Fukaya Categories and Applications

深谷类别和应用

• 批准号: 0405516
• 负责人: Paul Seidel
• 金额: $ 12.7万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405516&HistoricalAwards=false
• 关键词: Fukaya; Categories; Applications

AbstractAward: DMS-0405516Principal Investigator: Paul SeidelThe main aim of the project is to harness the power of Fukayacategories for use in symplectic topology. This includesdeveloping computational tools, building a repertoire ofnontrivial examples, and also studying the relationship betweenFukaya categories and Gromov-Witten invariants. Concreteexamples that will be looked at include: Fukaya categories ofRiemann surfaces; Lagrangian submanifolds in cotangent bundles;and the symplectic geometry of Khovanov's linkinvariants. Various aspects of mirror symmetry will be importantsources of intuition, and the experience gained in this wayshould ultimately feed back into a better understanding of mirrorsymmetry itself, especially its homological aspects.From a wider perspective, the project is part of the reaction ofthe scientific community to recent advances in theoreticalphysics. The contribution that mathematics can make is to verifythe soundness and consistency of physical intuition, and toprepare the general ground on which further development canoccur. This is particularly important in those situations wheredevelopments in physics suggest the presence of deep and complexstructures, which are difficult to detect by direct experiment. Anice aspect is that the same circle of ideas allows one to answersome purely mathematical (geometric) questions inspired by the"old" physics of classical mechanics.

AbstractAward：DMS-0405516首席研究员：Paul Seidel该项目的主要目的是利用辛拓扑中使用的拓扑类别的力量。这包括开发计算工具，建立一个非平凡的例子的剧目，并研究之间的关系，类和Gromov-Witten不变量。 具体的例子，将被视为包括：福谷类别的黎曼曲面;拉格朗日子流形的余切丛;和辛几何的Khovanov的linkinvariants。镜像对称的各个方面都将是直觉的重要来源，而通过这种方式获得的经验最终将反馈到对镜像对称本身的更好理解中，特别是它的同调方面。从更广泛的角度来看，该项目是科学界对理论物理学最新进展的反应的一部分。数学所能做出的贡献是验证物理直觉的合理性和一致性，并为进一步发展奠定总体基础。这在物理学的发展表明存在很深和复杂的结构的情况下特别重要，这些结构很难通过直接实验检测到。一个有趣的方面是，同样的思想循环允许人们回答一些纯粹的数学（几何）问题，这些问题受到经典力学的“旧”物理学的启发。