Abstract: A cohomology ring is a ring associated to a topological space that encodes geometric properties of the space---for instance, the dimension, number of puncture holes, number of connected components, and and other invariants of the space. Cohomology is an incredibly useful tool, but also frequently difficult to compute. We will describe a combinatorial method to compute cohomology rings for a family of topological spaces, one that removes many of the computational barriers to hands-on calculations. Surprisingly, we'll do this by computing apparently more complicated rings called equivariant cohomology rings. The construction we describe is often known as GKM theory; the underlying combinatorial objects also generalize splines, piecewise polynomial functions that are important in geometric combinatorics and in applications. We'll conclude with a collection of open questions.