CAREER: Combinatorial Methods in Low-Dimensional Topology

拓扑学泛指对形状的研究，而低维拓扑学专门指对1到4维形状的研究。从人类的角度来看，这些维度是特殊的，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，因为它们所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大不相同。许多用于研究这些现象的技术都涉及到组合学，即对离散结构的研究。本研究项目的一个中心目标是推进这些技术的组合方面，着眼于该领域的具体问题。除了研究部分，PI还提出了将他的研究兴趣与教育和培训计划结合起来的活动，这些活动的受众从高中水平到博士后研究人员。例如，PI通过汉普郡学院夏季数学研究和数学阶梯公司积极参与高中水平的数学丰富。在这些项目和其他指导活动的背景下，他试图激发发现过程，并帮助展示美丽的数学。该项目的主要外展活动是一个研究生暑期学校，它将展示低维拓扑的一个中心主题，即Dehn手术的几个不同观点。在对低维拓扑产生影响的各种技术中，有表面相交图，以Gordon和Luecke的工作为例，还有Heegaard Floer同源，由Ozsváth和Szabó定义和发展。这两种技术都在低维拓扑的主要问题上取得了令人瞩目的进展。这两种方法对该主题提供了非常不同的视角，它们具有互补的优点和缺点。曲面相交技术更直接，依靠图论工具的发展来得出拓扑结论。花同源性方法不太直接，但应用重型机械来解决大量问题。PI特别寻求将这些技术和其他技术的组合思想结合起来，以解决低维拓扑中的一些驱动问题，涵盖的主题包括结图、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.