Holomorphic Invariants of Knots and Contact Manifolds

结和接触流形的全纯不变量

基本信息

批准号：
2003404
负责人：
Lenhard Ng
金额：
$ 36万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003404&HistoricalAwards=false
关键词：
Holomorphic Invariants Knots Contact Manifolds

项目摘要

Symplectic geometry is an area of mathematics that dates back to the 19th century, with its roots in physics and Newtonian mechanics. In the past few decades, symplectic geometry has emerged as an exciting and fundamental area of mathematical research, due in part to close connections with many other parts of mathematics as well as physics. It has had especially striking recent applications to topology, the study of geometric shapes and spaces, and particularly the theory of knots, loops of string that are tied together at their ends. The Principal Investigator will apply ideas from symplectic geometry to construct and study invariants of knots as well as geometric spaces in three and higher dimensions. Preliminary past work suggests that these invariants provide a surprising and unexpected bridge between several modern areas of mathematics (symplectic geometry, algebraic geometry, and quantum knot theory) and physics (string theory, a model for the fundamental forces that shape the universe). This project will explore this bridge, with the goal of creating and strengthening new lines of two-way communication between mathematics and theoretical physics. As part of this project, the Principal Investigator will also promote the training of early-career researchers in mathematics, especially through research experiences in mathematics for undergraduate students. In addition the project will provide research training opportunities for graduate students.The project supported by this award will focus on several related lines of research, all centered around holomorphic curves, which have become central to the modern study of symplectic geometry since work of Gromov in the 1980s. One direction builds on the construction of Fukaya categories, algebraic structures associated to symplectic manifolds that play a key role in Homological Mirror Symmetry. The present project will construct a version of the Fukaya category for contact manifolds, built out of knots in a contact manifold. A key motivation for studying this category for contact manifolds is that it can serve as a central repository for holomorphic-curve invariants of the contact manifold. In particular, it will be a key intermediary in a larger picture that brings together other categories, both geometric (infinitesimal Fukaya categories) and algebraic (microlocal sheaf categories). Another aspect of the project deals with a package of knot invariants called knot contact homology, which has been studied by the Principal Investigator in previous work. This project will investigate the connection between knot contact homology and certain other knot invariants such as HOMFLY-PT polynomials, using recent progress in topological string theory. Goals in this direction include developing a formula for colored HOMFLY-PT polynomials in terms of holomorphic curves and quantizing the augmentation variety, a knot invariant devised from knot contact homology, to produce a recurrence relation for these polynomials.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.

象征性几何形状是数学领域，它的历史可以追溯到19世纪，其根源是物理学和牛顿力学。在过去的几十年中，象征性的几何形状已成为数学研究的令人兴奋的基本领域，部分原因是与数学的许多其他部分以及物理学之间的密切联系。它特别引起了拓扑，几何形状和空间的研究，尤其是打结理论，弦乐循环的末端。首席研究者将应用象征几何形状的思想来构建和研究结的不变性以及三个和更高维度的几何空间。过去的过去工作表明，这些不变的人在几个现代数学领域（符号几何，代数几何和量子结理论）和物理学（弦理论，一种模型的模型，是塑造宇宙塑造宇宙的基本力）之间的令人惊讶且出乎意料的桥梁。该项目将探索这座桥梁，以创建和加强数学和理论物理学之间的双向交流新线路。作为该项目的一部分，首席研究人员还将促进对数学早期研究人员的培训，尤其是通过针对本科生的数学研究经验。此外，该项目还将为研究生提供研究培训机会。该奖项支持的项目将重点介绍几条相关的研究渠道，所有这些都围绕着尸体形态曲线，这些曲线已成为自1980年代Gromov工作以来现代对象征性几何学的核心。一个方向建立在福卡亚类别的构建基础上，即与同型歧管相关的代数结构，这些结构在同源镜对称性中起着关键作用。本项目将构建福卡亚类别的版本，用于触点歧管，该版本是由触点歧管中的结构建的。研究此类别的接触歧管的一个关键动机是，它可以作为接触歧管的全体形态曲线不变的中心存储库。特别是，它将是更大范围的关键中介，将其他类别（无限福卡亚类别）和代数（微局部分层类别）汇总在一起。该项目的另一个方面涉及一包称为结的结的包装，称为结触点同源性，该问题已由首席研究员在先前的工作中进行了研究。该项目将使用拓扑字符串理论的最新进展研究打结接触同源性与某些其他结不变的连接。在这个方向上的目标包括在圆形曲线方面开发一个为多项式曲线的多项式制定公式，并量化增强品种，这是一个从结的结合式设计的结，为这些多项式提供了复发性关系，该奖项反映了NSF的法定任务，反映了通过评估的范围来审查构成的依据。