Abstract: Given a hyperplane arrangement, we associate two projective varieties: the wonderful compactification and the matroid Schubert variety. Adiprasito, Huh and Katz used the Hodge-Riemann relations of the wonderful compactification (and their combinatorial generalizations) to prove that the coefficients of their characteristic polynomials form a log-concave sequence. In a joint work with Huh, we proved Dowling and Wilson's Top-heavy conjecture for realizable matroids by applying the hard Lefschetz theorem to the matroid Schubert varieties. In a more recent work with Braden, Huh, Matherne and Proudfoot, we proved the Top-heavy conjecture to arbitrary matroids. In this talk, I will go over some of the key ideas about the proof of the Top-heavy conjectures. If time permits, I will also mention some on-going works with my students Colin Crowley and Connor Simpson towards generalizations to type A Coxeter matroids.