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Papers by Alexandru Suciu

Non UBCUnreviewedAuthor affiliation: Northeastern UniversityFacult

The FASEB Journal, 2018

There are currently over 14,300 Structural Genomics (SG) protein structures deposited in the PDB ... more There are currently over 14,300 Structural Genomics (SG) protein structures deposited in the PDB by protein structure initiatives. However, most of these SG proteins have unknown or putative functi...

This Special Issue is devoted to the activities surrounding ``Arrangements in Boston: A Conferenc... more This Special Issue is devoted to the activities surrounding ``Arrangements in Boston: A Conference on Hyperplane Arrangements", which was held at Northeastern University, 12--15 June 1999.}, URL = {http://www.sciencedirect.com/science/journal/01668641/118/1-2}, DOI = {10.1016/S0166-8641(01)00055-4}, gsid = {2628550103319635517

Transactions of the American Mathematical Society, 2006

If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π... more If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 1 (X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H * (X, k), viewed as a module over the exterior algebra E on A: θ k (G) = dim k Tor E k−1 (A, k) k , for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θ k (G) = (k − 1) r≥1 h r r+k−1 k , where h r is the number of r-dimensional components of the projective resonance variety R 1 (A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R 1 (A) and a localization argument, we establish the inequality θ k (G) ≥ (k − 1) r≥1 h r r + k − 1 k , for k 0, for arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R 1 (A), such that θ k (G) = P(k), for all k 0.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

For each integer n ≥ 2, we construct an irreducible, smooth, complex projective variety M of dime... more For each integer n ≥ 2, we construct an irreducible, smooth, complex projective variety M of dimension n, whose fundamental group has infinitely generated homology in degree n + 1 and whose universal cover is a Stein manifold, homotopy equivalent to an infinite bouquet of n-dimensional spheres. This non-finiteness phenomenon is also reflected in the fact that the homotopy group π n (M), viewed as a module over Zπ 1 (M), is free of infinite rank. As a result, we give a negative answer to a question of Kollár on the existence of quasi-projective classifying spaces (up to commensurability) for the fundamental groups of smooth projective varieties. To obtain our examples, we develop a complex analog of a method in geometric group theory due to Bestvina and Brady.

Forum Mathematicum, 2010

Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodr... more Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more general result about iterated group extensions. As an application, we obtain new criteria for formality of spaces, and 1-formality of groups, illustrated by bundle constructions and various examples from low-dimensional topology and singularity theory.

Algebraic & Geometric Topology, 2003

In a recent paper, Dimca and Némethi pose the problem of finding a homogeneous polynomial f such ... more In a recent paper, Dimca and Némethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p, there is a polynomial with p-torsion in the homology of the Milnor fiber. The techniques make use of properties of characteristic varieties of hyperplane arrangements.

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump... more We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;ℤ), with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a ... more Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a smooth, compact, connected [for short, closed] manifold, then π 1 (M) admits a finite presentation: π 1 (M) = x 1 ,. .. x p | r 1 ,. .. , r q. Conversely, every finitely presented group G can be realized as G = π 1 (M) for a closed manifold M n of dimension n ≥ 4. M n can be chosen to be orientable. M n (n even) can be chosen to be symplectic (Gompf 1995). M n (n even, n ≥ 6) can be chosen to be complex (Taubes 1992). Requiring n = 3 puts severe restrictions on G, e.g.: G abelian 3-manifold group ⇐⇒

Forum Mathematicum

We explore the graded-formality and filtered-formality properties of finitely generated groups by... more We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

Protein science : a publication of the Protein Society, 2018

As a result of high-throughput protein structure initiatives, over 14,400 protein structures have... more As a result of high-throughput protein structure initiatives, over 14,400 protein structures have been solved by Structural Genomics (SG) centers and participating research groups. While the totality of SG data represents a tremendous contribution to genomics and structural biology, reliable functional information for these proteins is generally lacking. Better functional predictions for SG proteins will add substantial value to the structural information already obtained. Our method described herein, Graph Representation of Active Sites for Prediction of Function (GRASP-Func), predicts quickly and accurately the biochemical function of proteins by representing residues at the predicted local active site as graphs rather than in Cartesian coordinates. We compare the GRASP-Func method to our previously reported method, Structurally Aligned Local Sites of Activity (SALSA), using the Ribulose Phosphate Binding Barrel (RPBB), 6-Hairpin Glycosidase (6-HG), and Concanavalin A-like Lectins...

