CAREER: Floer Homology and Low-Dimensional Topology

本项目研究低维流形的拓扑结构和几何结构，以及其中嵌入的结点和曲面。具体的目标包括在手术下产生简单流形的三维球体中的结的分类，在复杂曲面内结合复杂曲线的结的拓扑特征，以及更深层次地理解结和3流形的共体群的结构。这项研究的主要工具是辛几何、规范理论和量子代数。这些工具中的许多以代数不变量的形式出现，例如花同调理论或组合结不变量，例如Khovanov同调。该项目还寻求进一步我们对不变量本身的理解，一个中心主题是确定它们忠实地代表它们所阴影的拓扑对象的程度。该项目的具体数学目标是由针对研究生和本科教育的具体举措以及针对国际拓扑学家和几何学者的活动来补充的。例如，将设计和实施一门新的研究生课程，其双重重点是在各种情况下发展拓扑广度和沟通技巧。将组织一所面向本科生的暑期学校和一场广泛兴趣的拓扑学会议。拓扑学和几何学是研究形状和空间的数学领域，而低维拓扑学关注的是那些在我们视野范围内或刚刚超出我们视野范围的形状。该项目对低维拓扑学做出了基础性的贡献。该项目解决的关键问题之一是“打结”的数学理论，这是一个试图理解和量化理想弦在空间中纠缠和打结的方法的领域。除了打结在低维拓扑中扮演的基本角色外，它的理论与许多看似无关的数学和物理领域，甚至与聚合物科学等领域有着深刻的相互作用。该项目的中心目标之一是了解结如何随着时间的推移而演变。想象一下，在一部电影中，一根绳子在空间中自由移动，随着时间的推移或多或少地缠在一起。进一步想象一下，在电影中的某些帧中出现了更激烈的现象，例如出现或消失了一个新的字符串循环或将字符串的两个部分粘合在一起。这样的电影被称为“和谐”，利用这个概念，我们可以用对待数字的方式来对待打结的绳子；也就是说，我们可以用代数的方法加减结点。理解结的算术对四维空间的研究有着深远的影响，四维空间是现代数学中最困难和最不为人所知的领域之一。除了其具体的数学目标外，该项目还将通过设计和组织创新课程、研讨会和会议，为研究生和本科教育以及全球研究社区做出重大贡献。

This project studies the topology and geometry of low-dimensional manifolds and the knots and surfaces embedded therein. Specific goals include a classification of knots in the 3-dimensional sphere which produce simple manifolds under surgery, topological characterization of knots which bound complex curves inside complex surfaces, and a deeper understanding of the structure of cobordism groups of knots and 3-manifolds. Primary tools for this study arise in symplectic geometry, gauge theory, and quantum algebra. Many of these tools come in the form of algebraic invariants e.g. Floer homology theories or combinatorial knot invariants such as Khovanov homology. The project also seeks to further our understanding of the invariants themselves, and a central theme is to determine the extent to which they faithfully represent the topological objects which they shadow. The specific mathematical goals of the project are complemented by concrete initiatives targeted at graduate and undergraduate education, together with activities aimed at an international community of topologists and geometers. For instance, a new graduate course will be designed and implemented whose dual focus is on the development of topological breadth and communication skills in a variety of scenarios. A summer school for undergraduates and a broad-interest conference on topology will be organized. Topology and geometry are mathematical fields which study shapes and spaces, and low-dimensional topology focuses on those shapes which are within or just out of reach of our vision. This project makes fundamental contributions to low-dimensional topology. One of the key problems that the project attacks is the mathematical theory of "knotting", an area which attempts to understand and quantify the method by which ideal strings become tangled and knotted in space. In addition to a fundamental role which knotting plays in low-dimensional topology, its theory has deep interactions with many seemingly unconnected areas of mathematics and physics, and even to areas such as polymer science. One of the central aims of the project is to understand how knots evolve over time. Imagine a movie in which a piece of string freely moves in space, becoming more or less tangled over time. Further imagine that at some frames in the movie more drastic phenomena occur such as the appearance or disappearance of a new loop of string or the gluing of two segments of the string together. Such a movie is called a "concordance", and using this idea one can treat knotted pieces of string in much the same way that we treat numbers; namely, one can add and subtract knots in an algebraic way. Understanding the arithmetic of knots turns out to have deep implications for the study of 4-dimensional space, one of the most difficult and least understood areas of modern mathematics. In addition to its specific mathematical aims, the project will also contribute in a significant way to graduate and undergraduate education and to a global research community through design and organization of innovative courses, workshops, and conferences.