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

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.

该项目的主要目标是开发新方法，以产生统一且系统的方法来理解和分类 3 流形中 Heegaard 分裂的同位素类别。 除了加强该领域的基础之外，这种方法还将带来新的成果，并向更广泛的受众开放该领域。 新方法利用最近结果中的几何直觉，将 Heegaard 分裂与双曲几何联系起来，以扩展和澄清两种现有方法：薄位置和双扫描/图形。 PI 最近在理解和扩展这些方法方面取得了重大突破，并建议进一步探索其潜在应用。自 1900 年代初引入以来，Heegaard 分裂一直是在可访问的环境中放置 3 流形的重要工具。 它们为年轻数学家提供了几何拓扑的良好介绍和活跃的研究领域。 目前，Heegaard 分裂理论的核心内容适合研究生入门。 然而，新的研究继续为主要定理提供更简单的证明，并为基本概念提供更直观的方法，因此该领域的部分内容正变得可供高年级本科生使用。 这里描述的研究项目最终将产生适合本科论文甚至 REU 的问题。 这将为学生进入代数和几何拓扑的其他领域提供一个门户。 应该指出的是，俄克拉荷马州立大学拥有大量美洲原住民和其他服务不足的少数民族人口，而俄克拉荷马州在地理上与该国的学术中心隔离。 通过参与 PI 的研究，俄勒冈州立大学的数学天才学生将有机会发展自己的才能，提高他们的知名度和信心，并为在数学和科学领域取得进一步的成功做好准备。