Holomorphic Invariants of Knots and Contact Manifolds

辛几何是一个可以追溯到19世纪的数学领域，它起源于物理学和牛顿力学。在过去的几十年里，辛几何已经成为数学研究的一个令人兴奋的基础领域，部分原因是由于与数学和物理学的许多其他部分的密切联系。最近，它在拓扑学、几何形状和空间的研究中有了特别引人注目的应用，特别是绳结理论，即末端绑在一起的绳环。首席研究员将应用辛几何的思想来构造和研究纽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(弦理论，塑造宇宙的基本力的模型)之间提供了一座令人惊讶和意想不到的桥梁。这个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双向交流的新渠道。作为该项目的一部分，首席调查员还将促进对初出茅庐的数学研究人员的培训，特别是通过为本科生提供数学研究经验。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在几个相关的研究领域，所有这些研究都围绕着全纯曲线，自20世纪80年代格罗莫夫的工作以来，这些研究已经成为辛几何的现代研究的中心。一个方向建立在Fukaya范畴的构造上，Fukaya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。本项目将构建接触流形的Fukaya类别的一个版本，该类别由接触流形中的结组成。研究接触流形的这一范畴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储存库。特别是，它将是汇集其他范畴的更大图景中的关键中介，既有几何范畴(无穷小的Fukaya范畴)，也有代数范畴(微局部层范畴)。该项目的另一个方面涉及称为纽结接触同调的纽结不变量包，首席研究员在之前的工作中已经研究过这一点。这个项目将利用拓扑弦理论的最新进展，研究纽结接触同调与某些其他纽结不变量之间的联系，如HOMFLY-PT多项式。在这个方向上的目标包括开发一个关于全纯曲线形式的有色HOMFLY-PT多项式的公式，并量化从纽结接触同调中设计的纽结不变量，以产生这些多项式的递归关系。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.