Richard P. Stanley Seminar in Combinatorics

The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini-Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences. This is a joint paper with Eva Feichter (math.CO/0411260).

In 1980, Orlik and Solomon proved that the cohomology ring of the complement of a complex hyperplane arrangement is determined by the intersection poset of the arrangement. The relationship between combinatorial and topological aspects of arrangements has subsequently become a focal point of the subject. In this talk, we investigate the extent to which the topology of the boundary manifold of an arrangement is combinatorially determined. Specifically, we discuss an analogue of the Orlik-Solomon theorem for the boundary manifold.