Dimers on a disc and Grassmannian cluster algebras

Dimer models are basically bipartite graphs on a surface (possibly with boundary). Cluster algebras are commutative rings generated by a (usually infinite) set of variables that are produced in an iterative manner. One starts with a seed (an initial set of variables and a matrix) and produce new seeds by a prescribed process called mutation. Doing all possible mutations gives the generating set of the cluster algebra. They have been pretty fashionable in the algebra and representation theory world for the last 10 years or so and they appear in a surprising number of different contexts.