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.

Abstract

Abstract