Bordism in Floer Theory

The first hint of an interaction between the mathematical fields of topology and analysis is the fact that every function on the circle must have both a minimum and a maximum, unlike, for example, functions on the line which need not have either. Over a century ago, Marston Morse discovered a vast generalization of this basic fact, which has been a cornerstone of subsequent development in topology. This project aims to develop a framework that makes it possible to take Morse's insight as a starting point for formulating a new framework in which to study algebraic structures. The broader impacts of this proposal include the training of future leaders of the field, as well as developing resources for understanding the relevance of topological notions beyond pure mathematics. The key technical notion in the proposal is that of a flow category. The PI plans to formulate algebraic structures geometrically at the level of the underlying flow categories in order to extend our ability to develop applications. In this way, the PI expects to make substantial advances in Floer homotopy theory and its applications both to symplectic topology, and to low-dimensional topology, via the study of Heegaard-Floer homology. 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.