Research: Past and Present

Figure 1: A line arrangement in the plane The study of configurations of lines in the plane or planes in space has a relatively short history. Some say the field arose from the cheese cutting problem: what is the maximum number of pieces one can produce from a fixed number of cuts? Others say that it arose through the history of braid groups: the pure braid group on n strands turns out to be the fundamental group of the complement of a certain configuration of hyperplanes in an n dimensional vector space (for general reference see [24]). Regardless, the last thirty years has seen an explosion of the field, usually called ‘arrangements of hyperplanes’ (which is a finite collection of linear spaces of codimension 1 in a finite dimensional vector space or projective space). This interest is grounded in the fact that many disparate areas of mathematics, for example algebra, topology, and combinatorics, interact through a hyperplane arrangement. Not only have we seen the development of many beautiful mathematical theorems about arrangements, but arrangements have also been shown to have uses in biology, topological robotics, mathematical physics, and statistical economics (for example see [10], [15], and [32]). Another attraction to the study of hyperplane arrangements originates from singularity theory, see for example [25]. Much of algebraic geometry is focused on the study of smooth varieties (solutions to polynomial equations) or schemes (varieties with extra local and global information) and comparatively little is known about varieties and schemes with singularities (i.e. the tangent space degenerates in some way). Since the intersection of two lines or two hyperplanes is one of the simplest singularities and a hyperplane is one of the simplest hypersurfaces, a finite collection of hyperplanes provides a relatively simple object to study complicated singular geometry (Figure 1 is such an example). The applicant’s research is centered on algebraic, combinatorial, and topological problems on hyperplane arrangements and related objects. The applicant is particularly interested in solving difficult problems in algebraic geometry and algebraic topology on hyperplane arrangements by utilizing combinatorial intersection information. These problems are very exciting because of their simplicity and the solutions are very exciting because they use seemingly unrelated material in inventive and creative ways. Also many of these topics are accessible to students for multiple levels of research projects. In this note the applicant discusses various selected results together with ongoing projects which are listed as problems. In Section 2 the applicants early work on vector fields and Terao’s conjecture is discussed along with related, ongoing problems. In Section 3 the focus is on commutative algebraic aspects of hyperplane arrangements. Section 4 discusses the applicants work in algebraic and combinatorial topology. Lastly Section 5 briefly discusses the applicants research on error-correcting codes associated to some subspace arrangements.