Geometry and Topology of Counting Curves in Projective Manifolds

射影流形中计数曲线的几何和拓扑

• 批准号: 0072182
• 负责人: Kefeng Liu
• 金额: $ 7.81万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072182&HistoricalAwards=false
• 关键词: Geometry; Topology; Counting; Curves; Projective

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.

AbstractAward：DMS-0072182主要研究员：刘科峰我的项目将侧重于理解镜像对称的几何和拓扑，特别是射影流形中的计数曲线方面。更准确地说，有三个突出的和密切相关的问题，我想工作，并希望解决。首先证明了任意射影流形及其Calabi-Yau子流形中有理曲线计数的最一般镜像原理;其次理解了局部镜像对称的几何和代数，特别是应用镜像原理计算中出现的椭圆曲线所起的令人惊讶的作用;第三发展了射影流形中高亏格曲线计数的镜像原理。我希望通过进一步扩展我们开发的技术来解决这些问题，以证明气球流形的镜像原理。这三个问题是同一个问题的不同方面：证明和理解镜像原理的最一般形式。超弦理论是科学中最雄心勃勃的理论之一，它是为了大统一自然定律而提出的。基于超弦理论，弦理论家们做出了许多令人难以置信的数学成就。镜像对称就是其中之一，这些对称性的证明不仅会给出漂亮的数学结果，而且有助于获得对物理理论的信心。我们提出的镜像原理部分地验证了其中的一些原理。它有许多有趣的数学结果，并且与物理理论非常相容。