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Introduction
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, Milnor fibrations, 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.
Current institution
Additional affiliations
July 2017 - June 2020
July 1992 - May 2000
September 1986 - June 1992
Education
July 1980 - June 1984
October 1977 - June 1980
September 1971 - June 1976
Publications
Publications (250)
The k-th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the algebraic torus (C^*)^n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V_k(A). For any arrangement A, we s...
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-for...
Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these top...
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 survey some of the connections between the duality
properties of these spaces and the Cohen-Macaulay property...
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...
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 t...
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...
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...
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...
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 $1\to K\to G\to Q\to 1$, where $Q$ is an abelian group acting trivially on $H_1(K;\mathbb{Z})$, with suitable modifications in the rational an...
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...
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 expl...
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 expl...
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 expl...
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 exp...
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 s...
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...
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...
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...
Each connected graded, graded-commutative algebra A of finite type over a field k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Bbbk $$\end{document} of characteristi...
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 (perha...
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...
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 wi...
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 som...
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...
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...
Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ 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...
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 functorial...
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 r...
In previous work, we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod $\cup_1$-products and compatible binomial operations. Given such an $R$-dga, $(A,d_A)$, defined over the ring $R=\mathbb{Z}$ or $\mathbb{Z}_p$ (for $p$ a prime), with $H^0(A)=R$ and with $H^1(A)$ a finitely generated...
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 exte...
I will present a study of the lower central series,
the Alexander invariants, and the cohomology jump loci of groups
occurring as extensions with trivial monodromy in first homology
with appropriate coefficients. I will illustrate these concepts
with examples arising from the Milnor fibration of the complement
of a hyperplane arrangement and w...
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 obje...
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 du...
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...
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 usef...
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 som...
Following Lazard, we study the $N$-series of a group $G$ and their associated graded Lie algebras. The main examples we consider are the lower central series and Stallings' rational and mod-$p$ versions of this series. Building on the work of Massuyeau and Guaschi-Pereiro, we describe these $N$-series and Lie algebras in the case when $G$ splits as...
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.
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...
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-produc...
The Hyperplane Arrangements and Singularities Conference
(also known as Hyper-JARCS) was held December 2-6, 2019
at the University of Tokyo, Japan. The Organizing Committee
for the conference consisted of Laurentiu~Paunescu (Chair),
Alexandru Dimca, Toshizumi Fukui, Toshitake Kohno, Alex Suciu,
and Masahiko Yoshinaga.
The conference built on...
We consider smooth, complex quasi-projective varieties 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 the va...
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
Novik...
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 explore here the geometry of these varieties and the de...
![Figure 2. Cochains for the Massey triple product [a], [a], [a] in K(Z 3...](profile/Alex-Suciu/publication/351841394/figure/fig1/AS:1027322744410113@1621944237389/Cochains-for-the-Massey-triple-product-a-a-a-in-KZ-3-1_Q320.jpg)
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-prod...
Braid groups, configuration spaces, and hyperplane arrangements have been intertwined for at least 60 years. I will discuss some recent advances in our understanding of fundamental
groups of complements of complex line arrangements, with emphasis on several of the Lie algebras associated to them.
A recurring theme in geometry and topology is to determine the duality and finiteness properties of spaces and groups. I will discuss some of the interplay between these properties, the structure of associated algebraic models, and the geometry of the corresponding cohomology jump loci. I will also outline some of the applications of this theory to...
In this talk I survey the work of Alex Dimca over more than four decades, highlighting some of his many research contributions to algebraic geometry and singularity theory.
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 and the fundamental group...
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 $1\to K\to G\to Q\to 1$, where $Q$ is an abelian group acting trivially on $H_1(K;\mathbb{Z})$, with suitable modifications in the rational an...
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.
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 Bieri--Neumann--Strebel--Renz invariants Sigma^q(X) ⊂ H^1(X,R) are the vanishing loci for the Novikov--Sikorav homology of a finite CW-complex X in degrees up to q. The characteristic varieties
V^q(X) ⊂ H^1(X,C^*)$ are the non-vanishing loci in degree q for homology with coefficients in rank 1 local systems. I will show that each BNSR invarian...
