My research interests are in Topology, and how it relates to Algebra, Geometry, and Combinatorics. I currently investigate cohomology jumping loci, and their applications to algebraic varieties, low-dimensional topology, and toric topology, such as the study of hyperplane arrangements, moment angle complexes, configuration spaces, and various classes of knots, links, and manifolds, as well as the homology and lower central series of discrete groups.
Journal für die Reine und Angewandte Mathematik (Crelle's Journal), 2024
The resonance varieties are cohomological invariants that are studied in a variety of topological... more The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.
A complex hyperplane arrangement A is said to be decomposable if there are no elements in the deg... more A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant B(G) also decomposes as a direct sum of "local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if B(G) is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.
To each multi-arrangement (A,m), there is an associated Milnor fibration of the complement M=M(A)... more To each multi-arrangement (A,m), there is an associated Milnor fibration of the complement M=M(A). Although the Betti numbers of the Milnor fiber F=F(A,m) 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 for b_1(F). In this talk, I will take a different tack: I will consider the "generic" case (in which b_1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations. In particular, I will describe ways to extract information on the cohomology jump loci and the lower central series quotients of the fundamental group of F, even when the abelianization of this groups is not torsion-free.
The influence of Fusarium ear infection on the maize yield and mycotoxin content (Transylvania-Ro... more The influence of Fusarium ear infection on the maize yield and mycotoxin content (Transylvania-Romania) Maize is the host for a large number of pathogens, which invade all of its organs from the germination until being harvest, ear and grain infection often persistent even during storage. Diseases, through their symptoms reduce significantly the quantity and the quality of yield, estimated between 7-17% but, in the favorable years for this disease, they can be much greater. Fusarium diseases reduce yield value and quality by massive accumulation of Fusarium mycelium biomass (about 85%) on grain and ears and by mycotoxin contamination such as deoxynivalenol (DON), zearalenone (ZEA) and fumonisins (FUM). In this paper are presented aspects regarding the reaction of some maize hybrids under Fusarium spp. natural and artificial infections; the effect of Fusarium ear infection on yield, grain chemical composition, and mycotoxin content; the correlation between ear rot disease degree and ...
Each connected graded, graded-commutative algebra A of finite type over a field k of characterist... more Each connected graded, graded-commutative algebra A of finite type over a field k of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When A " kx∆y is the exterior Stanley-Reisner algebra associated to a finite simplicial complex ∆, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.
Revue Roumaine de Mathématiques Pures et Appliquées, 2024
Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although ... more Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber F can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for b_1(F). We study here the "generic" case (in which b_1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of π_1(F). In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet nonisomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients.
In previous work we introduced the notion of binomial cup-one algebras, which are differential gr... more In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod ∪ 1-products and compatible binomial operations. Given such an R-dga, (A, d A), defined over the ring R = Z or Z p (for p a prime), with H 0 (A) = R and with H 1 (A) a finitely generated, free R-module, we show that A admits a functorially defined 1-minimal model, ρ : (M(A), d) → (A, d A), which is unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, G(A), which only depends on the 1-quasi-isomorphism type of A. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic (torsion-free) cohomology rings that share the same rational 1-minimal model, yet whose integral 1-minimal models are not isomorphic.
Under suitable finiteness assumptions, one may define the Koszul modules and the resonance variet... more Under suitable finiteness assumptions, one may define the Koszul modules and the resonance varieties of any differential graded algebra. When such a dga models a space or a group, the geometry of these varieties mirror topological and group-theoretical properties of those objects. In characteristic 0, rational homotopy theory methods provide a useful bridge between resonance and the various Lie algebras associated to a group. In characteristic 2, the action of the Bockstein homomorphism on the mod 2 cohomology ring yields new resonance varieties, which bear witness to orientability and Poincaré duality for closed manifolds.
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024
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 Ñ K Ñ G Ñ Q Ñ 1, where Q is an abelian group acting trivially on H 1 pK; Zq, 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.
We observe greater systolic freedom via Sullivan’s telescopes. Namely, given integers m and n suc... more We observe greater systolic freedom via Sullivan’s telescopes. Namely, given integers m and n such that 2 ≤ m < n, we use rational homotopy theory to prove that every n-dimensional manifold admits metrics of arbitrarily small total volume, and satisfying the following property: every m-dimensional orientable submanifold of less than unit m-volume is necessarily torsion in homology.
