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

一圈金属丝在肥皂水中浸泡和取出后，会留下一层薄薄的肥皂面。如果将金属丝打结，结果往往是美丽的，但要事先预测表面的形状是非常困难的。这些表面，以及允许以约束（“拉格朗日”）方式延伸到第四维的相关表面，也有一个美丽的数学理论。它们已经在不同的数学和物理学科中研究了很长一段时间。这个项目是关于计算拉格朗日曲面，结束于一个给定的结：调查员以前的工作导致了一个新的预测，有关他们的枚举“集群代数。解释这个预言需要弦理论、量子场论、组合学和代数几何学的工具。这个项目开发了一种方法来理解拉格朗日曲面，结束于一个固定的勒让德结，通过给出一个明确的模型，他们的模量。PI将显示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，簇由结的精确拉格朗日填充索引。当纽结是代数的时候，PI将通过非交换霍奇理论证明这些填充与黎曼曲面上的不规则连接理论有关。这种观点将导致新的结果在勒让德结理论，例如一个完整的枚举确切拉格朗日填充辫子积极勒让德结，并与代数几何的新的连接。

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.