The Geometry and Topology of Heegaard Splittings

这个项目的主要目标是开发新的方法，将产生一个统一的和系统的方法来理解和分类的Heegaard分裂的3-流形的合痕类。 除了加强该领域的基础外，这种做法将导致新的成果，并向更广泛的受众开放该领域。 新的方法使用几何直觉从最近的结果连接Heegaard分裂双曲几何，以扩大和澄清现有的两种方法：薄的位置和双扫出/图形。 PI最近在理解和扩展这些方法方面取得了重大突破，并建议进一步探索它们的潜在应用。自20世纪初引入以来，Heegaard分裂一直是将三维流形置于可访问环境中的重要工具。 他们提供了一个很好的介绍几何拓扑和一个活跃的研究领域的年轻数学家。 现在，Heegaard分裂理论的核心是适合刚开始研究生的。 然而，新的研究继续提供更简单的证明的主要定理和更直观的方法的基本概念，使该领域的部分正在成为先进的本科生。 这里描述的研究项目最终将导致适合本科论文甚至REU的问题。 这将为学生提供一个进入代数和几何拓扑学其他领域的门户。 应该指出的是，俄勒冈州立大学有大量的美国原住民和其他服务不足的少数民族，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，在俄勒冈州立大学数学天才的学生将有机会发展自己的才能，提高他们的知名度和信心，并准备自己在数学和科学的进一步成功。

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.