Mirror Symmetry and Geometry of Calabi-Yau Manifolds

摘要为NSF提案DMS-0103355：在这个项目中，主要研究者研究几何ofCalabi-Yau流形和镜像对称。首先，PI计划描述Calabi-Yau流形上Kahler-Einstein度量在大复结构极限附近的退化. 其次主要研究了Calabi-Yau流形上的特殊Lagrange子流形和变形Hermitian-Yang-米尔斯丛。第三，利用Fourier变换、Legendre变换及其变式，解释了镜像流形之间复几何和辛几何的镜像对偶性。Calabi-Yau流形的几何是数学中的一个重要课题，也是物理学中的重要课题。镜像对称起源于弦论，它是这种几何的对偶变换。这个项目是为了揭示这个惊人的对称原理。

Abstract for NSF proposal DMS-0103355:In this project, the principal investigator investigates the geometry ofCalabi-Yau manifolds and mirror symmetry. First the PI plans to describe thedegeneration of Kahler-Einstein metrics on Calabi-Yau manifolds near thelarge complex structure limit. Second the principal investigator studies special Lagrangian submanifolds and deformed Hermitian-Yang-Mills bundles on Calabi-Yau manifolds. Third, using Fourier transformation, Legendre transformation and their variations, the principal investigator explains the mirror duality of complexgeometry and symplectic geometry between mirror manifolds.The geometry of Calabi-Yau manifolds is an important subject in Mathematics,and also in Physics. Originated in String theory, mirror symmetry conjecturegives a duality transformation of this geometry. This project is towardsuncovering this amazing symmetry principle.