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126
A short survey of noncommutative geometry
 J. Math. Physics
, 2000
"... We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the RiemannHilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geom ..."
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Cited by 45 (3 self)
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We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the RiemannHilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It expresses the gamma five matrix as the pairing between the operator theoretic chern characters of e and D. It is of degree five in the idempotent and four in the Dirac operator which only appears through its commutant with the idempotent. It determines both the sphere and all its metrics with fixed volume form. We also show using the noncommutative analogue of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude on some questions related to string theory. I
Perturbative 3manifolds invariants by cutandpaste topology
, 1999
"... We give a purely topological definition of the perturbative quantum invariants of links and 3manifolds associated with ChernSimons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that t ..."
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Cited by 41 (1 self)
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We give a purely topological definition of the perturbative quantum invariants of links and 3manifolds associated with ChernSimons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.
The EckmannHilton argument, higher operads and Enspaces, available at http://www.ics.mq.edu.au
 mbatanin/papers.html of Homotopy and Related Structures
"... The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of ..."
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Cited by 33 (5 self)
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The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2category, then its Homset is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an noperad A in the author’s sense there exists a symmetric operad S n (A) called the nfold suspension of A such that the
Multivariable cochain operations and little ncubes
 J. Amer. Math. Soc
"... Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasiisomorphic to the normalized singular chains of the little ncubes operad. The case n = 2 leads t ..."
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Cited by 29 (1 self)
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Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasiisomorphic to the normalized singular chains of the little ncubes operad. The case n = 2 leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture. 1. Introduction. This paper has two goals. The first (see Theorem 2.15 and Remark 2.16(a)) is to construct a small E ∞ chain operad S which acts naturally on the normalized cochains S∗X of a topological space X. This is of interest in view of a theorem of Mandell [15, page 44] which states that if O is any E ∞ chain operad over Fp (the algebraic closure of the field with
Multiple ζvalues, Galois groups, and geometry of modular varieties
, 2000
"... We discuss two arithmetical problems, at first glance unrelated: 1) The properties of the multiple ζvalues ζ(n1,..., nm):= ∑ 1 nm> 1 (1) ..."
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Cited by 27 (6 self)
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We discuss two arithmetical problems, at first glance unrelated: 1) The properties of the multiple ζvalues ζ(n1,..., nm):= ∑ 1 nm> 1 (1)
Manifoldtheoretic compactifications of configuration spaces
 Selecta Math. (N.S
"... Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to FultonMacPherson and AxelrodSinger in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elemen ..."
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Cited by 24 (5 self)
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Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to FultonMacPherson and AxelrodSinger in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.
Operads and knot spaces
 J. Amer. Math. Soc
"... Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent ..."
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Cited by 24 (2 self)
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Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated