Ordered groups and 3-manifolds

有序群和 3 流形

• 批准号: RGPIN-2014-05465
• 负责人: Clay, Adam
• 金额: $ 1.38万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677024
• 关键词: Ordered; groups; manifolds

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.

我提出的研究涉及称为群体理论的两个数学领域与三维流形的拓扑之间的相互作用。 三维流形的空间是模拟数学自然发生的许多现象，但在自然界中也存在现象，例如宇宙的可能形状。 小组理论是一种代数，在早期，它主要涉及形状的对称性和刚性运动的特性，因此以其现代形式这种代数理论非常适应研究三维空间的形状。 目前的目标是使用小组理论更好地了解三维流形的可能形状和拓扑。 **一种常见的现代技术之一是采用三维流形，然后从中计算出所谓的“基本组”，这是一个代数对象，它带有大量有关您开始使用的歧管的对称性，形状和结构的数据。 挑战是在计算了一个歧管的基本组之后，是从计算的输出中提取所有可用信息，这个问题具有许多不同的方法，并且数十年来一直是一个持续的研究主题。 我的研究计划将着重于尝试一种新技术，以调查三维流形的基本组所携带的信息，通过订购基本组的要素（代数练习）（代数练习），并查看流派的属性（一个拓扑问题）（一个拓扑问题）（一个拓扑问题）（一个拓扑问题）反映了订单。产生令人惊讶的结果。 计算的输出似乎表明，这种新方法可能与称为Heegaard-loer同源性的强大理论相关，而相关的研究领域被称为叶面理论，尽管目前尚无已知原因，因此应该存在这种联系。 将这三个领域连接起来的发展将使思想，解决问题的技术和定理从一个领域转移到另一个领域，就像在某些更简单的情况下，这种情况已经理解了。