The Algebra of Flow Categories

流范畴代数

• 批准号: 2103805
• 负责人: Mohammed Abouzaid
• 金额: $ 54.4万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2103805&HistoricalAwards=false
• 关键词: Algebra; Flow; Categories

Should you draw on a piece of paper, without running back over your strokes, you will find that your figure has an even number of endpoints: one for each time you brought pen to paper, and one for each time you lifted it. This basic fact gives a relationship between the topology of one dimensional figures and algebra. Mathematicians have built an elaborate machine based on the original work of Pontryagin and Thom from the middle of the 20th century, extending this correspondence to higher dimensions. This project aims to formulate notions of algebraic structures on the geometric side of this correspondence, with intended applications in the study of topology. The project will also support the training of graduate students in the subject.The project's new formulations are centered around the notion of a flow category which has become central to modern approaches to symplectic topology, as well as to its interactions with other mathematical fields, such as algebraic geometry, low dimensional topology, and dynamical systems, as well as mathematical physics, because the output of Floer's theory exactly provides such a structure. The PI plans to formulate algebraic structures (algebras, modules, etc.) geometrically at the level of the underlying flow categories. In this way, the PI expects to make substantial advances in three areas: (i) Floer homotopy theory and its applications to symplectic topology, (ii) the study of Fukaya categories and its applications to mirror symmetry, and (iii) the interaction between differential and algebraic topology, via a geometric model of Waldhausen's A-theory, with intended applications to the study of Lagrangians embeddings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

如果您在一张纸上画画，而不会跑到笔触上，您会发现您的身影有均匀数量的端点：每次将笔纸带到纸上时，每次提起它。这个基本事实给出了一维数字的拓扑与代数之间的关系。数学家根据20世纪中叶的Pontryagin和Thom的原始作品建造了一台精心设计的机器，将其扩展到更高的维度。该项目旨在在该对应关系的几何侧提出代数结构的概念，并在拓扑研究中采用了预期的应用。 The project will also support the training of graduate students in the subject.The project's new formulations are centered around the notion of a flow category which has become central to modern approaches to symplectic topology, as well as to its interactions with other mathematical fields, such as algebraic geometry, low dimensional topology, and dynamical systems, as well as mathematical physics, because the output of Floer's theory exactly provides such a structure. PI计划在基础流类别的水平上以几何形式制定代数结构（代数，模块等）。 In this way, the PI expects to make substantial advances in three areas: (i) Floer homotopy theory and its applications to symplectic topology, (ii) the study of Fukaya categories and its applications to mirror symmetry, and (iii) the interaction between differential and algebraic topology, via a geometric model of Waldhausen's A-theory, with intended applications to the study of Lagrangians该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估，被认为值得支持。