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567
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 343 (2 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
Holomorphic Disks and Topological Invariants for Closed ThreeManifolds
 Ann. of Math
, 2000
"... The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relat ..."
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Cited by 149 (33 self)
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The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relative to certain totally real subspaces associated to U0 and U1.
GLUING TIGHT CONTACT STRUCTURES
, 2002
"... We prove gluing theorems for tight contact structures. As special cases, we rederive gluing theorems due to V. Colin and S. MakarLimanov and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a ..."
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Cited by 135 (22 self)
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We prove gluing theorems for tight contact structures. As special cases, we rederive gluing theorems due to V. Colin and S. MakarLimanov and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 handlebody which becomes overtwisted after Legendrian −1 surgery and study certain Legendrian surgeries on T³.
Equivariant GromovWitten invariants
 Internat. Math. Res. Notices
, 1996
"... The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a co ..."
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Cited by 92 (10 self)
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The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a compact Kähler manifold X, the equivariant GWtheory provides, as we will show in Section 3, the equivariant cohomology space H ∗ G (X) with a Frobenius structure (see [11]). We will discuss applications of the equivariant theory to the computation ([15], [18]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cupproduct operators (Sections 7, 8), to the S1equivariant Floer homology theory on the loop space LX (see Section 6 and [14], [13]), and to a “quantum ” version of the Serre duality theorem (Section 12). In Sections 9–11 we combine the general theory developed in Sections 1–6 with the fixedpoint localization technique [21], in order to prove the mirror conjecture (in the form suggested in [14]) for projective complete intersections. By the mirror conjecture, one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kähler CalabiYau nfold and, respectively, complex and symplectic geometry on another CalabiYau nfold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on CalabiYau 3folds (1991, see the theorem below) raised a number of new mathematical problems—challenging tests of maturity for modern methods of symplectic topology. On the other hand, in 1993 I suggested that the relation between symplectic and complex geometry predicted by the mirror conjecture can be extended from the class of CalabiYau manifolds to more general compact symplectic manifolds if one admits non
Symplectic FloerDonaldson theory and quantum cohomology
 in Proceedings of the Symposium on Symplectic Geometry, held at the Isaac Newton Institute in Cambridge in 1994, edited by C.B. Thomas, LMS Lecture Note Series
, 1996
"... The goal of this paper is to give in outline a new proof of the fact that the Floer cohomology groups of the loop space of a semipositive symplectic manifold (M;!) are naturally isomorphic to the ordinary cohomology of M . We shall then outline a proof that this isomorphism intertwines the quantum ..."
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Cited by 88 (10 self)
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The goal of this paper is to give in outline a new proof of the fact that the Floer cohomology groups of the loop space of a semipositive symplectic manifold (M;!) are naturally isomorphic to the ordinary cohomology of M . We shall then outline a proof that this isomorphism intertwines the quantum cupproduct structure on the cohomology of M with the pairofpants product on Floerhomology. One of the key technical ingredients of the proof is a gluing theorem for Jholomorphic curves proved in [20]. In this paper we shall only sketch the proofs. Full details of the analysis will appear elsewhere. 1 Introduction The Floer homology groups of a symplectic manifold (M;!) can intuitively be described as the middle dimensional homology groups of the loop space. The boundary loops of Jholomorphic discs in the symplectic manifold with center in a given homology class ff 2 H (M) (integral homology modulo torsion) form a submanifold of the loop space of roughly half dimension and should ther...
Virtual neighborhoods and pseudoholomorphic curves
 Turkish J. Math
"... Since Gromov introduced his pseudoholomorphic curve theory in 80’s, pseudoholomorphic curve soon became a predominant technique in symplectic topology. Many important theorems in symplectic topology have been proved by the technique of pseudoholomorphic curve, for example, the squeezing theorem [ ..."
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Cited by 83 (10 self)
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Since Gromov introduced his pseudoholomorphic curve theory in 80’s, pseudoholomorphic curve soon became a predominant technique in symplectic topology. Many important theorems in symplectic topology have been proved by the technique of pseudoholomorphic curve, for example, the squeezing theorem [Gr], the rigidity [E], the classifications of rational and ruled symplectic 4
Heegaard Floer homologies and contact structures
 Duke Math. J
"... Abstract. Given a contact structure on a closed, oriented threemanifold Y, we describe an invariant which takes values in the threemanifold’s Floer homology ̂ HF (in the sense of [10]). This invariant vanishes for overtwisted contact structures and is nonzero for Stein fillable ones. The construc ..."
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Cited by 82 (12 self)
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Abstract. Given a contact structure on a closed, oriented threemanifold Y, we describe an invariant which takes values in the threemanifold’s Floer homology ̂ HF (in the sense of [10]). This invariant vanishes for overtwisted contact structures and is nonzero for Stein fillable ones. The construction uses of Giroux’s interpretation of contact structures in terms of open book decompositions (see [4]), and the knot Floer homologies introduced in [14]. 1.