We explore various formality and finiteness properties in the differential graded algebra models ... more We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1-formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3-dimensional manifolds.
We use the action of the Bockstein homomorphism on the cohomology ring H^*(X, Z_2) of a finite-ty... more We use the action of the Bockstein homomorphism on the cohomology ring H^*(X, Z_2) of a finite-type CW-complex X in order to define the resonance varieties of X in characteristic 2. Much of the theory is done in the more general framework of the Maurer-Cartan sets and the resonance varieties attached to a finite-type commutative differential graded algebra. We illustrate these concepts with examples mainly drawn from closed manifolds, where Poincaré duality over Z_2 has strong implications on the nature of the resonance varieties.
We explore the relationship between a certain “abelian duality” property of spaces and the propag... more We explore the relationship between a certain “abelian duality” property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the “EPY property”. The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina–Brady groups. Our approach brings to the fore the relevance of the Cohen–Macaulay condition in this combinatorial context.
We relate the geometry of the resonance varieties associated to a commutative differential graded... more We relate the geometry of the resonance varieties associated to a commutative differential graded algebra model of a space to the finiteness properties of the completions of its Alexander-type invariants. We also describe in simple algebraic terms the non-translated components of the degree-one characteristic varieties for a class of non-proper complex manifolds.
Journal of the London Mathematical Society, Aug 4, 2007
Bestvina–Brady groups arise as kernels of length homomorphisms GΓ → ℤ from right-angled Artin gro... more Bestvina–Brady groups arise as kernels of length homomorphisms GΓ → ℤ from right-angled Artin groups to the integers. Under some connectivity assumptions on the flag complex ΔΓ, we compute several algebraic invariants of such a group NΓ, directly from the underlying graph Γ. As an application, we give examples of finitely presented Bestvina–Brady groups which are not isomorphic to any Artin group or arrangement group.
We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a bo... more We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.
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...
Journal für die Reine und Angewandte Mathematik (Crelle's Journal), 2024
The resonance varieties are cohomological invariants that are studied in a variety of topological... more The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.
A complex hyperplane arrangement A is said to be decomposable if there are no elements in the deg... more A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant B(G) also decomposes as a direct sum of "local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if B(G) is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.
To each multi-arrangement (A,m), there is an associated Milnor fibration of the complement M=M(A)... more To each multi-arrangement (A,m), there is an associated Milnor fibration of the complement M=M(A). Although the Betti numbers of the Milnor fiber F=F(A,m) 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 for b_1(F). In this talk, I will take a different tack: I will consider the "generic" case (in which b_1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations. In particular, I will describe ways to extract information on the cohomology jump loci and the lower central series quotients of the fundamental group of F, even when the abelianization of this groups is not torsion-free.
The influence of Fusarium ear infection on the maize yield and mycotoxin content (Transylvania-Ro... more The influence of Fusarium ear infection on the maize yield and mycotoxin content (Transylvania-Romania) Maize is the host for a large number of pathogens, which invade all of its organs from the germination until being harvest, ear and grain infection often persistent even during storage. Diseases, through their symptoms reduce significantly the quantity and the quality of yield, estimated between 7-17% but, in the favorable years for this disease, they can be much greater. Fusarium diseases reduce yield value and quality by massive accumulation of Fusarium mycelium biomass (about 85%) on grain and ears and by mycotoxin contamination such as deoxynivalenol (DON), zearalenone (ZEA) and fumonisins (FUM). In this paper are presented aspects regarding the reaction of some maize hybrids under Fusarium spp. natural and artificial infections; the effect of Fusarium ear infection on yield, grain chemical composition, and mycotoxin content; the correlation between ear rot disease degree and ...
Each connected graded, graded-commutative algebra A of finite type over a field k of characterist... more Each connected graded, graded-commutative algebra A of finite type over a field k of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When A " kx∆y is the exterior Stanley-Reisner algebra associated to a finite simplicial complex ∆, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.
Revue Roumaine de Mathématiques Pures et Appliquées, 2024
Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although ... more Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber F can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for b_1(F). We study here the "generic" case (in which b_1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of π_1(F). In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet nonisomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients.
