Mirror Symmetry and Geometry of Calabi-Yau Manifolds

美国国家科学基金会建议DMS-0103355的摘要：在这个项目中，首席研究员研究了Calabi-Yau流形的几何和镜像对称性。首先，PI计划描述Calabi-Yau流形上在大复数结构极限附近的Kahler-Einstein度规的退化。第二，主要研究了Calabi-Yau流形上特殊的Lagrangian子流形和变形的Hermitian-Yang-Mills丛。第三，利用傅立叶变换、勒让德变换及其变式，解释了镜像流形之间复几何和辛几何的镜像对偶性。Calabi-Yau流形的几何是数学和物理学中的一个重要课题。起源于弦理论的镜像对称猜想是这种几何的对偶变换。这个项目正朝着揭示这一惊人的对称性原理的方向发展。

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.