Lagrangian Surfaces, Legendrian Knots, and the Microlocal Theory of Sheaves

Boston College, Chestnut Hill MA

Investigators

Abstract

A loop of wire dipped in and taken out of soapy water leaves a thin surface of soap stretching across the wire. If the wire is knotted up, the result is often beautiful, but it is very difficult to predict the shape of the surface in advance. These surfaces, and related surfaces that are allowed to extend into a fourth dimension in a constrained ("Lagrangian") way, also have a beautiful mathematical theory. They have been studied in diverse mathematical and physical disciplines for a long time. This project is concerned with counting Lagrangian surfaces that end on a given knot: the investigator's previous work has lead to a new prediction relating their enumeration to "cluster algebras." Explaining the prediction requires tools from string theory, quantum field theory, combinatorics, and algebraic geometry. This project develops an approach to understanding Lagrangian surfaces that end on a fixed Legendrian knot, by giving an explicit model for their moduli. The PI will show that this model has a rich geometric and combinatorial structure: it is an affine complex symplectic manifold and a cluster variety, with the clusters indexed by exact Lagrangian fillings of the knot. When the knot is algebraic, the PI will show that these fillings are related via nonabelian Hodge theory to the theory of irregular connections on Riemann surfaces. This perspective will lead to new results in Legendrian knot theory, for instance a complete enumeration of exact Lagrangian fillings of braid-positive Legendrian knots, and to new connections with algebraic geometry.