CAREER: Combinatorial Methods in Low-Dimensional Topology

职业：低维拓扑中的组合方法

• 批准号: 1455132
• 负责人: Joshua Greene
• 金额: $ 42万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1455132&HistoricalAwards=false
• 关键词: CAREER; Combinatorial; Methods; Low; Dimensional

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.

拓扑广泛地指对形状的研究，而低维拓扑特指对一维到四维的研究，从人类的角度来看，这些维度很特殊，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。 ，由于它们表现出的现象以及用于研究它们的技术集合与更高维度的技术有很大不同，许多用于研究这些现象的技术都涉及组合学，因此该研究项目的一个中心目标是。推进这些的组合方面除了研究领域的具体问题外，PI还提出了将其研究兴趣与教育和培训计划相结合的活动，这些活动覆盖从高中到博士后研究人员的受众，例如，PI积极参与。通过汉普郡学院数学暑期研究和数学楼梯公司，在高中阶段丰富了数学知识。在这些项目和其他指导活动的背景下，他力求激发发现过程并帮助阐述美丽的数学。主要外展该项目的活动是一个研究生暑期学校，它将展示低维拓扑中一个中心主题的几种不同观点，即德恩手术。在低维拓扑中发挥作用的各种技术中包括表面相交图，例如Gordon 和 Luecke 的工作，以及 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性的进展。表面相交技术更加直接，并且依赖于图论工具的发展来得出拓扑结论，弗洛尔同调方法不太直接，但将重型机械应用于大量集合。 PI 特别寻求将源自这些技术和其他技术的组合思想与低维拓扑中的一些驱动问题相结合，涵盖的主题包括图结的研究、Dehn 手术和复合曲线。