Holomorphic Invariants of Knots and Contact Manifolds

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Symplectic geometry is an area of mathematics that dates back to the 19th century, with its roots in physics and Newtonian mechanics. In the past few decades, symplectic geometry has emerged as an exciting and fundamental area of mathematical research, due in part to close connections with many other parts of mathematics as well as physics. It has had especially striking recent applications to topology, the study of geometric shapes and spaces, and particularly the theory of knots, loops of string that are tied together at their ends. The Principal Investigator will apply ideas from symplectic geometry to construct and study invariants of knots as well as geometric spaces in three and higher dimensions. Preliminary past work suggests that these invariants provide a surprising and unexpected bridge between several modern areas of mathematics (symplectic geometry, algebraic geometry, and quantum knot theory) and physics (string theory, a model for the fundamental forces that shape the universe). This project will explore this bridge, with the goal of creating and strengthening new lines of two-way communication between mathematics and theoretical physics. As part of this project, the Principal Investigator will also promote the training of early-career researchers in mathematics, especially through research experiences in mathematics for undergraduate students. In addition the project will provide research training opportunities for graduate students.The project supported by this award will focus on several related lines of research, all centered around holomorphic curves, which have become central to the modern study of symplectic geometry since work of Gromov in the 1980s. One direction builds on the construction of Fukaya categories, algebraic structures associated to symplectic manifolds that play a key role in Homological Mirror Symmetry. The present project will construct a version of the Fukaya category for contact manifolds, built out of knots in a contact manifold. A key motivation for studying this category for contact manifolds is that it can serve as a central repository for holomorphic-curve invariants of the contact manifold. In particular, it will be a key intermediary in a larger picture that brings together other categories, both geometric (infinitesimal Fukaya categories) and algebraic (microlocal sheaf categories). Another aspect of the project deals with a package of knot invariants called knot contact homology, which has been studied by the Principal Investigator in previous work. This project will investigate the connection between knot contact homology and certain other knot invariants such as HOMFLY-PT polynomials, using recent progress in topological string theory. Goals in this direction include developing a formula for colored HOMFLY-PT polynomials in terms of holomorphic curves and quantizing the augmentation variety, a knot invariant devised from knot contact homology, to produce a recurrence relation for these polynomials.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.