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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.

辛几何是数学的一个领域，可以追溯到世纪，其根源是物理学和牛顿力学。在过去的几十年里，辛几何已经成为数学研究中一个令人兴奋的基本领域，部分原因是与数学和物理学的许多其他部分密切相关。它有特别引人注目的最近应用拓扑学，研究几何形状和空间，特别是理论的结，循环的字符串是绑在一起，在其两端。主要研究者将运用辛几何的思想来构建和研究结的不变量以及三维和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，代数几何和量子纽结理论）和物理学（弦理论，塑造宇宙的基本力的模型）之间提供了一个令人惊讶和意想不到的桥梁。该项目将探索这座桥梁，目标是建立和加强数学和理论物理之间的双向交流。作为该项目的一部分，首席研究员还将促进数学早期职业研究人员的培训，特别是通过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集中在几个相关的研究领域，所有研究都以全纯曲线为中心，自Gromov在20世纪80年代的工作以来，全纯曲线已成为辛几何现代研究的核心。一个方向建立在福谷范畴的建设上，这些范畴是与辛流形相关的代数结构，在同调镜像对称中起着关键作用。目前的项目将建立一个版本的福谷类别的接触流形，建立了一个接触流形的纽结。研究这类接触流形的一个关键动机是，它可以作为接触流形的全纯曲线不变量的中心存储库。特别是，它将是一个关键的中介，在一个更大的图片，汇集了其他类别，几何（无穷小福谷类别）和代数（微局部层类别）。该项目的另一个方面涉及一个包的结不变量称为结接触同源性，这已经研究了由首席研究员在以前的工作。本计画将利用拓扑弦理论的最新进展，研究纽结接触同调与其他纽结不变量（如HOMFLY-PT多项式）之间的关系。这一方向的目标包括开发一个公式的有色HOMFLY-PT多项式的全纯曲线和量化的增广品种，一个结不变设计从结接触同源，以产生一个递归关系，这些多项式。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。