The work of Braden, Huh, Matherne, Proudfoot, and Wang (Braden et al. 2020), which grew out of their 2020 preprint and has recently been accepted to the Journal of the American Mathematical Society, proves a far-reaching form of a classical incidence phenomenon.
Recall that a hyperplane in is a -dimensional affine subspace. More generally, if , their affine span is the smallest affine subspace of containing them; a finite point set is said to determine an affine subspace if is the affine span of some subset of .
A classical theorem in plane geometry says that points in the plane, not all lying on a single line, determine at least lines. In 1951, Motzkin (Motzkin 1951) proved a higher-dimensional version of the plane theorem: points in , not all lying on a single hyperplane, determine at least hyperplanes. In 1970, Greene (Greene 1970) strengthened this result by proving that one can choose these hyperplanes in a pointwise compatible way: if and is the set of hyperplanes determined by , then there is an injective map such that for every . In 2017, Huh and Wang (Huh and Wang 2017) generalized Greene’s theorem from points and hyperplanes to subspaces of arbitrary dimensions.
The case and recovers Greene’s theorem: the elements of are the points of , while the elements of are the hyperplanes determined by . Braden, Huh, Matherne, Proudfoot, and Wang (Braden et al. 2020) proved a much broader version of this phenomenon for arbitrary matroids.
Let be a matroid on a finite ground set . For a subset , its rank is and the rank of the matroid is . A subset is a flat of rank if and every element outside increases the rank by one, that is, Let denote the set of flats of rank in .
Taking in Theorem gives whenever . This inequality was conjectured by Dowling and Wilson in 1974 and became known as the Dowling–Wilson conjecture, or the top-heavy conjecture. The Huh–Wang theorem proves the conjecture for matroids realizable over a field, while Theorem proves it for all matroids.
The proof of Theorem follows the Hodge-theoretic strategy. The guiding analogy is with the cohomology of algebraic varieties: to a matroid one attaches a carefully constructed combinatorial substitute for a cohomology space, and then proves analogues of Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. In (Braden et al. 2020), the relevant object is the intersection cohomology module of a matroid. Once these Hodge-theoretic statements are established, the injective maps in Theorem and the top-heavy inequalities follow.