We explore the graded and filtered formality properties of a finitely-generated group by studying... more We explore the graded and filtered formality properties of a finitely-generated group by studying the various Lie algebras attached to such a group, including the associated graded Lie algebra, the holonomy Lie algebra, and the Malcev Lie algebra. We explain how these notions behave with respect to split injections, coproducts, and direct products, and how they are inherited by solvable and nilpotent quotients. For a finitely-presented group, we give an explicit formula for the cup product in low degrees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansion method. We also give a presentation for the Chen Lie algebra of a filtered-formal group, and discuss various approaches to computing the ranks of the graded objects under consideration. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups, and fundamental groups...

Pacific Journal of Mathematics, 1988

We show that, bar unexpected developments in 3-manifold theory, the fundamental group and the cho... more We show that, bar unexpected developments in 3-manifold theory, the fundamental group and the choice of framing determine the oriented homotopy type of spun 3-manifolds.

Topology and its Applications, 2002

We give examples of complex hyperplane arrangements A for which the top characteristic variety, V... more We give examples of complex hyperplane arrangements A for which the top characteristic variety, V 1 (A), contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus (C *) |A|. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik-Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.

Transactions of the American Mathematical Society, 2002

If M is the complement of a hyperplane arrangement, and A = H * (M, k) is the cohomology ring of ... more If M is the complement of a hyperplane arrangement, and A = H * (M, k) is the cohomology ring of M over a field k of characteristic 0, then the ranks, φ k , of the lower central series quotients of π 1 (M) can be computed from the Betti numbers, b ii = dim Tor A i (k, k) i , of the linear strand in a minimal free resolution of k over A. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b ij = dim Tor E i (A, k) j , of a minimal resolution of A over the exterior algebra E. From this analysis, we recover a formula of Falk for φ 3 , and obtain a new formula for φ 4. The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra A is Koszul if and only if the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, b i,i+1 , of the linear strand of the free resolution of A over E; if the lower bound is attained for i = 2, then it is attained for all i ≥ 2. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of A are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid subarrangements), we show that b i,i+1 is determined by the number of triangles and K 4 subgraphs in the graph.

ABSTRACT This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements,... more ABSTRACT This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements, Local systems and Singularities held at Galatasaray University, Istanbul during June 2007. The volume is intended for a large audience in pure mathematics, including researchers and graduate students working in algebraic geometry, singularity theory, topology and related fields. The reader will find a variety of open problems involving arrangements, local systems and singularities proposed by the lecturers at the end of the school.

Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a ... more Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a smooth, compact, connected [for short, closed] manifold, then π 1 (M) admits a finite presentation: π 1 (M) = x 1 ,. .. x p | r 1 ,. .. , r q. Conversely, every finitely presented group G can be realized as G = π 1 (M) for a closed manifold M n of dimension n ≥ 4. M n can be chosen to be orientable. M n (n even) can be chosen to be symplectic (Gompf 1995). M n (n even, n ≥ 6) can be chosen to be complex (Taubes 1992). Requiring n = 3 puts severe restrictions on G, e.g.: G abelian 3-manifold group ⇐⇒

Transactions of the American Mathematical Society, 2010

We study the spectral sequence associated to the filtration by powers of the augmentation ideal o... more We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X X . In the process, we identify the d 1 d^1 differential in terms of the coalgebra structure of H ∗ ( X , k ) H_*(X,\Bbbk ) and the k π 1 ( X ) \Bbbk \pi _1(X) -module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod p p cohomology of cyclic p p -covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X X . It also yields computable upper bounds on the ranks of the cohomology groups of X X , with coefficients in a prime-power order, rank one local system. When X X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of H ∗ ( X , k ) H^*(X,\Bbbk ...

Topology and its Applications, 2002

Non UBCUnreviewedAuthor affiliation: Northeastern UniversityFacult

The FASEB Journal, 2018

There are currently over 14,300 Structural Genomics (SG) protein structures deposited in the PDB ... more There are currently over 14,300 Structural Genomics (SG) protein structures deposited in the PDB by protein structure initiatives. However, most of these SG proteins have unknown or putative functi...

