Affine Structures, Non-archimedean Analytic Geometryand Mirror Symmetry

仿射结构、非阿基米德解析几何和镜像对称

• 批准号: 0504048
• 负责人: Yan Soibelman
• 金额: $ 11.3万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504048&HistoricalAwards=false
• 关键词: Affine; Structures; Non; archimedean; Analytic

AbstractAward: DMS-0504048Principal Investigator: Yan SoibelmanThe project is devoted to the study of the relationship betweenmanifolds with integral affine structure and Calabi-Yau manifolds overnon-archimedean fields. It offers a new approach to HomologicalMIrror Symmetry based on the ideas of collapsing Calabi-Yau manifoldsdeveloped jointly by P.I. and Maxim Kontsevich. Part of the researchis devoted to an unexpected analogy between classical integrablesystems and Calabi-Yau manifolds over non-archimedean fields. Thisanalogy leads to different approaches to integral affine structures:one is via a kind of Liouville integrability and another one via thenotion of skeleton introduced by P.I. and Kontsevich. The problem ofreconstructing a non-archimedean space from the skeleton with(singular) integral affine structure is raised. The solution is givenfor K3 surfaces. Non-commutative version of the above analogy is alsoa part of the project.One of the most challenging problems of modern physics is theunification of all known forces under the roof of onetheory. Most promising candidate for such unification is StringTheory. It is far from being finished. Proposed research isdevoted to mathematical foundations of Mirror Symmetry, which isone of the better understood parts of String Theory. It is alsodevoted to unexpected mathematical analogies motivated by MirrorSymmetry. Hopefully this researh will give new theoretical toolsfor study microstructure of our world via better understanding ofthe underlying local geometry.

项目负责人：Yan soibelman本项目主要研究具有整体仿射结构的流形与非阿基米德场的Calabi-Yau流形之间的关系。它基于P.I.和Maxim Kontsevich共同提出的Calabi-Yau流形的坍缩思想，提供了一种新的同调镜像对称方法。部分研究致力于经典可积系统与非阿基米德场上的Calabi-Yau流形之间的意想不到的类比。这种类比导致了研究整体仿射结构的不同方法：一种是通过一种Liouville可积性，另一种是通过P.I.和Kontsevich引入的骨架概念。提出了由具有（奇异）积分仿射结构的骨架重构非阿基米德空间的问题。给出了K3表面的解。上述类比的非交换版本也是该项目的一部分。现代物理学中最具挑战性的问题之一是把所有已知的力统一在一个理论的屋檐下。这种统一最有希望的候选者是StringTheory。它远未完成。拟议的研究致力于镜像对称的数学基础，这是弦理论中较容易理解的部分之一。它还致力于由MirrorSymmetry激发的意想不到的数学类比。希望这项研究能够通过更好地理解潜在的局部几何，为研究我们世界的微观结构提供新的理论工具。