Program

Figures produced in DPGraph by James W. Swift. A hyperplane arrangement is a finite collection of linear or affine hyperplanes in a fixed vector space. When the underlying field is ${\mathbb R}$ or $GF(q)$ there are natural enumerative, combinatorial and geometric questions that have been studied in great detail over much of the last century. On the other hand, the study of complex hyperplane arrangements really began around 1970 with work of Arnold, Brieskorn, Deligne, Aomoto, Hattori, and others, motivated by applications to the topology of configuration space and the discriminant, the algebraic structure of the braid group, and the construction of multivariable hypergeometric functions. By now there has developed a rich and extensive theory of hyperplane arrangements in general, which draws techniques and inspiration from diverse fields of mathematics. Indeed, since its beginnings the main feature of the theory has been its location at the intersection of combinatorics, topology, and algebra. Applications of the theory now include conformal field theory, braid group representations and knot invariants, general Artin groups, and many other areas. In addition, arrangements are intrinsically fascinating objects, simple enough in their definition to submit to explicit calculations, but complicated enough to display interesting and surprising properties. Recent progress has revealed unexpected connections between generalized hypergeometric functions, topological and algebraic invariants of arrangements, and several touchstones of classical mathematics: pencils of algebraic curves, Cartan matrices, classical point configurations, Latin squares. This semester-long program will focus on several inter-related aspects of current research in the field: Topology of the complement - characteristic and resonance varieties, fundamental groups, higher homotopy groups, cohomology of local systems. Commutative and skew-commutative algebra -- modules of derivations, resolutions, Orlik-Solomon algebras, Koszul duality and lower central series formulas. Applications -- generalized hypergeometric functions, Gauss-Manin connections, KZ equations, quantization, elliptic and $q$KZ equations, discriminants, flat connections and braid group representations, Artin groups, moduli spaces and Grassmann strata, subspace arrangements, wonderful models. Combinatorial methods -- lattice homology, group representations, combinatorial models, isotopy and homotopy type, random walks.