The Algebra of 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.