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.

辛几何是一个可以追溯到 19 世纪的数学领域，其根源在于物理学和牛顿力学。在过去的几十年里，辛几何已成为数学研究的一个令人兴奋的基础领域，部分原因是它与数学和物理学的许多其他部分的密切联系。它最近在拓扑学、几何形状和空间的研究，尤其是结理论、末端系在一起的绳环理论中有着特别引人注目的应用。首席研究员将应用辛几何的思想来构建和研究结的不变量以及三维及更高维度的几何空间。过去的初步工作表明，这些不变量在数学的几个现代领域（辛几何、代数几何和量子结理论）和物理学（弦理论，塑造宇宙的基本力的模型）之间提供了令人惊讶和意想不到的桥梁。该项目将探索这座桥梁，目标是创建和加强数学和理论物理之间双向交流的新路线。作为该项目的一部分，首席研究员还将促进对早期职业研究人员的数学培训，特别是通过本科生的数学研究经验。此外，该项目将为研究生提供研究培训机会。该奖项支持的项目将集中于几个相关的研究领域，所有这些研究领域都以全纯曲线为中心，自格罗莫夫的工作以来，全纯曲线已成为现代辛几何研究的核心。 20世纪80年代。其中一个方向建立在 Fukaya 范畴的构造之上，这是与辛流形相关的代数结构，在同调镜像对称中发挥着关键作用。目前的项目将构建一个接触流形 Fukaya 类别的版本，由接触流形中的结构建而成。研究接触流形这一类别的一个关键动机是它可以作为接触流形的全纯曲线不变量的中央存储库。特别是，它将成为更大图像中的关键中介，将其他类别汇集在一起，包括几何（无穷小深谷类别）和代数（微局部层类别）。该项目的另一个方面涉及称为结接触同源性的一组结不变量，首席研究员在之前的工作中已对其进行了研究。该项目将利用拓扑弦理论的最新进展，研究结接触同源性与某些其他结不变量（例如 HOMFLY-PT 多项式）之间的联系。这个方向的目标包括根据全纯曲线开发彩色 HOMFLY-PT 多项式的公式，并量化增广多样性（一种根据结接触同源性设计的结不变量），以产生这些多项式的递推关系。该奖项反映了 NSF 的法定使命和通过使用基金会的智力优点和更广泛的影响审查标准进行评估，该项目被认为值得支持。