If you take a finite collection of n distinct planes in R^3 such that the intersection of all of them is the origin, then the number of distinct lines you get by taking intersections of the planes is at least n. This is an example of a more general statement, called the “Top-Heavy Conjecture” for matroids, that Dowling and Wilson conjectured in 1974. This long-standing conjecture was recently settled by introducing and studying the intersection cohomology IH(M) of a matroid M.

In this talk, I will explain the basics of IH(M) and survey how it fits into the broader field of combinatorial Hodge theory. I will also discuss ongoing joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang that gives a new, simpler characterization of IH(M) which opens the door to several exciting new directions. No specific background knowledge is necessary to follow this talk, and many examples will be given throughout.