Knots and contact topology through holomorphic curves

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The subjects of mathematics and physics have always been closely intertwined, with each one motivating and informing progress in the other. This project focuses on one current area of interplay, between topology (the study of shapes) on the mathematical side and string theory on the physics side. The jumping-off point for the project is an intriguing and unexpected connection, recently discovered by the Principal Investigator and collaborators on both sides, between two separate algebraic structures associated to knots in space: one in topology developed by the Principal Investigator, and one in string theory that has been the focus of much research in the past few years. In the course of this project, the Principal Investigator will establish this connection, which is currently only supported circumstantially; it is hoped that this work will create and strengthen new lines of communication between mathematics and physics, introducing techniques from each discipline into the other. The Principal Investigator will also use this project to train future mathematicians at all levels, from contributing to the annual USA Mathematical Olympiad for high school students, to supervising the research of undergraduates, graduate students, and postdoctoral fellows, to organizing conferences and seminars for established researchers.In the past decade, the Principal Investigator has introduced and studied a package of knot invariants called knot contact homology, which arises by counting holomorphic curves in certain symplectic manifolds, using a method in symplectic geometry pioneered by Gromov and Floer and more recently culminating in the Symplectic Field Theory of Eliashberg, Givental, and Hofer. Previous work has shown that knot contact homology is a robust invariant that is effective at distinguishing knots and contains classical topological information about the knot. In 2012, it was discovered by Aganagic, Ekholm, Vafa, and the Principal Investigator that knot contact homology has an unexpected and potentially powerful relation to string theory and mirror symmetry: the augmentation polynomial, a knot invariant derived from knot contact homology, is conjectured to be equal to Aganagic and Vafa's Q-deformed A-polynomial, which arises in the context of topological strings. The Principal Investigator will approach this conjecture using Lagrangian fillings and Gromov-Witten potentials. This could have significant ramifications in different directions: in knot theory, it would establish a variant of the AJ conjecture; in mirror symmetry, it would produce a new approach via Symplectic Field Theory to constructing mirror Calabi-Yau 3-folds; and in topological string theory, it would provide a mathematical foundation for recent results. In related work, the Principal Investigator will develop and strengthen the algebraic framework underneath certain aspects of Symplectic Field Theory, including knot contact homology and symplectic homology. New algebraic tools in this context, such as representation theory for differential graded algebras, would enable one to more effectively attack problems in symplectic geometry, in particular by analyzing Weinstein structures on symplectic manifolds and Legendrian and transverse knots in contact manifolds.