The top-heavy conjecture for vectors and matroids

A 1948 theorem of de Bruijn and Erdős says that if nn points in a projective plane do not lie all on a line, then they determine at least n lines. More generally, Dowling and Wilson conjectured in 1974 that for any finite set of vectors spanning a dd-dimensional vector space, the number of kk-dimensional spaces that they span is at most the number of (d−k)(d−k)-dimensional spaces they span, for all kk smaller than d/2d/2. In fact, Dowling and Wilson conjectured this more generally for matroids, a combinatorial abstraction of incidence geometries which do not have to arise from actual vector configurations. The conjecture for vector configurations was proved in 2017 by Huh and Wang, using the hard Lefschetz Theorem for intersection cohomology applied to a singular algebraic variety associated to the vector configuration. I will discuss their proof, and also more recent joint work with Huh, Matherne, Proudfoot and Wang which proves the conjecture for all matroids, by constructing a combinatorial replacement for intersection cohomology which makes sense even for matroids which do not arise from a vector configuration.