On the log-concavity of the characteristic polynomial of a matroid

In this dissertation we address a long-standing conjecture, due to Heron, Rota and Welsh on the log-concavity of the characteristic polynomial of a matroid. After decades of attempts and a series of partial results, the conjecture was fully solved in 2018 by Adiprasito, Huh and Katz, using combinatorial analogues of several results in Algebraic Geometry concerning a particular cohomology ring called Chow ring. In February 2020, a new, simpler proof was announced by Braden, Huh, Matherne, Proudfoot and Wang. This dissertation is conceived to be a self-contained guide to support the reader in understanding these two papers, providing also the necessary background, a wide horizon ranging from Hodge Theory to Combinatorics to Toric Geometry. Moreover, we provide concrete and nontrivial examples of computations of Chow rings, of which we feel current literature is still lacking.