Stanley gave us necessary conditions that f-vectors of simplicial polytopes must satisfy by relating the problem to the hard Lefschetz theorem in algebraic geometry, an insurmountably deep and intimidating theorem in algebraic geometry. McMullen conjectured that these conditions are necessary in general, posing before us the problem of proving the hard Lefschetz theorem beyond what algebraic geometers would dream of.

I will talk about the revenge of combinatorics, and in particular discuss how Hall's marriage theorem (or rather, one of its proofs) provides a way to this deep algebraic conjecture.

Based on arxiv:1812.10454