This Special Issue is devoted to the activities surrounding ``Arrangements in Boston: A Conferenc... more This Special Issue is devoted to the activities surrounding ``Arrangements in Boston: A Conference on Hyperplane Arrangements", which was held at Northeastern University, 12--15 June 1999.}, URL = {http://www.sciencedirect.com/science/journal/01668641/118/1-2}, DOI = {10.1016/S0166-8641(01)00055-4}, gsid = {2628550103319635517

Transactions of the American Mathematical Society, 2006

If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π... more If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 1 (X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H * (X, k), viewed as a module over the exterior algebra E on A: θ k (G) = dim k Tor E k−1 (A, k) k , for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θ k (G) = (k − 1) r≥1 h r r+k−1 k , where h r is the number of r-dimensional components of the projective resonance variety R 1 (A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R 1 (A) and a localization argument, we establish the inequality θ k (G) ≥ (k − 1) r≥1 h r r + k − 1 k , for k 0, for arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R 1 (A), such that θ k (G) = P(k), for all k 0.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

For each integer n ≥ 2, we construct an irreducible, smooth, complex projective variety M of dime... more For each integer n ≥ 2, we construct an irreducible, smooth, complex projective variety M of dimension n, whose fundamental group has infinitely generated homology in degree n + 1 and whose universal cover is a Stein manifold, homotopy equivalent to an infinite bouquet of n-dimensional spheres. This non-finiteness phenomenon is also reflected in the fact that the homotopy group π n (M), viewed as a module over Zπ 1 (M), is free of infinite rank. As a result, we give a negative answer to a question of Kollár on the existence of quasi-projective classifying spaces (up to commensurability) for the fundamental groups of smooth projective varieties. To obtain our examples, we develop a complex analog of a method in geometric group theory due to Bestvina and Brady.

Forum Mathematicum, 2010

Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodr... more Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more general result about iterated group extensions. As an application, we obtain new criteria for formality of spaces, and 1-formality of groups, illustrated by bundle constructions and various examples from low-dimensional topology and singularity theory.

Algebraic & Geometric Topology, 2003

In a recent paper, Dimca and Némethi pose the problem of finding a homogeneous polynomial f such ... more In a recent paper, Dimca and Némethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p, there is a polynomial with p-torsion in the homology of the Milnor fiber. The techniques make use of properties of characteristic varieties of hyperplane arrangements.

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump... more We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;ℤ), with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a ... more Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a smooth, compact, connected [for short, closed] manifold, then π 1 (M) admits a finite presentation: π 1 (M) = x 1 ,. .. x p | r 1 ,. .. , r q. Conversely, every finitely presented group G can be realized as G = π 1 (M) for a closed manifold M n of dimension n ≥ 4. M n can be chosen to be orientable. M n (n even) can be chosen to be symplectic (Gompf 1995). M n (n even, n ≥ 6) can be chosen to be complex (Taubes 1992). Requiring n = 3 puts severe restrictions on G, e.g.: G abelian 3-manifold group ⇐⇒

Forum Mathematicum

We explore the graded-formality and filtered-formality properties of finitely generated groups by... more We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

Protein science : a publication of the Protein Society, 2018

As a result of high-throughput protein structure initiatives, over 14,400 protein structures have... more As a result of high-throughput protein structure initiatives, over 14,400 protein structures have been solved by Structural Genomics (SG) centers and participating research groups. While the totality of SG data represents a tremendous contribution to genomics and structural biology, reliable functional information for these proteins is generally lacking. Better functional predictions for SG proteins will add substantial value to the structural information already obtained. Our method described herein, Graph Representation of Active Sites for Prediction of Function (GRASP-Func), predicts quickly and accurately the biochemical function of proteins by representing residues at the predicted local active site as graphs rather than in Cartesian coordinates. We compare the GRASP-Func method to our previously reported method, Structurally Aligned Local Sites of Activity (SALSA), using the Ribulose Phosphate Binding Barrel (RPBB), 6-Hairpin Glycosidase (6-HG), and Concanavalin A-like Lectins...

We explore the graded and filtered formality properties of a finitely-generated group by studying... more We explore the graded and filtered formality properties of a finitely-generated group by studying the various Lie algebras attached to such a group, including the associated graded Lie algebra, the holonomy Lie algebra, and the Malcev Lie algebra. We explain how these notions behave with respect to split injections, coproducts, and direct products, and how they are inherited by solvable and nilpotent quotients. For a finitely-presented group, we give an explicit formula for the cup product in low degrees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansion method. We also give a presentation for the Chen Lie algebra of a filtered-formal group, and discuss various approaches to computing the ranks of the graded objects under consideration. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups, and fundamental groups...

