Mathematical Sciences: Geometric Topology and the Fundamental Group

数学科学：几何拓扑和基本群

• 批准号: 9204502
• 负责人: James Cannon
• 金额: $ 10.29万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1996-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204502&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Topology; Fundamental

Most (perhaps all) 3-manifolds are geometric. Geometric group theory seeks to tie together combinatorial and geometric properties of geometric isometry groups. The theory is burgeoning at the hands of Gromov, Epstein, Thurston, Holt, Bestvina, Feighn, Mess, Paterson, Gersten, Short, Shapiro, Parry, Floyd, Baumslag, Casson, Brick, Stallings, a whole school of continental mathematicians, Cannon, and others. The theory has produced the notions of automatic group, negatively curved group, complex of groups, almost convex groups, combable groups, isoperimetric and isodiametric inequalities for groups. Amid these many topics, Cannon, his students, and coworkers are currently concentrating on a cluster of projects related to the existence of hyperbolic structures on 3- manifolds. Is the generic finitely presented group negatively curved, as claimed without explicit proof by Gromov? Does every negatively curved closed 3-manifold admit a Riemannian metric of constant negative curvature? Can negative curvature be recognized algorithmically? Can the space at infinity be recognized algorithmically? Can potential constant curvature be recognized algorithmically? Cannon is using as the basis for these studies his combinatorial Riemann mapping theorem developed during the previous projects, the related work by Parry, and the computer Programs of Epstein and his collaborators. Groups are the appropriate algebraic structures for describing the notion of symmetry with precision. For this reason they play an extensive role in geometry (and topology). While the more usual interaction between group theory and geometry is the use of algebraic computations involving groups to prove theorems about geometric objects possessing symmetry, there is a very significant mathematical subdiscipline in which the roles are reversed and our intuition about geometric objects assists us in conjecturing and proving theorems about associated groups. A fascinating recent development is the influence of the computer in all this. Not only is it the ubiquitous timesaver that everyone knows, enabling one to undertake computations that would otherwise be too formidable, but it has actually influenced the questions in geometric group theory in a more fundamental way. Certain classes of groups have been defined in terms of the types of hypothetical machines that would be able to perform certain computations about them. Real hardware is not involved in this, and the same definitions could have been conceived long ago, but they were not, for the existence of computing machines has influenced the way we look at the world, the types of questions we ask about it, the kinds of properties that we find interesting.

大多数(或许所有)三维流形都是几何的。几何群论试图将几何等距群的组合性质和几何性质联系在一起。这一理论在格罗莫夫、爱泼斯坦、瑟斯顿、霍尔特、贝斯特维纳、费恩、梅斯、帕特森、格斯滕、肖特、夏皮罗、帕里、弗洛伊德、鲍姆斯拉格、卡森、布里克、斯塔林斯、整个欧洲数学家学派、加农和其他人手中迅速发展起来。这一理论产生了自动机群、负曲群、群的复数、几乎凸群、可梳群、群的等周等径不等式等概念。在这些主题中，Cannon、他的学生和他的同事们目前正专注于一系列与三维流形上双曲结构的存在有关的项目。一般的有限表示群是否如Gromov所声称的那样是负曲的，而没有明确的证明？每个负弯曲的闭3-流形都有常负曲率的黎曼度量吗？负曲率可以通过算法识别吗？无限大的空间可以通过算法识别吗？位势常曲率能被算法识别吗？坎农使用他在以前的项目中发展的组合黎曼映射定理、帕里的相关工作以及爱泼斯坦和他的合作者的计算机程序作为这些研究的基础。群是精确描述对称性概念的合适的代数结构。因此，它们在几何学(和拓扑学)中扮演着广泛的角色。虽然群论和几何之间更常见的互动是使用涉及群的代数计算来证明关于具有对称性的几何对象的定理，但有一个非常重要的数学分支学科，其中角色颠倒，我们对几何对象的直觉帮助我们猜测和证明关于相关群的定理。最近一个引人入胜的发展是计算机在这一切中的影响。它不仅是大家都知道的无处不在的省时方法，使人能够进行否则会过于艰巨的计算，而且它实际上以更根本的方式影响了几何群论中的问题。某些类别的组已经根据能够执行关于它们的某些计算的假设机器的类型来定义。真正的硬件并不涉及这一点，同样的定义很久以前就可能被构思出来，但它们并没有，因为计算机的存在影响了我们看待世界的方式，影响了我们对世界提出的问题的类型，以及我们感兴趣的属性的类型。