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Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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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 145 (12 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...
Tropical geometry and its applications
 the Proceedings of the Madrid ICM
"... Abstract. These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM. 1. ..."
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Cited by 142 (7 self)
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Abstract. These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM. 1.
Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
 Internat. Math. Res. Notices bf
, 1998
"... Abstract. We construct a generalization of the variations of Hodge structures on CalabiYau manifolds. It gives a Mirror partner for the theory of genus = 0 GromovWitten invariants. ..."
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Cited by 142 (4 self)
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Abstract. We construct a generalization of the variations of Hodge structures on CalabiYau manifolds. It gives a Mirror partner for the theory of genus = 0 GromovWitten invariants.
Notes On Stable Maps And Quantum Cohomology
, 1996
"... Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Varia ..."
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Cited by 141 (14 self)
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Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Variations 43 References 46 0. Introduction 0.1. Overview. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields nontrivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let N d be the number of degree d, rational plane curves passing through 3d \Gamma 1 general points in P . Since there is a un
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 141 (11 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
Gromov–Witten theory of Deligne–Mumford stacks
, 2006
"... 2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18 ..."
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Cited by 129 (10 self)
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2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18
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 126 (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
Hurwitz numbers and intersections on moduli spaces of curves
 Invent. Math
"... 1.1. Topological classification of ramified coverings of the sphere. For a compact connected genus g complex curve C let f: C → CP 1 be a meromorphic function. We treat this function as a ramified covering of the sphere. Two ramified coverings (C1; f1), (C2; f2) are called topologically ..."
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Cited by 118 (3 self)
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1.1. Topological classification of ramified coverings of the sphere. For a compact connected genus g complex curve C let f: C → CP 1 be a meromorphic function. We treat this function as a ramified covering of the sphere. Two ramified coverings (C1; f1), (C2; f2) are called topologically