We study the class of Lorentzian polynomials. The class contains homogeneous stable
polynomials as well as volume polynomials of convex bodies and projective varieties. We …

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral
tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the …

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative
ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a …

Many important sequences in combinatorics are known to be log-concave or unimodal, but
many are only conjectured to be so although several techniques using methods from …

We introduce certain torus-equivariant classes on permutohedral varieties which we call
“tautological classes of matroids” as a new geometric framework for studying matroids …

We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The
decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem …

We give a self-contained proof of the strongest version of Mason's conjecture, namely that
for any matroid the sequence of the number of independent sets of given sizes is ultra log …

This monograph studies the interplay between various algebraic, geometric and
combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and …

A variety of probabilistic notions of network reliability of graphs and digraphs have been
proposed and studied since the early 1950s. Although grounded in the engineering and …

Proceedings of the International Congress of Mathematicians (ICM 2018) : COMBINATORIAL
APPLICATIONS OF THE HODGE–RIEMANN RELAT Page 1 P . I . C . M . – 2018 Rio de Janeiro …