Symmetries and Equivariance in Floer Homology and Homotopy

Topology is the study of abstract shapes and configurations. Of particular interest are knots, surfaces, and higher-dimensional spaces (called manifolds). Understanding how to describe and visualize these spaces has broadly impacted fields such as data science (by introducing tools for studying high-dimensional spaces of data), biology (by understanding knotted objects such as DNA), and physics. The investigator will use and develop tools drawn from topological quantum field theories to study knots and surfaces. Inspired by theoretical physics, these topological quantum field theories have led to a new area of low-dimensional topology (called Floer theory) which provides powerful new methods for understanding the properties of three- and four-dimensional spaces, as well as knots and surfaces inside of them. The investigator will use these advances to study fundamental problems about the shape of three- and four-dimensional spaces, the nature of knottedness, and relations between quantum field theories and other areas of mathematics. More specifically, the investigator will employ symmetries of knots and manifolds in conjunction with Floer homology and homotopy in order to develop new sliceness obstructions, find novel techniques for constructing and detecting exotic phenomena, and answer questions about equivariant bordism. The investigator will undertake fundamental research aimed at computing and understanding the output of different Floer theories, such as Seiberg-Witten Floer homotopy. The project will also seek to explore exciting new connections between Floer homotopy and singularity theory and will continue long-running research aimed at understanding the structure of various cobordism and concordance groups. 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.