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 …

Hyperplane arrangement theory is a very active area of research, combining ideas from
combinatorics, algebraic topology and algebraic geometry in a blend that is both tasty and …

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

In a recent paper, the first author proved the log-concavity of the coefficients of the
characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a …

Matroids are ubiquitous in modern combinatorics. As discovered by Gel'fand, Goresky,
MacPherson and Serganova there is a beautiful connection between matroid theory and the …

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

We show that the maximum likelihood degree of a smooth very affine variety is equal to the
signed topological Euler characteristic. This generalizes Orlik and Terao's solution to …

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 …

Combinatorial reciprocity is a very interesting phenomenon, which can be described as
follows: A polynomial, whose values at positive integers count combinatorial objects of some …