In previous work we introduced the notion of binomial cup-one algebras, which are differential gr... more In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod ∪ 1-products and compatible binomial operations. Given such an R-dga, (A, d A), defined over the ring R = Z or Z p (for p a prime), with H 0 (A) = R and with H 1 (A) a finitely generated, free R-module, we show that A admits a functorially defined 1-minimal model, ρ : (M(A), d) → (A, d A), which is unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, G(A), which only depends on the 1-quasi-isomorphism type of A. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic (torsion-free) cohomology rings that share the same rational 1-minimal model, yet whose integral 1-minimal models are not isomorphic.
Under suitable finiteness assumptions, one may define the Koszul modules and the resonance variet... more Under suitable finiteness assumptions, one may define the Koszul modules and the resonance varieties of any differential graded algebra. When such a dga models a space or a group, the geometry of these varieties mirror topological and group-theoretical properties of those objects. In characteristic 0, rational homotopy theory methods provide a useful bridge between resonance and the various Lie algebras associated to a group. In characteristic 2, the action of the Bockstein homomorphism on the mod 2 cohomology ring yields new resonance varieties, which bear witness to orientability and Poincaré duality for closed manifolds.
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024
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 Ñ K Ñ G Ñ Q Ñ 1, where Q is an abelian group acting trivially on H 1 pK; Zq, 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.
We observe greater systolic freedom via Sullivan’s telescopes. Namely, given integers m and n suc... more We observe greater systolic freedom via Sullivan’s telescopes. Namely, given integers m and n such that 2 ≤ m < n, we use rational homotopy theory to prove that every n-dimensional manifold admits metrics of arbitrarily small total volume, and satisfying the following property: every m-dimensional orientable submanifold of less than unit m-volume is necessarily torsion in homology.
We explore various formality and finiteness properties in the differential graded algebra models ... more We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1-formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3-dimensional manifolds.
We use the action of the Bockstein homomorphism on the cohomology ring H^*(X, Z_2) of a finite-ty... more We use the action of the Bockstein homomorphism on the cohomology ring H^*(X, Z_2) of a finite-type CW-complex X in order to define the resonance varieties of X in characteristic 2. Much of the theory is done in the more general framework of the Maurer-Cartan sets and the resonance varieties attached to a finite-type commutative differential graded algebra. We illustrate these concepts with examples mainly drawn from closed manifolds, where Poincaré duality over Z_2 has strong implications on the nature of the resonance varieties.
We explore the relationship between a certain “abelian duality” property of spaces and the propag... more We explore the relationship between a certain “abelian duality” property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the “EPY property”. The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina–Brady groups. Our approach brings to the fore the relevance of the Cohen–Macaulay condition in this combinatorial context.
We relate the geometry of the resonance varieties associated to a commutative differential graded... more We relate the geometry of the resonance varieties associated to a commutative differential graded algebra model of a space to the finiteness properties of the completions of its Alexander-type invariants. We also describe in simple algebraic terms the non-translated components of the degree-one characteristic varieties for a class of non-proper complex manifolds.
Journal of the London Mathematical Society, Aug 4, 2007
Bestvina–Brady groups arise as kernels of length homomorphisms GΓ → ℤ from right-angled Artin gro... more Bestvina–Brady groups arise as kernels of length homomorphisms GΓ → ℤ from right-angled Artin groups to the integers. Under some connectivity assumptions on the flag complex ΔΓ, we compute several algebraic invariants of such a group NΓ, directly from the underlying graph Γ. As an application, we give examples of finitely presented Bestvina–Brady groups which are not isomorphic to any Artin group or arrangement group.
We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a bo... more We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.
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...
To each multi-arrangement (A,m), there is an associated Milnor fibration of the complement M=M(A)... more To each multi-arrangement (A,m), there is an associated Milnor fibration of the complement M=M(A). Although the Betti numbers of the Milnor fiber F=F(A,m) 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 for b_1(F). After introducing these notions and explaining some of the known results, I will consider the "generic" case, in which 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 the Milnor fiber in this situation.
A complex hyperplane arrangement A is said to be decomposable if there are no elements in the deg... more A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras, and all the nilpotent quotients of G are combinatorially determined. Under some additional hypothesis, we show that the Alexander invariant of G also decomposes as a direct sum of "local" invariants. Consequently, the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.
