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Low dimensional topology and invariants from symplectic geometry, gauge theory, and quantum algebra

辛几何、规范理论和量子代数的低维拓扑和不变量

基本信息

批准号：
0706979
负责人：
Tomasz Mrowka
金额：
$ 7.84万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2007
资助国家：
美国
起止时间：
2007-09-01 至 2009-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706979&HistoricalAwards=false
关键词：
Low dimensional topology invariants symplectic

项目摘要

The theme of this project is to study invariants coming from symplectic geometry, gauge theory, and quantum algebra and apply these invariants to questions in low-dimensional topology and geometry. A central objective is to understand the topological quantum field theoretic aspects of various Floer homology theories in dimensions (2+1). Goals of the project include the discovery of formulas for the Floer homology of three-manifolds obtained by gluing together manifolds with boundary, the definition of relative Floer invariants for knots and surfaces in the context of contact and symplectic homology theories, and the discovery of topological characterizations of three-manifolds and knots which arise as the boundaries of complex surfaces and curves, respectively. As the invariants studied have already proved to be quite powerful, many possible applications could arise. Among these applications are the study of Dehn surgery problems, smooth concordance and unknotting numbers of knots, and the study of algebraic curves.The study of three- and four-dimensional spaces, and knotted curves and surfaces within them, is a central task to our understanding of both large and small scale aspects of the universe. Determining the shape of the universe depends upon a mathematical understanding of the possible shapes that could occur and the properties these shapes have. These properties are known as invariants, and the project furthers our understanding of space by the discovery of new invariants and the study of existing invariants. In this pursuit it has been quite fruitful to examine the way in which curves and surfaces can be tied in knots. This kind of knotting is not only relevant to understanding the shape of space, but has recently become significant in the study of DNA. Confined to a small space, long strands of DNA naturally become knotted, and certain processes depend upon an understanding of the complexity of these knots. Applications of this project include effective ways of measuring different types of complexity of knots.

这个项目的主题是研究来自辛几何、规范理论和量子代数的不变量，并将这些不变量应用于低维拓扑和几何中的问题。一个中心目标是理解(2+1)维的各种Floer同调理论的拓扑量子场理论方面。该项目的目标包括发现通过将流形和边界粘合在一起而得到的三维流形的Floer同调的公式，在接触和辛同调理论的背景下定义纽结和曲面的相对Floer不变量，以及发现分别作为复杂曲面和曲线边界的三维流形和纽结的拓扑特征。由于所研究的不变量已经被证明是相当强大的，因此可能会出现许多可能的应用。这些应用包括Dehn运算问题的研究，纽结的光滑协调和解结数目的研究，以及代数曲线的研究。研究三维和四维空间，以及其中的纽结曲线和曲面，是我们理解宇宙大小尺度方面的中心任务。确定宇宙的形状取决于对可能出现的形状以及这些形状所具有的性质的数学理解。这些性质被称为不变量，该项目通过发现新的不变量和研究现有的不变量，进一步加深了我们对空间的理解。在这个追求中，研究曲线和曲面可以打结的方式是相当有成效的。这种打结不仅与理解空间的形状有关，而且最近在DNA研究中变得重要起来。在一个很小的空间里，DNA的长链自然会打结，而某些过程取决于对这些打结的复杂性的理解。该项目的应用包括测量不同类型的结的复杂程度的有效方法。