We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted groupâ€¦ (More)

The kth Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, Vk(A), of the algebraic torus (Câˆ—)n. In the combinatoriallyâ€¦ (More)

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, Hâ‰¤2(X), to the second nilpotent quotient, G/G3. We define invariants of G/G3 by counting normalâ€¦ (More)

We investigate the relationship between the geometric Bieriâ€“Neumannâ€“ Strebelâ€“Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems overâ€¦ (More)

A finite simplicial graph Î“ determines a right-angled Artin group GÎ“, with generators corresponding to the vertices of Î“, and with a relation vw = wv for each pair of adjacent vertices. We computeâ€¦ (More)

The Chen groups of a finitely-presented group G are the lower central series quotients of its maximal metabelian quotient, G/Gâ€²â€². The direct sum of the Chen groups is a graded Lie algebra, withâ€¦ (More)

We study the homotopy types of complements of arrangements of n transverse planes in R4, obtaining a complete classification for n â‰¤ 6, and lower bounds for the number of homotopy types in general.â€¦ (More)

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artinâ€™s braid group Bn. Using Hansenâ€™s polynomial covering spaceâ€¦ (More)

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable,â€¦ (More)

Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, Î± : Fs â†’ Pn. Using the Gassner representation of theâ€¦ (More)