Knots, Sheaves, and Mirrors

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This project aims to connect several areas of mathematics that relate to modern mathematical physics. The theory of knots -- ordinary, tangled loops of string in three-dimensional space -- goes back to Lord Kelvin, who proposed that the periodic table was made from different knotted forms of the ether. Since that time, knot theory became a well-established subject in mathematics, even as the physics of Kelvin's proposal came into disfavor. The main challenge of knot theory in mathematics is to recognize when two knots are the same or different. More recently, knots have reappeared in the mathematical physics relating to supersymmetric gauge theory and string theory. Through their reappearance in mathematical physics, knots have re-emerged in mathematics in sophisticated form. This project will elucidate the interplay among these fields within math and physics, leading to a more comprehensive understanding of the subject. In addition, the principal investigator will train graduate students, support a research seminar, run a math circle, and perform other activities aimed at developing young mathematicians."Knots, Sheaves, and Mirrors" refers first to a connection between Legendrian knots and constructible sheaves, and second to a study of the moduli spaces involved under the identification. A Legendrian knot can arise as a boundary-at-infinity of a Lagrangian submanifold, so given the micro localization theorem (proved by the principal investigator and Nadler) relating sheaves and the Fukaya category, it is perhaps not surprising that Legendrians have a purely sheaf-theoretic description. In fact, the exact relationship is somewhat subtle, but in the end one can equate augmentations of the Chekanov-Eliashberg differential graded algebra with constructible sheaves on the base plane containing the front diagram. With this equivalence in hand, it is then natural, following the ideas of mirror symmetry, to study the moduli spaces of "rank-one" objects --- and these prove to be interesting spaces, related to Bott-Samelson resolutions, from which one can recover categorized HOMFLY invariants of the knot. The project will also study the sheaf-theoretic description of Legendrian conormals of topological knots (along the lines of Aganagic-Ekholm-Ng-Vafa), wild Hitchin systems corresponding to algebraic singularities whose link is the knot.