-dimer model to resolve an open conjecture from enumerative geometry. To do so, we prove that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions. Our proof is combinatorial, and no prior knowledge of enumerative geometry (or the dimer model) is required to understand the talk.