Every graded, graded-commutative algebra (such as the cohomology ring of a space) determines a fa... more Every graded, graded-commutative algebra (such as the cohomology ring of a space) determines a family of cochain complexes, parametrized by the elements in degree one of the algebra. The resonance varieties are the loci where the cohomology of these cochain complexes jumps. In this talk, I will discuss some of the constraints imposed by Poincaré duality on the geometry of these resonance varieties.
There are several topological invariants that one may
associate to a finitely generated group G ... more There are several topological invariants that one may associate to a finitely generated group G -- the characteristic varieties, the resonance varieties, and the Bieri–Neumann–Strebel invariants -- which keep track of various finiteness properties of certain subgroups of G. These invariants are interconnected in ways that makes them both more computable and more informative. I will describe one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.
The cohomology jump loci of a space are of several types: the characteristic varieties, defined i... more The cohomology jump loci of a space are of several types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems; the resonance varieties, constructed from information encoded in the cohomology ring; and the complements to the Bieri-Neumann-Strebel-Renz invariants, which are defined in terms of Novikov-Sikorav homology. In this talk, I will explore the geometry of these sets and the delicate interplay among them, especially in the context of compact, orientable 3-manifolds.
I will present a study of the lower central series, the Alexander invariants, and the cohomology ... more I will present a study of the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as extensions with trivial monodromy in first homology with appropriate coefficients.
The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defin... more The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X. We will explore in this talk the geometry of these varieties and the delicate interplay between them in the context of compact, orientable 3-manifolds. In the process, we will arrive at a fairly precise geometric description of the resonance varieties R^i_k(A) of a 3-dimensional Poincaré duality algebra A.
In this talk, I discuss the resonance varieties, the lower central series ranks, the Chen ranks, ... more In this talk, I discuss the resonance varieties, the lower central series ranks, the Chen ranks, and the formality properties of several families of braid-like groups: the pure braid groups P_n, the welded pure braid groups wP_n, the virtual pure braid groups vP_n, as well as their `upper' variants, wP_n^+ and vP_n^+. I will also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives. This is joint work with He Wang.
I will describe a relationship between the classical representation
theory of a complex, semis... more I will describe a relationship between the classical representation
theory of a complex, semisimple Lie algebra \g and the resonance
varieties R(V,K)\subset V^* attached to irreducible \g-modules V
and submodules K\subset V\wedge V. In the process, I will give a
precise roots-and-weights criterion insuring the vanishing of these
varieties In the case when \g= \sl_n(\C) or \sp_{2g}(\C), our approach
yields a unified proof of two vanishing results for the resonance varieties
of the (outer) Torelli groups of finitely generated free groups and surface groups. In turn, these vanishing results reveal certain homological finiteness properties of the Johnson filtration. This is joint work with Stefan Papadima.
A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates th... more A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. I will revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. When the spaces in question are either smooth, quasi-projective varieties, or closed three-dimensional manifolds, the respective jump loci are much more tightly connected, albeit in different ways.
I will present several results relating fundamental groups of compact Kaehler manifolds and smoot... more I will present several results relating fundamental groups of compact Kaehler manifolds and smooth, quasi-projective varieties to fundamental groups of 3-dimensional manifolds with empty or toroidal boundary. The bridge between the two settings is provided by the Alexander polynomial and its various generalizations.
We revisit and generalize a result of Eisenbud, Popescu, and Yuzvinsky, which says that the reson... more We revisit and generalize a result of Eisenbud, Popescu, and Yuzvinsky, which says that the resonance varieties of a hyperplane arrangement complement propagate. It turns out that the topological underpinning for this phenomenon is a certain abelian duality property, coupled with the minimality property enjoyed by arrangement complements and related spaces, such as Cohen-Macaulay toric complexes. The key ingredient in the proof is a general, cohomological vanishing result for spaces that admit suitable `combinatorial' covers.
This is joint work with Graham Denham and Sergey Yuzvinsky.
Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay b... more Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement. A key bridge between the two is provided by the resonance varieties of the cohomology ring and by the characteristic varieties of the fundamental group. This approach helps explain how certain homological invariants associated to the Milnor fibration of the complement can be computed in terms of multinets supported by the underlying matroid.
