Fukaya Categories and Applications

摘要：dms -0405516项目负责人：Paul seidel项目的主要目的是利用fukaycategories在辛拓扑中的应用。这包括开发计算工具，建立一个重要的示例库，以及研究fukaya类别和Gromov-Witten不变量之间的关系。具体的例子包括：riemann曲面的Fukaya分类；共切束中的拉格朗日子流形以及Khovanov联系不变量的辛几何。镜像对称的各个方面将是直觉的重要来源，以这种方式获得的经验将最终反馈到更好地理解镜像对称本身，特别是其同调方面。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。数学所能做的贡献是验证物理直觉的正确性和一致性，并为进一步的发展奠定基础。在物理学的发展表明存在难以通过直接实验检测到的深层复杂结构的情况下，这一点尤为重要。一个好的方面是，同样的思想循环允许人们回答一些受经典力学的“旧”物理学启发的纯数学（几何）问题。

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.