Abstract: One of the central questions in the theory of hyperplane arrangements is whether the module of logarithmic derivations of an arrangement is a free module over the underlying polynomial ring (such an arrangement is called free). Terao posed the question of whether this property is combinatorial, that is, whether freeness of an arrangement can be determined from the intersection lattice of the arrangement. In relation to this, we will discuss several obstructions to freeness, including Terao's celebrated factorization theorem, the notion of $k$-formality (due to Brandt and Terao), and Ziegler's multi-restriction. This latter obstruction leads naturally to multi-arrangements. Time permitting we will discuss examples which illustrate the non-combinatorial nature of free multi-arrangements and a recent homological characterization of freeness which extends the notion of $k$-formality. No prior knowledge of arrangements will be assumed.
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