Holomorphic Invariants in Symplectic Topology

辛几何是数学的一个领域，它可以追溯到19世纪和牛顿力学的现代物理公式。在过去的几十年里，它已经成为一个令人兴奋的数学研究的基础领域，部分原因是与数学和物理学的许多其他部分密切相关。辛几何最近在低维拓扑（三维和四维空间的数学研究）中有特别引人注目的应用。首席研究员将沿着这条路线，在绳结理论的背景下，或在两端系在一起的绳圈，追求一种特别有前途的技术。这种技术已被证明是物理界感兴趣的，它提供了一个尚未被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）的部分。本项目将致力于揭示这一框架，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，全纯曲线已经成为辛几何的核心工具，将强大的解析和几何技术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各个领域有着许多意想不到和有趣联系的主题。最近的结果为详细探索这些联系打开了大门，这将在本项目中进行。在辛几何中，结接触同调激发了一种新型的Floer理论（部分包裹的Floer同调）的深入研究；在结理论中，它与塞弗特属和调和等拓扑概念推测相关；在拓扑弦理论中，它被推测是由一个特定的Calabi-Yau流形决定的，这个流形近年来成为许多研究的对象。除了解决这些猜想之外，首席研究员还将进行一个相关的项目，根据最近发现的可构造轴（来自代数几何）与全纯曲线之间的联系。本项目将发展这种联系，特别是致力于定义接触流形的深谷范畴的模拟，并将其应用于深谷范畴和镜像对称的计算。

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.