Backing up a bit, the subject of our poster is the interplay between the topology, geometry, and combinatorics of subspace arrangements. Our research draws on a variety of techniques from algebraic and geometric topology, algebraic geometry, singularities, combinatorial group theory, and computational algebra. The emphasis is on finding effectively computable invariants of the complements of arrangements, with a view towards a complete homotopy classification. We focus on configurations of skew lines in 3-space, and on the corresponding arrangements of planes in 4-space. In the process, we discover subtle and varied differences between such arrangements and complex hyperplane arrangements. These phenomena may be thought of as manifestations of the non-complex nature of real arrangements, revealed through the geometry of the fundamental group of the complement.

In its simplest manifestation, an arrangement is merely a collection of lines in the plane. These lines cut the real plane into pieces, and understanding the topology of the complement is an elementary exercise, which amounts to counting those pieces. In the case of lines in the complex plane (or, for that matter, hyperplanes in complex n-space), the complement is of one piece. But, unlike a disk, for example, this does not mean that it can be drawn back over itself until it shrinks down to a single point: An algebraic invariant that measures this failure is the fundamental group, which, roughly speaking, consists of those loops that can not be shrunk in the complement. A particularly important example is the braid arrangement of ``diagonal" hyperplanes in complex n-space. In that case, loops in the complement can be viewed as braids, and the fundamental group can be identified with the (pure) braid group. For arbitrary arrangements, the identification of the fundamental group is more complicated, but it can be done in an algorithmic way, using the theory of braids. This theory, in turn, is intricately connected with the theory of knots and links in 3-space, with its wealth of algebraic and combinatorial invariants, and its varied applications to biology, chemistry, and physics.