During the last decade, several important long-standing conjectures about matroids, as the Heron-Rota-Welsh conjecture, has been solved thanks to the development of the combinatorial Hodge theory by Huh and his collaborators. Classical Hodge theory is about the cohomology of complex varieties. For matroids representable over the complex field, this theory applied to some complex varieties associated to the matroids implies the Heron-Rota-Welsh conjecture. For a general matroid, Adiprasito, Huh and Katz achieved to develop a combinatorial Hodge theory for (Chow rings of) matroids which works as if one can associate a complex variety to the matroid, though this is not the case. The proof is very clever but does not give much insight into why this combinatorial Hodge theory works in general.

Actually, every matroid is in some sense representable over the tropical hyperfield. Moreover, in a joint work with Amini, we developed a tropical Hodge theory. Hence, to every matroid one can associate a tropical variety (the canonically compactified Bergman fan), and the Hodge properties of this variety imply the Heron-Rota-Welsh conjecture. We thus get a geometric proof of the conjecture, as well as an extension of the applicability of the combinatorial Hodge theory. The heart of our proof relies on a very interesting induction, based on the deletion-contraction induction.