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.
A 3-dimensional manifold is a topological space that models the 3-dimensional universe in which we live. Heegaard splittings are topological structures that allow one to see such a space as a pair of simple pieces that have been glued together in a possibly complicated manner. While Heegaard splittings have been studied for over a century, our knowledge of Heegaard splittings has only in the last two decades matured to the point where they can be thoroughly understood. They are now an integral part of understanding geometric structures on 3-manifolds and knots. The goal of this project was to carry out research that would strengthen our understanding the relationship between these topological decompositions of three-dimensional spaces and the the hyperbolic geometry that has come to play a large role in the study of three-dimensional topology over the past few decades. During the course of the grant, the PI completed a number of research projects that greatly strengthen our understanding in these regards, the results of which have been submitted for publication and made publicly available in preprint form. One undexpected outcome of this research was the discovery of a connection between the techniques used to study Heegaard splittings and a common approach to a certain data analysis problem called clustering. A clustering algorithm is a method for dividing a data set into smaller subsets of closely related data points. One common approach involves minimizing a quantity called the normalized mean cut, which is closely related to a geometric measure called the Cheeger constant. Because a certain technique (called thin position) commonly used for studying Heegaard splittings is known to be related to the Cheeger constant, the PI was able to translate this technique into a clustering algorithm. A paper descibing this algorithm has since been published in the Proceedings volume of a computer science conference (The 2013 SIAM International Conference on Data Mining.)