## Symplectic Geometry and Mirror Symmetry: Proceedings of the 4th KIAS Annual International Conference, Korea Institute for Advanced Study, Seoul, South Korea, 14-18 August 2000In 1993, M Kontsevich proposed a conceptual framework for explaining the phenomenon of mirror symmetry. Mirror symmetry had been discovered by physicists in string theory as a duality between families of three-dimensional Calabi-Yau manifolds. Kontsevich's proposal uses Fukaya's construction of the Aì-category of Lagrangian submanifolds on the symplectic side and the derived category of coherent sheaves on the complex side. The theory of mirror symmetry was further enhanced by physicists in the language of D-branes and also by Strominger-Yau-Zaslow in the geometric set-up of (special) Lagrangian torus fibrations. It rapidly expanded its scope across from geometry, topology, algebra to physics.In this volume, leading experts in the field explore recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov-Witten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of Aì-categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by Fukaya-Oh-Ohta-Ono which takes an important step towards a rigorous construction of the Aì-category. There follow a number of contributions on the homological mirror symmetry, D-branes and the Gromov-Witten invariants, e.g. Getzler shows how the Toda conjecture follows from recent work of Givental, Okounkov and Pandharipande. This volume provides a timely presentation of the important developments of recent years in this rapidly growing field. |

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### Contents

Estimated transversahty in symplectic geometry and projective maps | 1 |

Local mirror symmetry and fivedimensional gauge theory | 31 |

The Toda conjecture | 51 |

Examples of special Lagrangian fibrations | 81 |

Linear models of supersymmetric Dbranes | 111 |

The connectedness of the moduli space of maps to homogeneous spaces | 187 |

Homological mirror symmetry and torus fibrations | 203 |

Genus1 Virasoro conjecture on the small phase space | 265 |

Obstruction to and deformation of Lagrangian intersection Floer cohomology | 281 |

Topological open pbranes | 311 |

Lagrangian torus fibration and mirror symmetry of CalabiYau manifolds | 385 |

More about vanishing cycles and mutation | 429 |

Moment maps monodromy and mirror manifolds | 467 |

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Aco-category Aco-pre-category action functional algebra asymptotically holomorphic boundary condition boundary interaction bracket BV master equation Calabi-Yau manifolds canonical chiral superfield codimension compact complex structure consider constant construction coordinates corresponding critical points curves D-branes defined Definition deformation denote derived derived category differential dimension equivalent F-term fiber fibration field theory Floer cohomology formula Fukaya category gauge theory geometry ghost number given graded graph Hamiltonian homology homotopy hypersurface intersection invariant isomorphic Kähler Kontsevich Lagrangian submanifold Lagrangian torus fibration Lemma line bundle linear master equation Math metric mirror symmetry moduli space moment map monodromy morphisms Morse Nucl parameter Phys proof Proposition quantum cohomology satisfies sequence sigma model singular fibres singular locus smooth special Lagrangian fibrations stratification string theory superpotential supersymmetry symplectic manifold Theorem transversality trivial vanishing vector field Virasoro conjecture Witten zero