A conjecture of Read predicts that the coefficients of the chromatic polynomial of any graph form a log-concave sequence. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. Both conjectures are special cases of the famous Rota conjecture asserting the log-concavity of the coefficients of the characteristic polynomial of any matroid. The recent story of these problems starts in 2010, when June Huh proved Rota's conjecture for the special case of hyperplane arrangements by identifying the Whitney coefficients with mixed multiplicities of its Jacobian ideal. It subsequently emerged that virtually all proofs we could come up with for this case use nontrivial geometric facts about the arrangement and/or Hodge theory for projective varieties, and the more general conjecture of Rota for possibly "nonrealizable" configurations/matroids remained open until recently. I will discuss how to extend Hodge theory beyond the classical setting to general matroids, starting with the surprising joint work with Björner on Lefschetz theorems for Mikhalkin's p,q-groups, and then discuss the proof of the "Kähler package" for general matroidal fans, which proves the Rota and Welsh conjecture in full generality. All proofs are purely combinatorial, and do not rely on analytifications or projective algebraic geometry, although there are some useful relations I will mention.
Based on joint work with June Huh and Eric Katz.