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.

象征性几何形状是数学领域，其历史可以追溯到19世纪，也可以追溯到牛顿力学物理学中的现代表述。在过去的几十年中，它已成为数学研究的一个令人兴奋和基本的领域，部分原因是与数学和物理学的许多其他部分紧密联系。符合性几何形状尤其引起了对低维拓扑的最新应用，即三维空间和四维空间的数学研究。首席研究人员将在这些方面，打结理论或在其末端绑在一起的弦乐循环来追求一种特别有前途的技术。事实证明，这项技术对物理界引起了人们的关注，提供了一个尚未发现的框架的诱人线索，该框架结合了一部分数学（尤其是符号几何形状）和理论物理学（尤其是弦乐理论，为塑造宇宙的基本力量提供了模型）。本项目将有助于发现该框架，从而促进数学和物理学之间的思想交换。作为该项目的一部分，首席研究员还将促进对本科生和当地高中生的未来数学家的培训，进行研究计划和数学竞赛。该项目中的统一方法统一方法由Holomorphic Curves提供。自1980年代格罗莫夫（Gromov）开创性的工作以来，全态曲线已成为符号几何形状的中心工具，将强大的分析和几何技术与可计算的组合风味相结合。该奖项支持的研究将全体形态曲线应用于结的设置。首席研究员和合作者的先前工作导致了结的发展，这是在象征性领域理论精神上不变的一个强大结，该理论不断发展，后者已经演变成一个与数学和物理学各个领域有许多意外且有趣的联系的主题。最近的结果为对这些连接的详细探索打开了大门，该连接将在该项目中进行。在象征性的几何形状中，结触点同源性激励了对新型的浮子理论（部分包裹的浮子同源性）的仔细研究；在结理论中，它与Seifert属和一致性等拓扑概念有关。在拓扑弦理论中，它的猜想是由近年来研究的某些calabi-yau歧管决定的。除了解决这些猜想之外，首席研究人员还将追求一个相关的项目，此前在最近发现的一方面可构造的滑轮（从代数几何形状）与另一侧的全体形态曲线之间的联系之后。该项目将建立这种联系，特别是致力于定义福卡亚类别的类似物，以进行触点歧管，并将其应用以促进福卡亚类别中的计算和镜像对称性。