CAREER: Floer Homology and Low-Dimensional Topology

这个项目研究低维流形的拓扑和几何以及嵌入其中的纽结和曲面。具体目标包括对在外科手术下产生简单流形的3维球体中的纽结进行分类，对束缚复杂曲面中的复杂曲线的纽结的拓扑特征进行描述，以及更深入地理解纽结和3-流形的协边群的结构。这项研究的主要工具是辛几何、规范理论和量子代数。其中许多工具以代数不变量的形式出现，如Floer同调理论或组合纽结不变量，如Khovanov同调。这个项目还试图加深我们对不变量本身的理解，一个中心主题是确定它们在多大程度上忠实地表示它们所映射的拓扑对象。该项目的具体数学目标得到了针对研究生和本科教育的具体倡议的补充，以及针对国际拓扑学家和地质学家社区的活动。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能。将为本科生组织暑期班和拓扑学广泛兴趣的会议。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觉范围内或只是超出我们视线范围的形状。该项目对低维拓扑学做出了根本性的贡献。该项目研究的关键问题之一是“打结”的数学理论，该领域试图理解和量化理想弦在空间中缠绕和打结的方法。除了纽结在低维拓扑学中扮演的基本角色外，它的理论还与许多看似互不相连的数学和物理领域，甚至是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是如何随着时间的推移而演变的。想象一部电影，其中一根线在空间中自由移动，随着时间的推移或多或少地变得缠绕在一起。再想象一下，在电影中的一些画面中，会出现更剧烈的现象，比如新的一圈线的出现或消失，或者两段线的粘合在一起。这样的电影被称为“和声”，使用这个想法，人们可以像处理数字一样处理打结的绳子段；也就是说，人们可以用代数的方法来加减结。理解纽结的算术被证明对研究4维空间具有深远的意义，4维空间是现代数学中最困难、最不被理解的领域之一。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，对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.