The Geometry and Topology of Heegaard Splittings

Heegaard 分裂的几何和拓扑

• 批准号: 1006369
• 负责人: Jesse Johnson
• 金额: $ 11.64万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006369&HistoricalAwards=false
• 关键词: Geometry; Topology; Heegaard; Splittings; Geometry; Topology; Heegaard; Splittings

The principal objective of this project is to develop new methods that will produce a unified and systematic approach to understanding and classifying isotopy classes of Heegaard splittings in 3-manifolds. In addition to strengthening the foundations of the field, such an approach will lead to new results, as well as opening up the field to a wider audience. The new approach uses geometric intuition from recent results connecting Heegaard splittings to hyperbolic geometry in order to expand and clarify two existing methods: thin position and double sweep-outs/graphics. The PI has recently made significant breakthroughs in understanding and expanding these methods in this direction and proposes to further explore their potential applications.Since their introduction in the early 1900s, Heegaard splittings have been a vital tool for placing 3-manifolds in an accessible context. They provide a good introduction to geometric topology and an active area of research for young mathematicians. Right now, the core of the theory of Heegaard splittings is appropriate for beginning graduate students. However, new research continues to provide simpler proofs of the main theorems and more intuitive approaches to the fundamental concepts, so that parts of the field are becoming accessible to advanced undergraduates. The research project described here will eventually lead to problems that are appropriate for an undergraduate thesis or even an REU. This will provide a gateway for students into other areas of algebraic and geometric topology. It should be noted that OSU has a substantial population of Native American and other underserved minorities, and Oklahoma is geographically isolated from the academic centers of the country. Through involvement in the PI's research, mathematically talented students at OSU will have the opportunity to develop their talents, increase their visibility and confidence and prepare themselves for further success in mathematics and science.

该项目的主要目的是开发新方法，该方法将产生一种统一的，系统的方法来理解和分类3个manifolds中Heegaard分裂的同位素类别。 除了加强该领域的基础外，这种方法还将带来新的结果，并向更广泛的受众开放领域。 这种新方法使用几何直觉，从将Heegaard分裂连接到双曲线几何形状的最新结果，以扩展和阐明两种现有方法：薄位置和双重扫​​描/图形。 PI最近在理解和扩展这些方法方面取得了重大突破，并提议进一步探索其潜在的应用。由于他们在1900年代初期的引入，Heegaard分裂一直是将​​3个元件放置在可访问环境中的至关重要的工具。 它们很好地介绍了几何拓扑和年轻数学家的积极研究领域。 目前，Heegaard分裂理论的核心适合初学者。 但是，新的研究继续为主要定理提供了更简单的证据，并为基本概念提供了更直观的方法，以便该领域的某些部分可以被高级本科生访问。 此处描述的研究项目最终将导致适合本科论文甚至REU的问题。 这将为学生进入代数和几何拓扑的其他领域。 应当指出的是，OSU拥有大量美洲原住民和其他服务不足的少数民族，俄克拉荷马州与该国的学术中心隔绝了俄克拉荷马州。 通过参与PI的研究，OSU的数学才华横溢的学生将有机会发展自己的才能，提高其知名度和信心，并为在数学和科学方面的进一步成功做好准备。