Gauge Theory and Symplectic Geometry in Dimensions 3 and 4

近三十年来，规范理论在三维和四维流形的研究中发挥了重要作用。从规范理论对不变量的研究揭示了这些流形的许多有趣的几何和拓扑性质。同时，利用辛几何中的伪全纯曲线引入了其他密切相关的不变量。本提案中概述的项目被组织为五个主题，每个主题都利用最近从Seiberg-Witten规范理论和辛几何中证明的三维和四维流形不变量之间的对应关系来揭示更多关于这些流形的几何和拓扑。更明确地说，其中两个主题研究了三维流形的接触拓扑和某些类型的四维流形上辛结构的存在性/唯一性。另一个主题是关于光滑四维流形的Seiberg-Witten和Ozsvath-Szabo不变量之间的推测对应关系。其余两个课题分别从Seiberg-Witten规范理论和辛几何中发展和研究结点和三维流形的flower同调不变量。规范理论和辛几何都起源于理论物理学，它们被用作描述经典或量子动力系统的数学框架。特别是，量子化规范理论通过所谓的标准模型在理解宇宙动力学方面起着核心作用。我们的宇宙是一个四维空间——三维空间和一维时间——随着时间的推移，确定宇宙形状的努力创造了数学和物理学之间的成功合作。数学家在抽象的背景下对物理学家所使用的这些数学框架进行研究，不仅创造了新的数学研究方向，而且为物理学家的研究提供了新的视角。本提案中概述的研究旨在通过继续研究Seiberg-Witten规范理论与辛几何之间的相互作用来促进这种合作。

Gauge theory has played an important role in the study of three- and four-dimensional manifolds in the last three decades. Investigation of invariants from gauge theory has revealed many interesting geometric and topological properties of these manifolds. Meanwhile, pseudo-holomorphic curves in symplectic geometry have been used to introduce other closely related invariants. The projects outlined in this proposal are organized into five topics, each of which exploit the recently proved correspondences between invariants of three- and four-dimensional manifolds from Seiberg-Witten gauge theory and symplectic geometry to reveal more about the geometry and topology of these manifolds. To be more explicit, two of the topics investigate the contact topology of three-dimensional manifolds and the existence/uniqueness of symplectic structures on certain types of four-dimensional manifolds. Another topic concerns the conjectured correspondence between the Seiberg-Witten and Ozsvath-Szabo invariants of smooth four-dimensional manifolds. The remaining two topics develop and study Floer homological invariants of knots and three-dimensional manifolds with boundary from Seiberg-Witten gauge theory and symplectic geometry, respectively. Both gauge theory and symplectic geometry have their origins in theoretical physics, where they are used as mathematical frameworks to describe a classical or quantum dynamical system. In particular, quantized gauge theory plays a central role in understanding the dynamics of our universe through what is known as the standard model. The effort to determine the shape of our universe, which we perceive as a four-dimensional space--three spatial dimensions and one time dimension--has over time created a successful collaboration between mathematics and physics. The study in the abstract setting by mathematicians of these mathematical frameworks used by physicists not only creates new research directions in mathematics but also provides new perspectives for physicists to employ in their research. The research outlined in this proposal aims to contribute to this collaboration by continuing the study of interactions between Seiberg-Witten gauge theory and symplectic geometry.