Embedded and Immersed Surfaces in Three-Dimensional Topology

三维拓扑学的指导问题是三维流形的分类问题--构造一个列表，其中每个同胚类型只出现一次。PI将解决这一问题，并探索由此产生的方法在三个组成部分的大型数据集的拓扑结构中的应用：首先，PI将扩展和完善最近由Minsky，Namazi，Souto等人开发的技术，通过Teichmuller理论将双曲几何与拓扑结构联系起来。其次，PI将推广这些技术来回答最近证明的虚拟哈肯猜想所提出的三维流形的有限覆盖问题。第三，PI将探索这些拓扑技术在数据分析中的应用，以开发和改进对广泛领域的科学家有用的新算法。Heegaard分裂是一种拓扑结构，它允许人们将这样的空间视为一对以可能复杂的方式组合在一起的简单片段。虽然Heegaard分裂已经研究了一个世纪，但我们对Heegaard分裂的认识只是在最近二十年才成熟到可以彻底理解的程度。它们现在是理解三维流形和纽结上的几何结构的一个组成部分。此外，最近的发展表明，在Heegaard分裂的抽象研究中使用的技术（特别是一种称为薄位置的技术）可以适用于应用数学中的某些问题，特别是大型数据集的分析。由该补助金资助的研究将发展这一领域的抽象和应用方面。

The guiding question in three-dimensional topology is the classification problem for three-dimensional manifolds - to construct a list in which every homeomorphism type appears exactly once. The PI will both address this question and explore applications of the resulting methods to the topology of large data sets in three components: First, the PI will expand and refine techniques developed recently by Minsky, Namazi, Souto and others to related hyperbolic geometry to topology via Teichmuller theory. Second, the PI will generalize these techniques to answer questions about finite covers of three-dimensional manifolds suggested by the recent proof of the Virtual Haken Conjecture. Third, the PI will explore applications of these topological techniques to data analysis in order to develop and refine new algorithms that will be useful to scientists in a broad range of fields.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 combined 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. Moreover, recent developments have demonstrated that techniques (particularly one called thin position) used in the abstract study of Heegaard splittings can be adapted to work on certain problems in applied mathematics, particularly the analysis of large data sets. The research funded by this grant will develop both the abstract and the applied aspects of this field.