Physical interpretation of Khovanov-Rozansky homology led to many new predictions and connections between various areas, which include knot contact homology, gauge theory, and algebras of interfaces, just to name a few. One of such connections is an intriguing relation between knot Floer homology and Khovanov homology. While the former is defined via symplectic geometry and the latter diagrammatically, these two homology theories exhibit many similarities that can be explained with the help of spectral sequences and embedding the two theories in a larger framework suggested by physics. It turns out that there is a 3-manifold version of this relation. To explain it, we will first need concrete and explicit 3-manifold variants of knot Floer homology and Khovanov homology. This talk is based on recent work with Marino, Putrov, and Vafa:
https://arxiv.org/pdf/1602.05302.pdf
https://arxiv.org/pdf/1605.07615.pdf
and work in progress.
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