9504457 Orlik This work applies tools developed in the theory of arrangements to several problems arising from the theory of Aomoto-Gelfand hypergeometric functions. A twisted de Rham complex is of special interest. It is particularly important to compute the dimension of the top cohomology group and to find an explicit basis for it, because this cohomology group may be identified with the space of hypergeometric functions. The present approach is by using similar complexes with different and simpler coboundary maps. In particular, a complex of logarithmic forms and the Orlik-Solomon algebras are studied. The calculation of the cohomology of these complexes is usually easier than the twisted de Rham complex itself. The aim is to prove that under certain conditions these cohomology groups are isomorphic to the twisted de Rham cohomology groups. Another goal is to use Morse theory to find twisted cycles that give a basis for the twisted de Rham homology. The study of arrangements of hyperplanes is a new branch of mathematics on the interface between topology, algebra, algebraic geometry, combinatorics, and analysis. Applications include coding theory, computer science, spline functions, and physics. The present work is related to hypergeometric functions, representations of Lie groups, Knizhnik-Zamolodchikov equations, and conformal field theory. These last topics are of great interest to mathematicians and theoretical physicists. ***
