Let $\mathcal{A}$ be a Coxeter hyperplane arrangement, that is the arrangement of reflecting hyperplanes of an irreducible finite Coxeter group. A deformation of $\mathcal{A}$ is an affine arrangement each of whose hyperplanes is parallel to some hyperplane of $\mathcal{A}$. We survey some of the interesting combinatorics of classes of such arrangements, reflected in their characteristic polynomials.

We relate the cohomology of the Orlik–Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik–Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its Orlik–Solomon algebra.

We review the regular tilings of $d$-sphere, Euclidean $d$-space, hyperbolic $d$-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd $m \ge 7$, star-honeycombs $\{m,m/2\}$ are embeddable while $\{m/2,m\}$ are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension $d \gt 2$ are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, $\{4, 3, 5\}$ and $\{4, 3, 3, 5\}$, of those 11 have compact both, facets and vertex figures).

In “On the homotopy theory of arrangements” published in 1986 the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.

Consider the set of surface-curve pairs $(X,\mathcal{C})$, where $X$ is a normal surface and $\mathcal{C}$ is an algebraic curve. In this paper, we define a family $\mathcal{F}$ of normal surface-curve pairs, which is closed under coverings, and which contains all smooth surface-curve pairs $(X, \mathcal{C})$, where $X$ is smooth and $\mathcal{C}$ has smooth irreducible components with normal crossings. We give a modification of W. Neumann’s definition of plumbing graphs, their associated 3-dimensional graph manifolds, and intersection matrices, and use this construction to describe rational intersection matrices and boundary manifolds for regular branched coverings.

The classical harmonic functions are characterized in terms of the mean value property with respect to the unit ball. Replacing the ball by a polytope, we are led to the notion of polyhedral harmonic functions, i.e., those continuous functions which satisfy the mean value property with respect to a given polytope. The study of polyhedral harmonic functions involves not only analysis but also algebra, including combinatorics of polytopes and invariant theory for finite reflection groups. We give a brief survey on this subject, focusing on some recent results obtained by the author.

For an arrangement with complement $X$ and fundamental group $G$, we relate the truncated cohomology ring, $H^{\le2} (X)$, to the second nilpotent quotient, $G/G_3$. We define invariants of $G/G_3$ by counting normal subgroups of a fixed prime index $p$, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik–Solomon algebra mod $p$. As an application, we establish the cohomology classification of 2-arrangements of $n \le 6$ planes in ${\mathbb{R}}^4$.

We consider a tensor product of a Verma module and the basic linear representation of $sl(n + 1)$. We prove that the corresponding phase function, which is used in the solutions of the KZ equation with values in the tensor product, has a unique critical point and show that the Hessian of the logarithm of the phase function at this critical point equals the Shapovalov norm of the corresponding Bethe vector in the tensor product.

We use the critical points of a multivalued holomorphic function and Morse theory to find a basis for a local system homology group defined on the complement of an arrangement of hyperplanes. This generalizes results of Kohno [4] and Douai–Terao [2] from complexified real arrangements to all arrangements. We also show that the set of critical points satisfies the same recursion with respect to deletion and restriction as the $\beta$nbc set of the arrangement.

Let $W$ be a finite group generated by unitary reflections and $\mathcal{A}$ be the set of reflecting hyperplanes. We will give a characterization of the logarithmic differential forms with poles along $\mathcal{A}$ in terms of anti-invariant differential forms. If $W$ is a Coxeter group defined over $\mathbf{R}$, then the characterization provides a new method to find a basis for the module of logarithmic differential forms out of basic invariants.