Floer Homology and Immersed Curve Invariants in Low Dimensional Topology

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This research project in low-dimensional topology develops tools for classifying three-dimensional spaces as well as surfaces and knotted curves within them. Applications of these topological objects range from exploring the possible shapes of the three-dimensional spatial universe we live in to understanding the knotting of polymers and DNA. These three-dimensional objects are studied by analyzing certain invariant properties which capture essential characteristics of the objects. A successful family of such invariants, which has lead to numerous applications since it was first developed two decades ago, is called Heegaard Floer homology. The definition of Heegaard Floer homology draws on sophisticated techniques from multiple fields, including topology, geometry, and analysis. This project aims to reformulate some of these invariants in a way that both sheds light on their underlying structure and also provides computational tools and leads to new applications. In particular, the PI aims to translate bordered Heegaard Floer invariants for 3-manifolds with boundary, which take the form of complicated algebraic objects, into geometric objects built from collections of curves in a surface.This project focuses on the structure of Heegaard Floer homology with the goal to extend and better understand the Topological Quantum Field Theory (TQFT)-like structure in Heegaard Floer theory and to apply it to problems in low-dimensional topology. First, the PI aims to extend the earlier results to the stronger “minus” version of Heegaard Floer homology, which carries more information and also has interesting connections to four-dimensional invariants. The PI will also consider the case of manifolds with higher genus boundary or with multiple boundary components. In a related direction, the PI will explore the behavior of invariants for knots under the satellite operation and concordance. A long-term goal is to study the Fukaya categories of symmetric products of surfaces and morphisms between them, finding practical geometric descriptions of these objects which fit into a description of Heegaard Floer homology as a (2+1+1)-dimensional Topological Quantum Field Theory.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.