I will discuss recent progress in understanding the
topology of the complement of an arrangemen... more I will discuss recent progress in understanding the
topology of the complement of an arrangement of complex
hyperplanes in terms of the combinatorics of the intersection
lattice of the arrangement. More specifically, I will explain how
certain homological invariants associated to the Milnor fibration
of the complement can be computed in terms of multinets on
the underlying matroid.
The automorphism group of a group $G$ comes endowed with a natural filtration: an automorphism be... more The automorphism group of a group $G$ comes endowed with a natural filtration: an automorphism belongs to the $k$-th term of this ``Johnson filtration" if it has the same $k$-jet as the identity, with respect to the lower central series of $G$. In this talk, I will discuss the Johnson filtration of the automorphism group of a finitely generated free group, and that of the mapping class group of a surface, with emphasis on the homological finiteness properties of the first few terms in these filtrations. A key ingredient in this approach is a rather surprising relationship between the classical representation theory of a complex, semisimple Lie algebra $\mathfrak{g}$ and the resonance varieties $R(V,K)\subset V^*$ attached to irreducible $\mathfrak{g}$-modules $V$ and submodules $K\subset V\wedge V$. In the case when $\mathfrak{g}= \mathfrak{sl}_2(\mathbb{C})$, this relationship sheds new light on certain modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves. This is joint work with Stefan Papadima (arXiv:1011.5292, 1207.2038).
The purpose of this workshop was to bring together researchers from the two different fields ment... more The purpose of this workshop was to bring together researchers from the two different fields mentioned in the title, and to create more interac- tion and connections between these fields. Among the topics which appear in both subjects are Lusternik-Schnirelmann category, Bieri-Neumann-Strebel invariants and a spectral sequence introduced by Farber and Novikov.
Lecture notes from the CIMPA Summer School held at Galatasaray University, Istanbul, June 11-22,... more Lecture notes from the CIMPA Summer School held at Galatasaray University, Istanbul, June 11-22, 2007
Every finite-type graded algebra defines a complex
of finitely generated, graded modules over a... more Every finite-type graded algebra defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology modules are called the Koszul modules of the given algebra. Particularly interesting in a variety of contexts is the geometry of the support loci of these modules, known as the resonance schemes of the algebra. In this talk, I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley-Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced, and give bounds on the regularity and projective dimension of the Koszul modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
A complex hyperplane arrangement A is said to be decomposable if there are no elements in the deg... more A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant B(G) also decomposes as a direct sum of "local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if B(G) is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.
A complex hyperplane arrangement A is said to be decomposable if there are no elements in the deg... more A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the I-adic completion of the Alexander invariant B(G) also decomposes as a direct sum of ``local" invariants and the Chen ranks of G are the sums of the local contributions. Moreover, if B(G) is separated, then the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.
The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, w... more The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.
The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, w... more The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.
The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, w... more The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.
Master class on Hodge theory and cohomological jump loci, Stockholm University, Sweden, 2024
The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, w... more The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.
It has long been recognized that there are many fruitful connections between the protagonists of ... more It has long been recognized that there are many fruitful connections between the protagonists of this workshop: hyperplane arrangements and Artin groups. I will discuss in this talk a connection between two kinds of fibrations that occur in these contexts. One is the Milnor fibration of the complement of an arrangement of complex hyperplanes (perhaps taken with multiplicities) over the circle. The other is an "algebraic fibration" of the right-angled Artin group associated to a graph (perhaps with weights assigned to the vertices) over the integers. In either case, one is interested in the topological invariants of the fiber (or the kernel), be it the Milnor fiber of the (multi-)arrangement, the Bestvina-Brady group of the graph, or the Artin kernel of the weighted graph. After a brief overview of these topics, I will focus on the case when the monodromy of the aforementioned fibrations is trivial in homology, and I will outline several new results in this direction, some of which can be stated in a unified way.
