Ordered groups and 3-manifolds

我提议的研究涉及两个数学领域之间的相互作用，即群论和三维流形的拓扑结构。 三维流形是一种空间，它模拟了数学中自然发生的许多现象，也模拟了自然界中的现象，例如宇宙的可能形状。 群论是一种代数，在其早期，主要处理形状的对称性和刚性运动的性质，因此在其现代形式中，这种代数理论非常适合于研究三维空间的形状。 目前的目标是使用群论来更好地理解三维流形的可能形状和拓扑。 一个常见的现代技术是采取一个三维流形，并从它，计算什么是所谓的“基本组”，一个代数对象，它携带着令人难以置信的大量数据的对称性，形状和结构的流形你开始。 在计算了流形的基本群之后，挑战在于从计算的输出中提取所有可用的信息，这个问题有许多不同的方法，几十年来一直是一个持续的研究课题。 我的研究计划将侧重于尝试一种新的技术，通过对基本群的元素进行排序（代数练习）来研究三维流形的基本群所携带的信息，并看看排序反映了流形的哪些属性（拓扑问题）。 目前，这一研究方向特别有趣，因为虽然对群进行排序的想法已经存在多年，但对基本群进行排序的想法是新的，并且正在产生令人惊讶的结果。 计算的结果似乎表明，这种新方法可能与一个强大的理论Heegaard-Floer同源性和一个相关的研究领域称为叶理理论有关，尽管目前还没有已知的原因为什么这种联系应该存在。 将这三个领域联系起来的发展将使思想、解决问题的技术和定理从一个领域到另一个领域的富有成效的转移成为可能，在一些比较简单的情况下，这种联系已经得到了理解。

My proposed research deals with the interplay between two fields of mathematics known as group theory and the topology of three-dimensional manifolds. Three-dimensional manifolds are spaces that model many phenomena that occur naturally in mathematics, but also phenomena in nature, such as the possible shapes of the universe. Group theory is a type of algebra that, in its early days, dealt primarily with the properties of symmetry and rigid motions of shapes, and so in its modern form this algebraic theory is well-adapted to studying the shapes of three-dimensional spaces. The goal at hand is to use group theory to better understand the possible shapes and topologies of three-dimensional manifolds. One of the common modern techniques is to take a three-dimensional manifold, and from it, calculate what is called its `fundamental group', an algebraic object that carries with it an incredible amount of data about the symmetries, shape and structure of the manifold you started with. Having computed the fundamental group of a manifold, the challenge is to extract all the available information from the output of your calculation, a problem which has many different approaches and has been an ongoing topic of research for decades. My research program will focus on trying a new technique of investigating the information carried by the fundamental group of a three dimensional manifold, by ordering the elements of the fundamental group (an algebraic exercise) and seeing what properties of the manifolds (a topological question) are reflected by the orderings. This line of research is particularly interesting at present, because while the idea of ordering groups has been around for years, the idea of ordering fundamental groups is new and is producing surprising results. The outputs of the calculations are seeming to indicate that this new approach may be connected to a powerful theory called Heegaard-Floer homology, and a related area of study known as foliation theory, although for the time being there is no known reason why such connections should exist. Developments linking these three fields will allow for a fruitful transfer of ideas, problem-solving techniques and theorems from one area to another, as has already been the case in some of the simpler situations where the connection is understood.