The Geometry and Topology of Heegaard Splittings

Oklahoma State University, Stillwater OK

Investigators

Abstract

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.