The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A ⊆ Cn; it is the quotient of an exterior algebra Λ(V ) on |A| generators. In [9], Orlik and Terao introduced a commutative analog Sym(V ∗)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice L(A). In [6], Falk and Randel...

The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation ...

Let X be a nonsingular complex projective variety and D a locally quasihomogeneous free divisor in X. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on X with respect to D, and the Chern-Schwartz-MacPherson class of the complement of D in X. Our result confirms a conjectural formula for these classes, at least after push-forward to pr...

We define a novel, basic, unsupervised learning problem learning the lowest density homogeneous hyperplane separator of an unknown probability distribution. This task is relevant to several problems in machine learning, such as semi-supervised learning and clustering stability. We investigate the question of existence of a universally consistent algorithm for this problem. We propose two natura...

In a previous work, we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the homology of such a subcomplex is concentrated in degree k − 1. This homology group supports a natural action of the Coxeter group W (D n) of type D. In this paper, we explicitly determine the characters (ov...

The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algeb...