Hyperplane arrangements: computations and conjectures

This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as Koszul and Lie algebra methods, homological techniques, and the Bernstein-Gelfand-Gelfand correspondence, all illustrated with concrete calculations. We also explore connections of arrangements to other areas, such as De Concini-Procesi wonderful models, the Feichtner-Yuzvinsky algebra of an atomic lattice, fatpoints and blowups of projective space, and plane curve singularities.