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

该项目的主题是研究辛几何、规范理论和量子代数中的不变量，并将这些不变量应用于低维拓扑和几何问题。一个中心目标是了解拓扑量子场论方面的各种花同调理论在维度（2+1）。本项目的目标包括：发现带边界的流形粘接得到的三流形的Floer同调的公式；在接触和对称同调理论的背景下定义结点和面的相对Floer不变量；发现分别作为复杂曲面和曲线边界的三流形和结点的拓扑特征。由于所研究的不变量已经被证明是非常强大的，因此可能会出现许多应用。这些应用包括Dehn手术问题的研究，平滑一致性和解结数的结，以及代数曲线的研究。研究三维和四维空间，以及其中的弯曲曲线和曲面，是我们理解宇宙大尺度和小尺度方面的核心任务。确定宇宙的形状取决于对可能出现的形状和这些形状所具有的属性的数学理解。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。在这一追求中，研究曲线和曲面可以打结的方式是相当富有成效的。这种打结不仅与理解空间的形状有关，而且最近在DNA的研究中也变得重要起来。被限制在一个小空间内，长链DNA自然地打结，某些过程依赖于对这些结的复杂性的理解。这个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。

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.