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

将一圈电线浸入肥皂水中，然后取出，在电线上留下一层薄薄的肥皂表面。如果电线打结，结果往往很漂亮，但很难提前预测表面的形状。这些曲面，以及允许以受限（拉格朗日）方式扩展到第四维的相关曲面，也有一个美丽的数学理论。长期以来，人们在各种数学和物理学科中对它们进行了研究。这个项目关注的是结束于给定结的拉格朗日曲面的计数：研究者之前的工作已经导致了一个新的预测，将它们的计数与“簇代数”联系起来。解释预言需要弦理论、量子场论、组合学和代数几何的工具。本项目通过给出其模的显式模型，开发了一种理解以固定的Legendrian结结束的拉格朗日曲面的方法。PI将表明该模型具有丰富的几何和组合结构：它是一个仿射复辛流形和簇的变化，簇由结点的精确拉格朗日填充索引。当结是代数的时，PI将表明这些填充通过非阿贝尔霍奇理论与黎曼表面上的不规则连接理论相关。这一视角将导致在legendrin结理论中的新结果，例如辫状正legendrin结的精确拉格朗日填充的完整枚举，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.