The Algebra of Flow Categories

如果你在一张纸上画画，而不回顾你的笔画，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放在纸上，另一个端点代表你每次拿起笔。这个基本事实给出了一维图形的拓扑与代数之间的关系。数学家们根据20世纪中期庞特里亚金和托姆的原始工作，制造了一台精密的机器，将这种对应关系扩展到更高的维度。这个项目的目的是在这种对应的几何方面形成代数结构的概念，在拓扑学的研究中有预期的应用。该项目还将支持该学科研究生的培训。该项目的新公式围绕着流动范畴的概念展开，这一概念已经成为辛拓扑的现代方法的核心，以及它与其他数学领域的相互作用，如代数几何、低维拓扑和动力系统，以及数学物理，因为Floer理论的输出正好提供了这样的结构。PI计划在底层流类别的层次上以几何方式形成代数结构（代数、模块等）。通过这种方式，PI期望在三个领域取得实质性进展：(i)花同伦理论及其在辛拓扑中的应用，（ii）深谷范畴的研究及其在镜像对称中的应用，以及（iii）微分和代数拓扑之间的相互作用，通过瓦尔德豪森a理论的几何模型，有意应用于拉格朗日嵌入的研究。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.