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238
Formal (non)commutative symplectic geometry, from
 The Gelfand Mathematical Seminars, 1990–1992”, Birkhäuser
, 1993
"... and I had tried to understand a remark of J. Stasheff [15] on open string theory and higher associative algebras [16]. Then I found a strange construction of cohomology classes of mapping class groups using as initial data any differential graded algebra with finitedimensional cohomology and a kin ..."
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Cited by 150 (4 self)
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and I had tried to understand a remark of J. Stasheff [15] on open string theory and higher associative algebras [16]. Then I found a strange construction of cohomology classes of mapping class groups using as initial data any differential graded algebra with finitedimensional cohomology and a kind of Poincare ́ duality. Later generalizations to the commutative and Lie cases appeared. In attempts to formulate all this I have developed a kind of (non)commutative calculus. The commutative version has fruitful applications in topology of smooth manifolds in dimensions ≥ 3. The beginnings of applications are perturbative ChernSimons theory (S. Axelrod and I.M. Singer [1] and myself), V. Vassiliev’s theory of knot invariants and discriminants (see [19], new results in [2]) and V. Drinfeld’s works on quasiHopf algebras (see [6]), also containing elements of Lie calculus. Here I present the formal aspects of the story. Theorem 1.1 is the main motivation for my interest in noncommutative symplectic geometry. Towards the end the exposition becomes a bit more vague and informal. Nevertheless, I hope that I will convince the reader that noncommutative calculus has every right to exist. I have benefited very much from conversations with B. Feigin, V. Retakh,
Twisted equivariant Ktheory with complex coefficients
, 2008
"... Using a global version of the equivariant Chern character, we describe an effective method for computing the complexified twisted equivariant Ktheory of a space ..."
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Cited by 50 (6 self)
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Using a global version of the equivariant Chern character, we describe an effective method for computing the complexified twisted equivariant Ktheory of a space
Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments
 Ann. of Math
"... The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene, see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with a special attentio ..."
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Cited by 40 (9 self)
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The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene, see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with a special attention to the case of hyperplane arrangements as discussed
On the structure of cofree Hopf algebras
 J. reine angew. Math
"... Abstract. We prove an analogue of the PoincaréBirkhoffWitt theorem and of the CartierMilnorMoore theorem for noncocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞algebra. We construct a universal enveloping functor U2 from nondifferential B∞algebr ..."
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Cited by 35 (4 self)
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Abstract. We prove an analogue of the PoincaréBirkhoffWitt theorem and of the CartierMilnorMoore theorem for noncocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞algebra. We construct a universal enveloping functor U2 from nondifferential B∞algebras to 2associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the description of the free 2asalgebra in terms of planar trees to unravel the operad associated to nondifferential B∞algebras.
HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 32 (10 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
Topology and geometry of cohomology jump loci
 DUKE MATHEMATICAL JOURNAL
, 2009
"... Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk a ..."
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Cited by 30 (21 self)
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Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic, if the group is 1formal; in particular, the tangent cone to Vk at 1 equals Rk. These new obstructions to 1formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasiprojective algebraic variety. This sheds new light on a classical problem of J.P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
Operads, homotopy algebra, and iterated integrals for double loop spaces
 15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146
, 1995
"... Chen's theory of iterated integrals provides a remarkable model for the differential forms on the based loop space M of a differentiable manifold M (Chen [10]; see also HainTondeur [23] and GetzlerJonesPetrack [21]). This article began as an attempt to nd an analogous model for 2 the complex ..."
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Cited by 29 (0 self)
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Chen's theory of iterated integrals provides a remarkable model for the differential forms on the based loop space M of a differentiable manifold M (Chen [10]; see also HainTondeur [23] and GetzlerJonesPetrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of differentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding twodimensional topological field theories. Our approach is to use the formalism of operads. Operads can be defined in any symmetric monoidal category, although we will mainly be concerned with dgoperads (differential graded operads), that is, operads in the category of chain complexes with monoidal structure defined by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative  we give a precise definition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a nonlinear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the kth tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or aalgebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)