Third Floer: Refined invariants in low-dimensional topology

The flight path of an airplane, the shape of a DNA strand, and the arc of a bridge are all described by curves in 3-space. In each case, it is natural, and often important, to understand how those curves can move or vary. Over time, curves that move, and perhaps coalesce, sweep surfaces. These surfaces lie not in the familiar 3-dimensional space, but in 4-space, with time as the fourth coordinate. This project develops several tools, all centered around understanding curves in 3-space and surfaces in 4-space. One project studies techniques called Floer homology, that use the global behavior of partial differential equations to study curves. Those tools encode a lot of information, but are hard to compute; the project studies properties that let one extract information from them in a practical way. Another project uses techniques from high-dimensional topology to study a particular class of curves that arise from wave fronts, called Legendrian knots. A third project focuses on the information contained in an algebraic invariant of surfaces, Khovanov homology. Broader impacts of this project include graduate training, seminars, and writing a textbook that would be a resource for students and early researchers in Khovanov homology. Heegaard Floer homology, Khovanov homology, and Floer homotopy theory continue to transform the understanding of low-dimensional topology. Their impact has faced certain limits, however. While the Khovanov invariants for surfaces distinguish some surfaces not visible to knot Floer homology, there are few tools for computing these Khovanov invariants, other than brute force and good fortune. There is also a lack of tools for computing the most refined Heegaard Floer invariants (e.g., the concordance invariant Upsilon) in natural settings, like for satellite knots. There is a lack of explicit examples in Floer homotopy theory, particularly of higher structures like the spectral Fukaya category, making it harder to understand, in detail, what structures should be sought and what information they likely contain. This project seeks to address these difficulties, by developing formal properties of the Khovanov invariants of surfaces, an extension of the "minus" variant of Heegaard Floer homology to 3-manifolds with boundary, and a stable homotopy refinement of Legendrian contact 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.