Gauge Theory and Symplectic Geometry in Dimensions 3 and 4

3 维和 4 维规范理论和辛几何

• 批准号: 1308597
• 负责人: Cagatay Kutluhan
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2013-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308597&HistoricalAwards=false
• 关键词: Gauge; Theory; Symplectic; Geometry; Dimensions

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.

在过去的三十年里，规范理论在三维和四维流形的研究中扮演了重要的角色。从规范理论研究不变量揭示了这些流形的许多有趣的几何和拓扑性质。同时，辛几何中的伪全纯曲线也被用来引入其他密切相关的不变量。本提案中概述的项目分为五个主题，每个主题都利用了最近证明的Seiberg-Witten规范理论和辛几何中三维和四维流形不变量之间的对应关系，以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。另一个主题涉及光滑四维流形的Seiberg-Witten和Ozsvath-Szabo不变量之间的固定对应。剩下的两个主题分别从Seiberg-Witten规范理论和辛几何的角度发展和研究纽结和三维边界流形的Floer同调不变量。规范理论和辛几何都起源于理论物理学，在那里它们被用作描述经典或量子动力学系统的数学框架。特别是，量子化规范理论在通过所谓的标准模型来理解我们宇宙的动力学方面发挥着核心作用。确定我们的宇宙形状的努力，我们认为这是一个四维空间-三个空间维度和一个时间维度-随着时间的推移，创造了数学和物理学之间的成功合作。数学家在抽象背景下对物理学家所使用的这些数学框架的研究，不仅开创了数学研究的新方向，而且为物理学家提供了新的研究视角。本提案中概述的研究旨在通过继续研究Seiberg-Witten规范理论和辛几何之间的相互作用来促进这种合作。