  We survey some of the mechanisms used to prove that naturally defined
sequences in combinatorics are log-concave. Among these mechanisms are
Alexandrov's inequality for mixed discriminants, the Alexandrov Fenchel
inequality for mixed volumes, Lorentzian polynomials, and the Hard Lefschetz
theorem. We use these mechanisms to prove some new log-concavity and extremal
results related to partially ordered sets and matroids. We present joint work
with Ramon van Handel and Xinmeng Zeng to give a complete characterization for
the extremals of the Kahn-Saks inequality. We extend Stanley's inequality for
regular matroids to arbitrary matroids using the technology of Lorentzian
polynomials. As a result, we provide a new proof of the weakest Mason
conjecture. We also prove necessary and sufficient conditions for the
Gorenstein ring associated to the basis generating polynomial of a matroid to
satisfy Hodge-Riemann relations of degree one on the facets of the positive
orthant.
