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

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问题。一个中心目标是了解拓扑量子场论方面的各种弗洛尔同源理论的尺寸（2+1）。 该项目的目标包括发现公式的弗洛尔同源性的三个流形获得胶合在一起的流形与边界，相对弗洛尔不变量的定义为结和表面的背景下，接触和辛同源理论，并发现拓扑特征的三个流形和结出现的边界复杂的表面和曲线，分别。 由于研究的不变量已经被证明是相当强大的，许多可能的应用可能会出现。这些应用包括研究Dehn手术问题、光滑一致性和解结数，以及研究代数曲线。三维和四维空间以及其中的打结曲线和曲面的研究，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可能形状以及这些形状所具有的属性的数学理解。 这些属性被称为不变量，该项目通过发现新的不变量和研究现有的不变量来进一步加深我们对空间的理解。 在这种追求中，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。 DNA长链被限制在一个很小的空间里，自然会打结，而某些过程取决于对这些结的复杂性的理解。 该项目的应用包括测量不同类型的复杂的结的有效方法。

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.