UTA Department of Mathematics

Mathematics and Statistics Colloquium

Date/Time/Room: Friday (2/10/2006) at 2:00pm in 304 Pickard Hall

Speaker: Dr. Hal Schenck, Associate Professor
Department of Mathematics, Texas A&M University

"Geometry, Algebra, and Topology of Hyperplane Arrangements"

Abstract: A hyperplane arrangement A is a collection of hyperplanes in a fixed vector space V. Arrangements can be studied from many different viewpoints: geomentry, combinatorics, topology, and algebra all have roles to play. In the first half of the talk I'll give an overview of the field, fill in the necessary background, work out some good examples, and discuss some of the major open problems in the area.

In the second half of the talk, we'll tackle a hard topic: investigate the fundamental group of the complement X = V-A. 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 H^*(X). The ring H^*(X) has a beautiful and simple description; it is a quotient of an exterior algebra by a combinatorially determined ideal. For certain classes of arrangements there is a striking formula giving the LCS ranks in terms of H^*(X). I will describe this formula, give examples, and report on progress in extending it.