Geometry seminar: Hyperplane complements and non-positive curvature of cell complexes

A (complex) hyperplane arrangement is a finite collection of hyperplanes in C^n. Since these hyperplanes are complex codimension 1 (and thus real codimension 2), the complement in C^n of the union of the hyperplanes is a connected manifold. It is a broad, longstanding problem with many connections to different areas of mathematics to determine the arrangements for which this space is aspherical (has contractible universal cover). A subset of this problem, commonly attributed to Arnol’d, Brieskorn, Thom, and Pham, concerns arrangements arising from reflection groups in R^n. For these arrangements, one approach is to study a certain cell complex which is homotopy equivalent to the complement and carries a natural (singular) metric, which is conjectured by Charney and Davis to be non-positively curved. This approach is deceivingly hard and has seen very little progress since it was proposed in 1995. I will discuss joint work with Amy Herron showing that this complex is indeed non-positively curved for a class of reflection arrangements, and more generally, 3-dimensional cell complexes which satisfy a strictly combinatorial condition.