Mathematical Sciences: RUI: The Homotopy Theory of Arrangements

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Michael Falk will continue his research into the interplay of combinatorics and algebra with the topology of the complement of an arrangement of hyperplanes over the field of complex numbers. The emphasis will be on two related problems: the classification problem, and the K(pi,1) problem. The classification centers on the following question and its converse: is the topology of the complement of an arrangement determined by the intersection pattern of the original hyperplanes? The K(pi,1) problem asks for necessary and/or sufficient conditions for the vanishing of certain topological obstructions in the complement. This simple object, the complement of a collection of (n-1)- dimensional linear subspaces in n-dimensional space, displays a surprisingly rich structure. The variety of methods which have been found useful for analyzing it come from combinatorics, differential equations, and K-theory.