Holomorphic Invariants in Symplectic Topology

辛几何是一个可以追溯到19世纪的数学领域，也是牛顿力学在物理学中的现代表述。在过去的几十年里，它已经成为数学研究的一个令人兴奋的基础领域，部分原因是它与数学和物理学的许多其他部分有着密切的联系。辛几何最近在低维拓扑，即三维和四维空间的数学研究中有着特别引人注目的应用。首席调查员将沿着这些路线探索一种特别有希望的技术，即在末端绑在一起的绳结理论的背景下。这项技术已被证明是物理学界感兴趣的，它提供了一个尚未发现的框架的诱人线索，该框架结合了数学(特别是辛几何)和理论物理(特别是弦理论，它为塑造宇宙的基本力提供了一个模型)。本项目将致力于揭示这一框架，促进数学和物理之间的思想交流。作为这个项目的一部分，首席研究员还将促进培养未来的数学家，为本科生和本地高中生举办研究项目和数学竞赛。在这个项目中，辛几何的统一方法是由全纯曲线提供的。自20世纪80年代Gromov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巧与可计算的组合风格结合在一起。该奖项支持的研究将全纯曲线应用于结的设定。主要研究人员和合作者之前的工作导致了纽结接触同调的发展，这是辛场论精神中的一个强大的纽结不变量，它已经演变成一个与数学和物理的各个领域有许多意想不到的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，纽结接触同调激发了对一种新型的Floer理论(部分包裹Floer同调)的深入研究；在纽结理论中，它被猜想地与Seifert亏格和调和等拓扑概念有关；在拓扑弦理论中，它被猜想由近年来被大量研究的某个Calabi-Yau流形所决定。除了解决这些猜想，首席调查员还将继续一个相关的项目，遵循最近发现的一侧可构造的滑轮(来自代数几何)与另一侧的全纯曲线之间的联系。这个项目将发展这种联系，特别是为接触流形定义Fukaya范畴的一个类似，并应用它来促进Fukaya范畴和镜像对称性的计算。

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.