Enhanced invariants of Legendrian links

This project is centered in an area of mathematics called symplectic geometry, which has a long history dating back to the advent of Newtonian mechanics in physics. The central objects of study in this project are Legendrian links, which may be visualized as certain loops of string in three-dimensional space tied together at their ends. Legendrian links play a key role in the modern study of symplectic geometry in three and four dimensions.

The project seeks to develop and explore new algebraic structures associated to Legendrian links, by combining an established framework, developed over the past few decades, with recent ideas from other areas of mathematics such as combinatorics and algebraic geometry. This research will create new algebraic structures that can be applied to answer some long-unsolved questions in symplectic geometry.

The Principal Investigator will also train future mathematicians through his work organizing undergraduate summer research programs at Duke University and other extracurricular activities for students.

The project is built around an algebraic invariant of Legendrian links called Legendrian contact homology, which was originally developed roughly three decades ago and has emerged as one of the premier tools in modern symplectic topology. The Principal Investigator will develop a number of interrelated enhancements of Legendrian contact homology. Some are combinatorial in nature, including a quantization of Legendrian contact homology that has a conjectural relationship to mirror symmetry; others are more geometric, including a topological stable homotopy type that lifts Legendrian contact homology into the realm of Floer homotopy theory.

These enhancements, which can be viewed as strengthened versions of existing invariants, are motivated by recently-discovered connections between symplectic geometry and other mathematical areas such as cluster theory in combinatorics. This project will apply the enhanced invariants to study open questions in symplectic topology, notably the question of classifying certain surfaces in four dimensions, called Lagrangian fillings, with prescribed boundary conditions.

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.