Abstract: 

Hodge-Riemann relations are a rich source of log-concave sequences. For example, Alexandrov's inequality for mixed discriminants and the Alexandrov-Fenchel inequality for convex bodies can both be deduced from Hodge-Riemann relations. After reviewing the classical theory, I will present recent extensions of Hodge-Riemann relations to ample vector bundles and an application to the log-concavity of the derivative sequence of Schur polynomials. This talk is partially based on joint work with Qing Lu.