Geometry and Topology of Counting Curves in Projective Manifolds

摘要奖：DMS-0072182首席研究员：刘克峰我的项目将集中于对镜像对称的几何和拓扑的理解，特别是射影流形中曲线计数的方面。更确切地说，我想要解决并希望解决三个悬而未决的密切相关的问题。首先证明了任何射影流形及其Calabi-Yau子流形中计算有理曲线的最一般的镜像原理；其次是利用镜像原理理解局部镜像对称的几何和代数，特别是椭圆曲线在我们的计算中所起的惊人作用；第三是发展了射影流形中高亏格曲线的镜像原理。我希望通过进一步扩展我们开发的证明气球歧管的镜面原理的技术来解决这些问题。这三个问题是一个问题的不同方面：证明和理解镜像原理的最一般形式。超弦理论是科学界最雄心勃勃的理论之一，它是为了实现自然规律的大统一而提出的。基于这个理论，弦理论家做出了许多令人惊叹的数学猜想。镜像对称性就是其中之一。这些猜想的证明不仅会给出美丽的数学结果，而且还有助于获得对物理理论的信心。我们发展的镜像原理部分地证实了其中一些猜想。它有许多有趣的数学结果，并且与物理理论非常相容。

AbstractAward: DMS-0072182Principal Investigator: Kefeng LiuMy project will focus on the understanding of the geometry andtopology of mirror symmetry, in particular the aspect of countingcurves in projective manifolds. More precisely there are threeoutstanding and closely related problems that I would like towork on and hope to solve. The first is to prove the most generalmirror principle for counting rational curves in any projectivemanifold and its Calabi-Yau submanifolds; the second is tounderstand the geometry and algebra of the local mirror symmetry,in particular, of the surprising role played by the ellipticcurves appeared in our computation by applying mirror principle;the third is to develop a mirror principle for counting curves ofhigher genus in projective manifolds. I hope to solve theseproblems by further extending the techniques we developed toprove the mirror principle for balloon manifolds. These threeproblems are the different aspects of a single problem: to proveand understand the mirror principle in its most general form.Superstring theory, one of the most ambitious theory in sciences,is proposed for the grand unification of the laws of the nature.Based on this theory, string theorists have made many remarkablemathematical conjectures. Mirror symmetry is among one of them.The proofs of these conjectures will not only give beautifulmathematical results, but also help gain confidence in thephysical theory. The mirror principle we developed has partiallyverified some of these conjectures. It has many interestingmathematical consequences and is remarkably compatible with thephysical theory.