In recent years there have been a number of papers where people construct monomial ideals with a given resolution type. Many of these constructions can be seen to be instances of ``coordinatizing" a finite atomic lattice. In this talk I will explain what this means and demonstrate this fact using the constructions pertaining to monomial ideals whose minimal resolutions are supported on trees. Furthermore I will give a discussion on how linking these two constructions can give us insight into how to study resolutions of monomial ideals in general by studying the space of all finite atomic lattices with a fixed number of atoms (which is itself a finite atomic lattice).
No prior knowledge on commutative algebra or more specifically free resolutions will be assumed. All of the necessary notions can be explained purely in the context of combinatorics, as they will be in the talk.