![BNS invariants Σ1(G)\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/351433453/figure/fig1/AS:11431281116720307@1675306317620/BNS-invariants-S1Gdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
The Bieri–Neumann–Strebel–Renz invariants \(\Sigma ^q(X,\mathbb {Z})\subset H^1(X,\mathbb {R})\) of a connected, finite-type CW-complex X are the vanishing loci for Novikov–Sikorav homology in degrees up to q, while the characteristic varieties \(\mathcal {V}^q(X) \subset H^1(X,\mathbb {C}^{\times })\) are the nonvanishing loci for homology with co...
The Bieri--Neumann--Strebel--Renz invariants Sigma^q(X) of a connected, finite-type CW-complex X are the vanishing loci for the Novikov--Sikorav homology of X in degrees up to q. These invariants live in the unit sphere inside H^1(X, R); this sphere can be thought of as parametrizing all free abelian covers of X, while the \Sigma-invariants keep tr...
The Bieri-Neumann-Strebel-Renz invariants $\Sigma^q(X,\mathbb{Z})\subset H^1(X,\mathbb{R})$ of a connected, finite-type CW-complex $X$ are the vanishing loci for Novikov-Sikorav homology in degrees up to $q$, while the characteristic varieties $\mathcal{V}^q(X) \subset H^1(X,\mathbb{C}^{\times})$ are the nonvanishing loci for homology with coeffici...
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.
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 explore here the geometry of these varieties and the de...
The Bieri--Neumann--Strebel--Renz invariants $\Sigma^q(X)$
of a connected, finite-type CW-complex $X$ are the vanishing
loci for the Novikov--Sikorav homology of $X$ in degrees up to $q$.
These invariants live in the unit sphere inside $H^1(X,\mathbb{R})$;
this sphere can be thought of as parametrizing all free abelian covers
of $X$, while the...
Let G be a finitely generated group, and let kG be its group algebra over a field of characteristic 0. A Taylor expansion is a certain type of map from G to the degree completion of the associated graded algebra of kG which generalizes the Magnus expansion of a free group. The group G is said to be filtered-formal if its Malcev Lie algebra is isomo...
We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of Grigory Rybnikov in a more general framework and leads to an applicati...
Editorial for a volume dedicated to the memory of Stefan Papadima (1953--2018)
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 varieti...
In this article we develop the theory of residually finite rationally p (RFRp) groups, where p is a prime. We first prove a series of results about the structure of finitely generated RFRp groups (either for a single prime p, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furt...
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 connectio...
Does a space or a group admit a differential graded algebra model with certain desirable finiteness or formality properties? In this talk I will outline several obstructions for this to happen. Some will involve minimal models and Malcev Lie algebras, while others will
involve generalized Massey triple products.
The Bieri--Neumann--Strebel--Renz invariants Sigma^i(X) of a connected, finite-type CW-complex X are the vanishing loci for the Novikov homology of X in degrees up to i. The Sigma-invariants live in the unit sphere S(X) inside H^1(X, R); this sphere can be thought of as parametrizing all free abelian covers of X, while the Sigma-invariants keep tra...
We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a 3-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties R^i_k(A).
Let M be a compact smooth manifold supporting an almost free action by a compact, connected Lie group K. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe a commutative differential graded algebra model for M with commensurate finiteness and partial formality...
The Bieri--Neumann--Strebel--Renz invariants Sigma^i(X) of a connected, finite-type CW-complex X are the vanishing loci for the Novikov homology of X in degrees up to i. The Sigma-invariants live in the unit sphere S(X) inside H^1(X, R); this sphere can be thought of as parametrizing all free abelian covers of X, while the Sigma-invariants keep tra...
We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a 3-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties R^i_k(A).
Twenty-some-odd years ago, Donu Arapura wrote the paper "Geometry of cohomology support loci for local systems." To this day, this seminal paper provides inspiration and guides the work of geometers and topologists alike. I will discuss some recent developments in the theory, especially in regards to duality and finiteness properties of spaces and...
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 ap...
