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Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 529 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Moduli of Coassociative Submanifolds and Semiflat Coassociative Fibrations
, 2009
"... We show that the moduli space of deformations of a compact coassociative submanifold L has a natural local embedding as a submanifold of H 2 (L, R). We show that a G2manifold with a T 4action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3manifold ..."
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We show that the moduli space of deformations of a compact coassociative submanifold L has a natural local embedding as a submanifold of H 2 (L, R). We show that a G2manifold with a T 4action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3manifold
Coassociative 4folds with Conical Singularities
, 2006
"... This paper is dedicated to the study of deformations of coassociative 4folds in a ..."
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Cited by 7 (5 self)
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This paper is dedicated to the study of deformations of coassociative 4folds in a
Coassociative Cones that are Ruled by 2Planes
, 2005
"... Abstract. It is shown that coassociative cones in R 7 that are roriented and ruled by 2planes are equivalent to CRholomorphic curves in the oriented Grassmanian of 2planes in R 7. The geometry of these CRholomorphic curves is studied and related to holomorphic curves in S 6. This leads to an eq ..."
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Cited by 3 (0 self)
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Abstract. It is shown that coassociative cones in R 7 that are roriented and ruled by 2planes are equivalent to CRholomorphic curves in the oriented Grassmanian of 2planes in R 7. The geometry of these CRholomorphic curves is studied and related to holomorphic curves in S 6. This leads
VERTEX COALGEBRAS, COMODULES, COCOMMUTATIVITY AND COASSOCIATIVITY
, 801
"... Abstract. We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skewsymmetry, and an endomorphism, D ∗ , which hold on vertex coalgebras. The former two properties require grading. We then discuss ..."
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Cited by 1 (0 self)
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Abstract. We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skewsymmetry, and an endomorphism, D ∗ , which hold on vertex coalgebras. The former two properties require grading. We then discuss
Coassociativity breaking and oriented graphs 1
, 2002
"... Abstract: With each coassociative coalgebra, we associate an oriented graph. The coproduct ∆, obeying the coassociativity equation ( ∆ ⊗ id) ∆ = (id ⊗ ∆) ∆ is then viewed as a physical propagator which can convey information. We notice that such a coproduct is non local. To recover locality we hav ..."
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Abstract: With each coassociative coalgebra, we associate an oriented graph. The coproduct ∆, obeying the coassociativity equation ( ∆ ⊗ id) ∆ = (id ⊗ ∆) ∆ is then viewed as a physical propagator which can convey information. We notice that such a coproduct is non local. To recover locality we
Homotopy decompositions involving the loops of coassociative coHspaces
 School of Mathematics, University of Manchester, Manchester
"... Abstract. James gave an integral homotopy decomposition of ΣΩΣX, HiltonMilnor one for Ω(ΣX ∨ ΣY), and CohenWu gave plocal decompositions of ΩΣX if X is a suspension. All are natural. Using idempotents and telescopes we show that the James and HiltonMilnor decompositions have analogues when the s ..."
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Cited by 12 (4 self)
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the suspensions are replaced by coassociative coH spaces, and the CohenWu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative coH space. 1.
The CROCs, noncommutative deformations, and (co)associative bialgebras
, 2003
"... We compactify the spaces K(m, n) introduced by Maxim Kontsevich. The initial idea was to construct an L ∞ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic stru ..."
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Cited by 11 (2 self)
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We compactify the spaces K(m, n) introduced by Maxim Kontsevich. The initial idea was to construct an L ∞ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic
Deformations of Asymptotically Cylindrical Coassociative Submanifolds with Fixed
 Boundary, Geometry&Topology
, 2005
"... Abstract. In an earlier paper, [5], we proved that given an asymptotically cylindrical G2manifold M with a Calabi–Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4fold C ⊂ M with a fixed special Lagrangian boundary L ⊂ X is a smooth man ..."
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Cited by 11 (1 self)
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Abstract. In an earlier paper, [5], we proved that given an asymptotically cylindrical G2manifold M with a Calabi–Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4fold C ⊂ M with a fixed special Lagrangian boundary L ⊂ X is a smooth
Results 1  10
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3,696