Mathematical Sciences: Geometry and Topology

数学科学：几何和拓扑

• 批准号: 9503616
• 负责人: Anatoly Libgober
• 金额: $ 5.1万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9503616&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Topology

Libgober 9503616 The proposed research lies in the interface of complex differential geometry and algebraic geometry. The investigator poses a number of problems on the phenomenon of mirror symmetry. This includes a study of the hypergeometric differential equations that are satisfied by the periods of families of Calabi-Yau manifolds. The mirror symmetry conjecture, first observed by physicists, asserts that each Calabi-Yau 3-fold has a 'mirror' (also a Calabi-Yau 3-fold) and that certain structures of a Calabi-Yau manifold are tightly related to seemingly very different structures of its mirror manifold. The phenomenon of mirror symmetry relates qualitatively different structures arising in theoretical physics, geometry and algebra. The rough idea is that there are structures in theoretical physics which are mirrored in structures in pure mathematics. This entire phenomenon, verified in special cases, remains an intriguing mathematical puzzle.

本文提出的研究方向是复杂微分几何与代数几何的交叉。研究者对镜像对称现象提出了许多问题。这包括对Calabi-Yau流形族周期所满足的超几何微分方程的研究。首先由物理学家观察到的镜像对称猜想断言，每个Calabi-Yau 3-fold都有一个“镜子”（也是Calabi-Yau 3-fold），并且Calabi-Yau流形的某些结构与其镜像流形的看似非常不同的结构紧密相关。镜像对称现象涉及理论物理、几何和代数中出现的性质不同的结构。大概的想法是理论物理中的结构是纯数学结构的镜像。这整个现象，在特殊情况下得到证实，仍然是一个有趣的数学难题。