Every finite-type graded algebra defines a complex of finitely generated, graded modules over a s... more Every finite-type graded algebra defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology modules are called the Koszul modules of the given algebra. Particularly interesting in a variety of contexts is the geometry of the support loci of these modules, known as the resonance schemes of the algebra. In this talk, I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley-Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced, and give bounds on the regularity and projective dimension of the Koszul modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
The resonance varieties are cohomological invariants that are studied
in a variety of topologica... more The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric settings. I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley--Reisner rings of simplicial complexes, this approach leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin groups. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
In previous work we introduced the notion of binomial cup-one algebras, which are differential gr... more In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod cup-one products and compatible binomial operations. Given such an R-dga, (A,d), defined over the ring R=Z or Z_p (for p a prime) and with H^1(A) a finitely generated, free R-module, we show that A admits a functorially defined 1-minimal model, unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, G(A), which only depends on the 1-quasi-isomorphism type of A. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic (torsion-free) cohomology that share the same rational 1-minimal model, yet whose integral 1-minimal models are not isomorphic. This is joint work with Richard Porter.
Much of the fascination with arrangements of complex
hyperplanes comes from the rich interplay ... more Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice, the algebraic topology of the complement and its Milnor fibration. A key bridge between these objects is provided by the geometry of two sets of algebraic varieties associated to the complement: the resonance varieties of the cohomology ring and the characteristic varieties of the fundamental group. I will discuss some recent advances in our understanding of these topics, illustrating with concrete examples aided by computer computations.
There are several interrelated Lie algebras associated with the complement and the Milnor fiber o... more There are several interrelated Lie algebras associated with the complement and the Milnor fiber of a complex hyperplane arrangement: the holonomy Lie algebra, the associated graded Lie algebra, the Chen Lie algebra, and the Malcev Lie algebra. In this talk, I will discuss these notions and present recent progress in our understanding of these objects.
We study the lower central series, the Alexander invariants, and the cohomology jump loci of grou... more We study the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as split extensions with trivial monodromy in first homology with appropriate coefficients. We use these techniques to gain further understanding of the Milnor fibration of the complement of a hyperplane arrangement.
I will present a study of the lower central series, the Alexander invariants, and the cohomology ... more I will present a study of the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as extensions with trivial monodromy in first homology with appropriate coefficients.
We study the lower central series, the Alexander invariants, and the cohomology jump loci of grou... more We study the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as split extensions with trivial monodromy in first homology with appropriate coefficients. We use these techniques to gain further understanding of the Milnor fibration of the complement of a hyperplane arrangement.
Given a finitely generated group π and a complex, linear algebraic group G, the representation va... more Given a finitely generated group π and a complex, linear algebraic group G, the representation variety Hom(π,G) has a natural filtration by the cohomology jump loci associated to a rational representation from G to GL(V). The infinitesimal counterpart of Hom(π,G) around the trivial representation is the space of g-valued flat connections on an appropriate commutative, differential graded algebra (A,d). This space admits a corresponding filtration by the resonance varieties associated to the tangential representation from g to gl(V) of the Lie algebra of G. I will explain how one can understand the local behavior of all these varieties, at least in some favorable situations of geometric interest. This approach works particularly well in the case when G=SL(2,C) and π is either a Kähler group or the fundamental group of a smooth, quasi-projective variety. This is joint work with the late Stefan Papadima.
The cohomology jumping loci of a space come in two
basic flavors: the characteristic varieties, ... more The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. The geometry of these varieties, and the interplay between them sheds new light on the topology of the original space and that of its abelian covers.
In these introductory lectures, I will explain the algebraic notions underlying these constructions, and present some structural results. I will illustrate the general theory with a number of examples, such as the computation of the resonance varieties of the Stanley-Reisner ring associated to a simplicial complex, or the Orlik-Solomon algebra associated to a hyperplane arrangement. I will conclude with a brief overview of some topics of current research, such as resonance varieties of CDGA models for spaces, the influence of formality, and the interplay between resonance and duality.
The polyhedral product is a functorial construction that assigns to each simplicial complex K on ... more The polyhedral product is a functorial construction that assigns to each simplicial complex K on n vertices, and to each pair of topological spaces, (X,A), a certain subspace, Z_K(X,A), of the cartesian product of n copies of X. I will discuss some of the relationships between the duality properties of these spaces and the Cohen-Macaulay property of the original simplicial complex. This is based on joint work with Graham Denham and Sergey Yuzvinsky.
Each connected graded, graded-commutative algebra A of finite type over a field k of characterist... more Each connected graded, graded-commutative algebra A of finite type over a field k of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When A " kx∆y is the exterior Stanley-Reisner algebra associated to a finite simplicial complex ∆, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.
