Finite covers of hyperbolic 3-manifolds

双曲3流形的有限覆盖

• 批准号: 0805828
• 负责人: Alan Reid
• 金额: $ 38.25万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805828&HistoricalAwards=false
• 关键词: Finite; covers; hyperbolic; manifolds

Motivated by considerations in topology, geometry, arithmetic and group theory the proposal intends to study real, complex and quaternionic hyperbolic manifolds. This will involve the study of discrete groups, their connections with number theory, and expanding graphs. We will also explore various interconnections between these topics with a view to better understanding the topology of hyperbolic 3-manifolds.Three dimensional manifolds are locally like the space we live in and understanding these objects have been one of the central themes of research in the last 30 years. The importance of these objects extends far beyond their intrinsic interest, since their study connects to mathematical physics, mathematical biology and computer science. One of the aims of this proposal is to explore some of these connections via families of "expanding graphs". These graphs are well-known in computer science because of their importance in building efficient networks. Remarkably, this is connected to the main study of the proposal, "finite covers of 3-manifolds".By their nature, many of the problems in the proposal are cross-disciplinary, and hence progress will have a broader impact. In addition, these have proved to be fertile grounds for the training of graduate students.

出于对拓扑、几何、算术和群论的考虑，该提案旨在研究实数、复数和四元双曲流形。这将涉及离散群的研究、它们与数论的联系以及扩展图。我们还将探索这些主题之间的各种相互联系，以更好地理解双曲 3 流形的拓扑结构。三维流形局部类似于我们居住的空间，理解这些对象一直是过去研究的中心主题之一。 30年。 这些对象的重要性远远超出了它们的内在兴趣，因为它们的研究与数学物理学、数学生物学和计算机科学相关。该提案的目的之一是通过“扩展图”系列来探索其中一些联系。这些图在计算机科学中众所周知，因为它们在构建高效网络方面的重要性。值得注意的是，这与该提案的主要研究“3-流形的有限覆盖”有关。从本质上来说，该提案中的许多问题都是跨学科的，因此进展将产生更广泛的影响。此外，这些也被证明是研究生培养的沃土。