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

一圈金属丝浸在肥皂水中，然后从肥皂水中取出，在金属丝上留下一层薄薄的肥皂表面。如果电线打结，结果往往是美丽的，但很难提前预测表面的形状。这些曲面，以及允许以约束(“拉格朗日”)方式延伸到第四维的相关曲面，也有一个美丽的数学理论。长期以来，人们在不同的数学和物理学科中对它们进行了研究。这个项目关注于计算结束于给定结点的拉格朗日曲面：研究人员之前的工作导致了一个新的预测，将它们的计数与“簇代数”联系起来。解释这一预测需要弦论、量子场论、组合学和代数几何的工具。这个项目开发了一种理解拉格朗日曲面的方法，通过给出它们的模数的显式模型，结束于固定的勒让德结上。PI将表明该模型具有丰富的几何和组合结构：它是仿射复辛流形和簇簇，簇由纽结的精确拉格朗日填充来索引。当纽结是代数的时，PI将表明这些填充通过非阿贝尔霍奇理论与黎曼曲面上的不规则连接理论相关联。这一观点将导致Legendrian纽结理论中的新结果，例如，对辫子正Legendrian纽结的精确Lagrangian填充的完全计数，以及与代数几何的新联系。

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.