CAREER: Combinatorial Methods in Low-Dimensional Topology

拓扑学泛指对形状的研究，而低维拓扑学具体指的是从一维到四维的研究。从人类学的角度来看，这些维度是特别的，因为它们模拟了我们对物理世界的日常感知，从数学的角度来看，因为它们展示的现象和用于研究它们的技术集合与更高维度的现象相当不同。许多用于研究这些现象的技术涉及组合学，即对离散结构的研究。这项研究项目的一个中心目标是推进这些技术的组合方面，以期解决该领域的具体问题。除了研究部分，PI还提出了一些活动，将他的研究兴趣与教育和培训倡议结合起来，覆盖从高中到博士后研究人员的受众。例如，PI通过汉普郡学院数学暑期学习和数学楼梯公司，积极参与高中阶段的数学丰富。在这些计划和其他指导活动的背景下，他试图启发发现过程并帮助展示美丽的数学。该项目的一个主要推广活动是一个暑期研究生班，它将展示关于低维拓扑学的一个中心主题Dehn Surgery的几个不同观点。低维拓扑学的各种技术是曲面交叉图，例如Gordon和Luecke的工作，以及由Ozsváth和Szabó定义和开发的Heegaard Floer同调。这两种技术都在低维拓扑的主要问题上取得了轰动的进展。这两种方法在这个问题上提供了非常不同的视角，它们有互补的优势和劣势。曲面求交技术更加直接，并且依赖于图论工具的发展来得出拓扑结论。Floer同调方法不那么直接，但将重型机械应用于大量问题。PI特别寻求融合这些技术和其他技术产生的组合思想，以期解决低维拓扑中的一些驱动问题，跨越的主题包括纽结图、Dehn手术和曲线复合体的研究。

Topology refers broadly to the study of shapes, and low-dimensional topology refers specifically to their study in dimensions one through four. These dimensions are special from an anthropic perspective, since they model our everyday perception of the physical world, and from a mathematical perspective, since the phenomena they exhibit and the collection of techniques used to study them are rather different from those in higher dimensions. Many of these techniques used to study these phenomena involve combinatorics, the study of discrete structures. A central goal of this research project is to advance the combinatorial aspects of these techniques with a view towards concrete problems in the field. Alongside the research component, the PI proposes activities that integrate his research interests with education and training initiatives that reach audiences from the high school level to postdoctoral researchers. For instance, the PI is actively involved with mathematics enrichment at the high school level through the Hampshire College Summer Studies in Mathematics and Mathematical Staircase, Inc. In the context of these programs and in other mentoring activities, he seeks to inspire the discovery process and aid in the exposition of beautiful mathematics. A chief outreach activity in the project is a graduate summer school that will showcase several different perspectives on one central theme in low-dimensional topology, Dehn surgery.Amongst the various techniques that come to bear on low-dimensional topology are graphs of surface intersections, exemplified by the work of Gordon and Luecke, and Heegaard Floer homology, defined and developed by Ozsváth and Szabó. Both techniques have led to sensational progress on the main problems in low-dimensional topology. The two approaches lend very different perspectives on the subject, and they have complementary strengths and weaknesses. The surface intersection techniques are more direct and rely on the development of graph theoretic tools in order to draw topological conclusions. Floer homology methods are less direct but apply heavy machinery to a vast collection of problems. The PI specifically seeks to blend the combinatorial ideas stemming from these techniques and others with a view towards some of the driving problems in low-dimensional topology, spanning topics including the study of knot diagrams, Dehn surgery, and the curve complex.