We use augmented commutative differential graded algebra (ACDGA) models to study G-representation varieties of fundamental groups π=π_1(M) and their embedded cohomology jump loci, around the trivial representation 1. When the space M admits a finite family of maps, uniformly modeled by ACDGA morphisms, and certain finiteness and connectivity assump...
The group of basis-conjugating automorphisms of the free group of rank $n$, also known as the McCool group or the welded braid group $P\Sigma_n$, contains a much-studied subgroup, called the upper McCool group $P\Sigma_n^+$. Starting from the cohomology ring of $P\Sigma_n^+$, we find, by means of a Gr\"obner basis computation, a simple presentation...
The group of basis-conjugating automorphisms of the free group of rank n, also known as the McCool group or the welded braid group PΣn, contains a much-studied subgroup, called the upper McCool group PΣ+n. Starting from the cohomology ring of PΣ+n, we find, by means of a Groebner basis computation, a simple presentation for the infinitesimal Alexan...
We explore the graded 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...
We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to...
Let $G$ be a finitely generated group, and let $\Bbbk{G}$ be its group algebra over a field of characteristic $0$. A Taylor expansion is a certain type of map from $G$ to the degree completion of the associated graded algebra of $\Bbbk{G}$ which generalizes the Magnus expansion of a free group. The group $G$ is said to be filtered-formal if its Mal...
Let G be a finitely generated group, and let kG be its group algebra over a field of characteristic 0. A Taylor expansion is a certain type of map from G to the degree completion of the associated graded algebra of kG which generalizes the Magnus expansion of a free group. The group G is said to be filtered-formal if its Malcev Lie algebra is isomo...
A recurring theme in topology is to determine the duality and finiteness properties of spaces and groups. I will discuss some of the interplay between these properties, the structure of algebraic models associated to them, and the geometry of the corresponding cohomology jump loci. Furthermore, I will outline some of the applications of this theory...
I will discuss some recent advances in our understanding of fundamental groups of complements of complex hyperplane arrangements, with emphasis on associated graded and holonomy Lie algebras, as well as Massey products in positive characteristic. The talk will be based on current joint work with Rick Porter and with He Wang.
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 holo...
Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1‐mi...
The Oberwolfach workshop "Topology of arrangements and representation stability'' brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and
the homological and representation stability of configuration
spaces and related families of spaces and discrete groups.
The participants were...
Recent developments exhibit a strong connection between low-dimensional topology and complex algebraic geometry. A common theme is provided by the Alexander polynomial and its many avatars. The mini-Workshop brought together at Oberwolfach groups of researchers working in mostly separate areas, but sharing common interests in a vibrant, emerging fi...
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 ge...
I will discuss some of the interplay between duality
and finiteness properties of spaces and groups, the structure
of differential graded algebra models associated to them,
and the geometry of the corresponding cohomology jump loci.
Furthermore, I will outline some of the applications of this theory
to complex algebraic geometry and low-dimens...
In this talk I survey the work of Stefan Papadima over the past four decades, highlighting some of his many research contributions to algebraic topology, algebraic geometry, group theory, and Lie theory, with special emphasis on cohomology jump loci, finiteness obstructions, and hyperplane arrangements.
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
$...
I will discuss some of the interplay between duality and finiteness properties of spaces and groups, the structure of differential graded algebra models associated to them, and the geometry of the corresponding cohomology jump loci.
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 function annotations. This accumulated structural information represents a tremendous contribution to structural biology and genomics. Still, the addition of a...
As a result of high‐throughput protein structure initiatives, over 14,300 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 la...
I will discuss some of the interplay between duality and finiteness properties of spaces and groups, the structure of differential graded algebra models associated to them, and the geometry of the corresponding cohomology jump loci.
In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups Pn, the welded pure braid groups wPn, the virtual pure braid groups vPn, as well as their ‘upper’ variants, wPn⁺ and...
Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1-mi...
A recurring theme in topology is to determine the geometric and homological finiteness properties of spaces and groups. A fruitful approach is to compare these finiteness properties to those of differential graded algebras that model topological objects of this sort. I will discuss several concrete questions that arise in this context, and explain...
