By what has been shown in previous talks, we have seen that we can show coefficients of the characteristic polynomial of a realizable matroid can be realized via specific computations in the Chow ring of its wonderful compactification. In this talk, we will introduce the notion of Poincare duality algebras, which are graded algebras with a degree function giving an isomorphism from the top degree to the base field that induces a non-degenerate pairing between complementary degrees of the algebra. Furthermore, we will introduce a notion of hard Lefschetz and Hodge-Riemann relations for such algebras. When a Poincare duality algebra satisfies a certain version of these properties, we can show that the log-concavity of its "volume polynomial" is equivalent to the eigenvalues of a symmetric form on the algebra arising from the Hodge-Riemann relations. Because the Hodge-Riemann relations in appropriate degree imply the log-concavity of the coefficients of the characteristic polynomial of the matroid, this framework gives us a program to establish the log-concavity result. Throughout this talk, I will attempt to provide intuition from the case of the Chow rings of smooth projective varieties.