Pacific Journal of Mathematics, 1988

We show that, bar unexpected developments in 3-manifold theory, the fundamental group and the cho... more We show that, bar unexpected developments in 3-manifold theory, the fundamental group and the choice of framing determine the oriented homotopy type of spun 3-manifolds.

Topology and its Applications, 2002

We give examples of complex hyperplane arrangements A for which the top characteristic variety, V... more We give examples of complex hyperplane arrangements A for which the top characteristic variety, V 1 (A), contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus (C *) |A|. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik-Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.

Transactions of the American Mathematical Society, 2002

If M is the complement of a hyperplane arrangement, and A = H * (M, k) is the cohomology ring of ... more If M is the complement of a hyperplane arrangement, and A = H * (M, k) is the cohomology ring of M over a field k of characteristic 0, then the ranks, φ k , of the lower central series quotients of π 1 (M) can be computed from the Betti numbers, b ii = dim Tor A i (k, k) i , of the linear strand in a minimal free resolution of k over A. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b ij = dim Tor E i (A, k) j , of a minimal resolution of A over the exterior algebra E. From this analysis, we recover a formula of Falk for φ 3 , and obtain a new formula for φ 4. The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra A is Koszul if and only if the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, b i,i+1 , of the linear strand of the free resolution of A over E; if the lower bound is attained for i = 2, then it is attained for all i ≥ 2. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of A are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid subarrangements), we show that b i,i+1 is determined by the number of triangles and K 4 subgraphs in the graph.

ABSTRACT This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements,... more ABSTRACT This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements, Local systems and Singularities held at Galatasaray University, Istanbul during June 2007. The volume is intended for a large audience in pure mathematics, including researchers and graduate students working in algebraic geometry, singularity theory, topology and related fields. The reader will find a variety of open problems involving arrangements, local systems and singularities proposed by the lecturers at the end of the school.

Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a ... more Fundamental groups and geometry Fundamental groups Realizing finitely presented groups If M is a smooth, compact, connected [for short, closed] manifold, then π 1 (M) admits a finite presentation: π 1 (M) = x 1 ,. .. x p | r 1 ,. .. , r q. Conversely, every finitely presented group G can be realized as G = π 1 (M) for a closed manifold M n of dimension n ≥ 4. M n can be chosen to be orientable. M n (n even) can be chosen to be symplectic (Gompf 1995). M n (n even, n ≥ 6) can be chosen to be complex (Taubes 1992). Requiring n = 3 puts severe restrictions on G, e.g.: G abelian 3-manifold group ⇐⇒

Transactions of the American Mathematical Society, 2010

We study the spectral sequence associated to the filtration by powers of the augmentation ideal o... more We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X X . In the process, we identify the d 1 d^1 differential in terms of the coalgebra structure of H ∗ ( X , k ) H_*(X,\Bbbk ) and the k π 1 ( X ) \Bbbk \pi _1(X) -module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod p p cohomology of cyclic p p -covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X X . It also yields computable upper bounds on the ranks of the cohomology groups of X X , with coefficients in a prime-power order, rank one local system. When X X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of H ∗ ( X , k ) H^*(X,\Bbbk ...

Topology and its Applications, 2002

Special Session on Singularities at the Joint Meeting of the NZMS, AustMS, and AMS, 2024

Each hyperplane arrangement A in C^d gives rise to a Milnor fibration of its complement, F → M → ... more Each hyperplane arrangement A in C^d gives rise to a Milnor fibration of its complement, F → M → C^*. Although the eigenvalues of the monodromy h: F → F acting on the homology groups H_i(F;C) can be expressed in terms of the jump loci for rank 1 local systems on M, explicit formulas are still lacking in full generality, even in degree i=1. In this talk, I will explain some of the results relating the combinatorics of the intersection lattice of A to the algebraic topology of the Milnor fiber F. In the case when b_1(F) is as small as possible, I will describe ways to extract information on the cohomology jump loci, the lower central series quotients, and the Chen ranks of the fundamental group of F.