A brief description of my research interests
I did my graudate study at the University of Wisconsin - Madison, receiving a Ph.D. in Mathematics in 1983. My thesis advisor was Peter Orlik, an algebraic topologist, expert in group actions on manofilds and singularity theory. At the time I started working with him he had just become interested in the general theory of complex hyperplane arrangements. Besides the broad background in many areas of algebra, topology, and singularity theory that I gained in graduate school, I have since developed some expertise in combinatorics, mostly in matroid theory, and projective algebraic geometry.
My research centers around the interplay between topological, algebraic, and combinatorial properties of arrangements of hyperplanes in complex vector spaces. This focus leads to my interest in a wide variety of topics, including, in no particular order, algebraic topology, Lie algebras and root systems, combinatorial group theory, homotopy theory, geometry, algebraic geometry, K-theory, singularities, representation theory, matroid theory, polytopes. I did some foundational work in the field in the late 1980's and early 1990's, some of it with my post-doctoral advisor Dick Randell. In the late 1990's I introduced the study of resonance varieties in arrangements theory. That concept has since been exported to other realms of group theory and topology. Lately I've been interested in several things, all of whoch seem to relate somehow to resonance varieties: qualitative properties of the (pure) braid group and other finitely-presented groups; the nature of critical points of products of powers of linear forms in several complex variables, especially their dependence on combinatorial data; applications of tropical algebraic geometry in arrangements theory; calculation of the cohomology of rak-one local systems over hyperplane complements, for resonant values of the parameters. The ultimate goal of my research is to cast the topology of complex hyperplane complements, and the associated topological invariants, into combinatorial terms - that is, to do everything discretely!