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Title:Hyperplane arrangements and free resolution
Abstract: A hyperplane arrangement A is a finite collection of hyperplanes in a fixed vector space V. Arrangements can be studied from many different viewpoints: combinatorics, topology, and algebra all have roles to play. In the first part of the talk I'll give an overview of the area; then we'll discuss a very difficult question: describe the topology of the complement X of the arrangement. The lower central series (LCS) filtration of the fundamental group of X gives rise to a graded Lie algebra, whose graded ranks can be computed from a free resolution of the residue field over the cohomology ring A(X) of the arrangement complement. A(X) has a beautiful and simple combinatorial description; it is a quotient of an exterior algebra. For certain classes of arrangements there is a striking formula (due to Kohno for Braid arrangements, and extended by Falk-Randell to fiber-type arrangements) giving the LCS ranks in terms of A(X). I will describe this formula, give examples, and report on progress in extending it. (Joint work with Alex Suciu, Northeastern Univ.) |