Title: Symbolic Computation in Algebra, Geometry, and Topology
Abstract: In this talk we'll examine an application of symbolic computation to algebraic topology; in particular, to the topology of the complement X of a complex hyperplane arrangement. Arrangements can be studied from many different viewpoints: combinatorics, topology, geometry 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 fundamental group of X. 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 giving the LCS ranks in terms of A(X). I will describe this formula, give examples, and report on progress in extending it. In the process, I'll give an introduction to the Bernstein-Gelfand-Gelfand correspondance, and will show how to compute examples using the Macaulay2 package. The talk will assume no specialized knowledge and will be accessible to all.
(Joint work with Alex Suciu, Northeastern Univ.)