Knot Homology and Moduli of Sheaves

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A knot is an embedding of a circle in three dimensional space. Knot invariants are quantities that are assigned to an embedded circle that do not change under continuous change of the embedding. Tools from quantum field theory yield conjectural constructions for new, algebraic, knot invariants known as knot homologies. In this project, the PI will give rigorous mathematical constructions of knot homologies, as well as efficient algorithms for computing these knot invariants. In particular, the PI will interpret knot homology in terms of the geometry of moduli spaces that are defined by polynomial equations and use methods in analytic geometry to study these moduli spaces, thus uncovering new properties of knot homology. In connection with this research, the PI will undertake research-training activities aimed at both undergraduate and doctoral students.More precisely, in this project the PI will study relations between moduli of sheaves and knot homology. In particular, the PI will study a conjecture that relates the HOMFLYPT homology of a link of a plane curve singularity with the homology of the Hilbert scheme of points on the curve. To compute the HOMFLYPT homology of a knot, the PI will to develop further a theory that interprets the HOMFLYPT homology of a link as a space of global sections a naturally defined coherent sheaf on the Hilbert scheme of points on the plane. The PI will also study Virasoro constraints for the descendant invariants of Pandharipande-Thomas moduli spaces for a general threefold, and the Gromov-Witten/Pandharipande-Thomas correspondence for invariants with descendants will be developed further, so that Virasoro constraints can be transported from Gromov-Witten 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.