The Algebra of Flow Categories

如果你在一张纸上画，而不回头看你的笔画，你会发现你的图形有偶数个端点：一个端点代表你每次把笔拿到纸上，一个端点代表你每次举起笔。这个基本事实给出了一维图形的拓扑和代数之间的关系。数学家们在庞特里亚金和托姆世纪中期的原始工作的基础上建造了一台精心制作的机器，将这种对应关系扩展到更高的维度。这个项目的目的是制定概念的代数结构的几何方面的这种对应关系，与预期的应用研究的拓扑结构。该项目还将支持研究生在该主题的培训。该项目的新配方是围绕流范畴的概念，这已成为核心的现代方法辛拓扑，以及它与其他数学领域的相互作用，如代数几何，低维拓扑，动力系统，以及数学物理，因为弗洛尔理论的输出恰好提供了这样一种结构。PI计划制定代数结构（代数，模块等）。在基本流类别的级别上几何地。通过这种方式，PI预计将在三个领域取得实质性进展：（i）Floer同伦理论及其在辛拓扑中的应用，（ii）福谷范畴的研究及其在镜像对称中的应用，（iii）微分拓扑和代数拓扑之间的相互作用，通过Waldhausen A理论的几何模型，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.