The resonance varieties are cohomological invariants that are studied in a variety of topological... more The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.
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 Ñ K Ñ G Ñ Q Ñ 1, where Q is an abelian group acting trivially on H 1 pK; Zq, 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.
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 H _1 (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.
Motivated by the construction of Steenrod cup-i products in the singular cochain algebra of a spa... more Motivated by the construction of Steenrod cup-i products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants. Contents
We study the germs at the origin of G-representation varieties and the degree 1 cohomology jump l... more We study the germs at the origin of G-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group G is either SL 2 pCq or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either G " SL n pCq for some n ě 3, or the depth is greater than 1, then certain natural inclusions of germs are strict.
We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algeb... more We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for computing the corresponding holonomy Lie algebra, using a Magnus expansion method. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of Seifert fibered manifolds.
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fundamental group of the Milnor fiber in this situation.
associate to a finitely generated group G -- the characteristic
varieties, the resonance varieties, and the Bieri–Neumann–Strebel
invariants -- which keep track of various finiteness properties
of certain subgroups of G. These invariants are interconnected
in ways that makes them both more computable and more informative.
I will describe one such connection, made possible by tropical
geometry, and I will provide examples and applications
pertaining to complex geometry and low-dimensional topology.
theory of a complex, semisimple Lie algebra \g and the resonance
varieties R(V,K)\subset V^* attached to irreducible \g-modules V
and submodules K\subset V\wedge V. In the process, I will give a
precise roots-and-weights criterion insuring the vanishing of these
varieties In the case when \g= \sl_n(\C) or \sp_{2g}(\C), our approach
yields a unified proof of two vanishing results for the resonance varieties
of the (outer) Torelli groups of finitely generated free groups and surface groups. In turn, these vanishing results reveal certain homological finiteness properties of the Johnson filtration. This is joint work with Stefan Papadima.
This is joint work with Stefan Friedl.
This is joint work with Graham Denham and Sergey Yuzvinsky.
topology of the complement of an arrangement of complex
hyperplanes in terms of the combinatorics of the intersection
lattice of the arrangement. More specifically, I will explain how
certain homological invariants associated to the Milnor fibration
of the complement can be computed in terms of multinets on
the underlying matroid.
of finitely generated, graded modules over a symmetric algebra,
whose homology modules are called the Koszul modules of the given
algebra. Particularly interesting in a variety of contexts is
the geometry of the support loci of these modules, known as
the resonance schemes of the algebra. In this talk, I will
describe several conditions that ensure the reducedness of
the associated projective resonance schemes and yield asymptotic
formulas for the Hilbert series of the corresponding Koszul modules.
For the exterior Stanley-Reisner algebra associated to a finite
simplicial complex, we show that the resonance schemes are reduced,
and give bounds on the regularity and projective dimension of the
Koszul modules. This leads to a relationship between resonance
and Hilbert series that generalizes a known formula for the Chen
ranks of a right-angled Artin group. Based on joint work with
Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
in a variety of topological, combinatorial, and geometric settings. I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley--Reisner rings of simplicial complexes, this approach leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin groups. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
hyperplanes comes from the rich interplay between the combinatorics
of the intersection lattice, the algebraic topology of the complement
and its Milnor fibration. A key bridge between these objects is provided
by the geometry of two sets of algebraic varieties associated to the
complement: the resonance varieties of the cohomology ring and the
characteristic varieties of the fundamental group. I will discuss
some recent advances in our understanding of these topics, illustrating
with concrete examples aided by computer computations.
basic flavors: the characteristic varieties, which are the jump loci
for homology with coefficients in rank 1 local systems, and the
resonance varieties, which are the jump loci for the homology
of cochain complexes arising from multiplication by degree 1
classes in the cohomology ring. The geometry of these varieties,
and the interplay between them sheds new light on the topology
of the original space and that of its abelian covers.
In these introductory lectures, I will explain the algebraic notions
underlying these constructions, and present some structural results.
I will illustrate the general theory with a number of examples, such
as the computation of the resonance varieties of the Stanley-Reisner
ring associated to a simplicial complex, or the Orlik-Solomon algebra
associated to a hyperplane arrangement. I will conclude with a brief
overview of some topics of current research, such as resonance varieties
of CDGA models for spaces, the influence of formality, and the interplay
between resonance and duality.