Holomorphic Invariants in Symplectic Topology

辛拓扑中的全纯不变量

基本信息

批准号：
1707652
负责人：
Lenhard Ng
金额：
$ 35.57万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-15 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1707652&HistoricalAwards=false
关键词：
Holomorphic Invariants Symplectic Topology

项目摘要

Symplectic geometry is an area of mathematics that dates back to the 19th century and the modern formulation in physics of Newtonian mechanics. In the past few decades, it has become an exciting and fundamental area of mathematical research, due in part to close connections with many other parts of mathematics as well as physics. Symplectic geometry has had especially striking recent applications to low-dimensional topology, the mathematical study of three- and four-dimensional spaces. The Principal Investigator will pursue one particularly promising technique along these lines, in the setting of the theory of knots, or loops of string that are tied together at their ends. This technique has proven to be of interest to the physics community, providing tantalizing clues of an as-yet-undiscovered framework that combines portions of mathematics (in particular, symplectic geometry) and theoretical physics (in particular, string theory, which provides a model for the fundamental forces that shape the universe). The present project will work to uncover this framework, facilitating the exchange of ideas between mathematics and physics. As part of this project, the Principal Investigator will also promote the training of future mathematicians, running research programs and mathematical competitions for both undergraduate students and local high school students.The unifying approach to symplectic geometry in this project is provided by holomorphic curves. Since pioneering work by Gromov in the 1980s, holomorphic curves have become a central tool in symplectic geometry, combining powerful analytical and geometric techniques with a computable combinatorial flavor. The research supported by this award applies holomorphic curves to the setting of knots. Previous work by the Principal Investigator and collaborators led to the development of knot contact homology, a powerful knot invariant in the spirit of Symplectic Field Theory, which has evolved into a subject that has many unexpected and intriguing connections to various areas of mathematics and physics. Recent results have opened the door to a detailed exploration of these connections, which will be carried out in this project. Within symplectic geometry, knot contact homology motivates a close study of a new type of Floer theory (partially wrapped Floer homology); in knot theory, it is conjecturally related to topological concepts like Seifert genus and concordance; in topological string theory, it is conjectured to be determined by a certain Calabi-Yau manifold that has been the object of much study in recent years. Besides tackling these conjectures, the Principal Investigator will pursue a related project, following on a recently discovered connection between constructible sheaves (from algebraic geometry) on one side, and holomorphic curves on the other. This project will develop this connection, in particular working to define an analogue of the Fukaya category for contact manifolds and applying this to facilitate computations in Fukaya categories and mirror symmetry.

辛几何是一个可以追溯到 19 世纪的数学领域，也是牛顿力学的现代物理学表述。在过去的几十年里，它已成为数学研究的一个令人兴奋的基础领域，部分原因是与数学和物理学的许多其他部分的密切联系。辛几何最近在低维拓扑（三维和四维空间的数学研究）中有着特别引人注目的应用。首席研究员将沿着这些思路，在结或末端系在一起的绳圈理论的设置中，追求一种特别有前途的技术。事实证明，这项技术引起了物理学界的兴趣，它为尚未发现的框架提供了诱人的线索，该框架结合了部分数学（特别是辛几何）和理论物理（特别是弦理论，它提供了一个模型）塑造宇宙的基本力）。当前的项目将致力于揭示这个框架，促进数学和物理学之间的思想交流。作为该项目的一部分，首席研究员还将促进未来数学家的培训，为本科生和当地高中生开展研究项目和数学竞赛。该项目中辛几何的统一方法是由全纯曲线提供的。自 Gromov 在 20 世纪 80 年代的开创性工作以来，全纯曲线已成为辛几何的核心工具，它将强大的分析和几何技术与可计算的组合风格相结合。该奖项支持的研究将全纯曲线应用于结的设置。首席研究员和合作者之前的工作导致了结接触同调的发展，这是辛场论精神中的一个强大的结不变量，它已经发展成为一个与数学和物理的各个领域有着许多意想不到和有趣的联系的学科。最近的结果为详细探索这些联系打开了大门，这将在这个项目中进行。在辛几何中，结接触同调激发了对新型 Floer 理论（部分包裹 Floer 同调）的深入研究；在纽结理论中，它推测与 Seifert 属和一致性等拓扑概念有关；在拓扑弦理论中，它被推测是由某个卡拉比-丘流形决定的，该流形近年来一直是许多研究的对象。除了解决这些猜想之外，首席研究员还将开展一个相关项目，追踪最近发现的一侧可构造滑轮（来自代数几何）与另一侧全纯曲线之间的联系。该项目将开发这种联系，特别是定义接触流形的 Fukaya 类别的类似物，并将其应用于促进 Fukaya 类别和镜像对称的计算。