We study the homology jump loci of a chain complex over an affine

We investigate the resonance varieties attached to a commutative differential graded algebra and to a representation of a Lie algebra, with emphasis on how these varieties behave under finite products and coproducts.

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 of the degree 1 characteristic varieties of a hyperplane

There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and pi_1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and pi_1(N) is a quasi-projective group, then all the prime components of N are graph manifolds.

The regular Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Omega^i_r(X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1

We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, translated components in the characteristic varieties affect

We examine the Johnson filtration of the (outer) automorphism group of a finitely generated group. In the case of a free group, we find a surprising result: the first Betti number of the second subgroup in the Johnson filtration is finite. Moreover, the corresponding Alexander invariant is a non-trivial module over the Laurent polynomial ring. In the process, we show that the first resonance variety of the outer Torelli group of a free group is trivial.

We exploit the classical correspondence between finitely generated abelian groups and abelian complex algebraic reductive groups to study the intersection theory of translated subgroups in an abelian complex algebraic reductive group, with special emphasis on intersections of (torsion) translated subtori in an algebraic torus.

The Dwyer-Fried invariants of a finite cell complex X are the subsets Omega^i_r(X) of the Grassmannian of r-planes in H^1(X,Q) which parametrize the regular Z^r-covers of X having finite Betti numbers up to degree i. In previous work, we showed that each Omega-invariant is contained in the complement of a union of Schubert varieties associated to a certain subspace arrangement in H^1(X,Q). Here, we identify a class of spaces for which this inclusion