Abstract: Hyperplane arrangements arise naturally in various contexts, ranging from the study of reflection groups, matroids, and configuration spaces. This sequence of lectures will emphasize the combinatorial and algebraic aspects of the theory, including some interactions with toric and tropical geometry. Topics will include: real hyperplane arrangements, reflection groups and enumerative aspects; hyperplane arrangements as linear matroid realizations; tropicalization of a linear space and the Bergman fan; various compactifications of arrangement spaces, and a combinatorial view of cohomology algebras associated with an arrangement.

Abstract: Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement and related spaces. These lectures will emphasize the topological and geometric aspects of the theory, including some interactions with low-dimensional topology, singularity theory, and group theory. Topics will include: Milnor fibration, boundary manifold, and branched covers, cohomology jump loci, and Lie algebras associated to the fundamental group. Several classes of arrangements will be discussed, with emphasis on concrete examples and computer-aided computations.

Given a projective line arrangement, a folklore conjecture (attributed to Salvetti) asserts that if the graph of double point intersections is connected then the algebraic monodromy of the Milnor fiber of the arrangement is trivial. The conjecture has been proved by Bailet under some combinatorial conditions using resonance varieties, by Salvetti and Serventi for real complexified line arrangements satisfying certain technical conditions, and by Venturelli for line arrangements containing two multiple points such that every line of the arrangement contains one of them. In this talk, we will present a proof of the conjecture for a certain class of real complexified line arrangements, based on connectivity properties of the real picture of such an arrangement. Joint work with Moisés Herradón Cueto and Laurentiu Maxim.

It is known that the topological type of the boundary manifold of a complex projective line arrangement is combinatorially determined; it is obtained by plumbing construction along the incidence graph. In this talk, we focus on its generalization to arbitrary combinatorial line arrangements, which we also call the boundary manifold. We construct a basis of the homology groups by constructing explicit cycles, and compute the intersection product. This allows us to generalize the doubling formula for the cohomology rings, originally proved by Cohen and Suciu for realizable cases.This talk is based on the preprint arXiv:2507.06728.