Algebraic and Geometric Topology

代数和几何拓扑

• 批准号: 0305712
• 负责人: Steven Kerckhoff
• 金额: $ 29.56万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305712&HistoricalAwards=false
• 关键词: Algebraic; Geometric; Topology

AbstractAward: DMS-0305712Principal Investigator: R.L. Cohen, S.P. Kerckhoff, R.J. MilgramThis project consists of research programs in a broad range oftopics in Algebraic and Geometric Topology. The principalinvestigators, Professors R.L. Cohen, S.P. Kerckhoff, andR.J. Milgram are senior topologists whose research interestscover a large number of areas of topology and related fields.Included in these projects are the study of core topologicalquestions such as the study of 3 - dimensional geometry, thestudy of the topology of complex and related structures onmanifolds, and topological invariants of finite groups. Inaddition it includes applications of topological methods toengineering questions in the field of Robotics.The topology of 3-dimensional manifolds is intimately connectedto differential geometry, particularly to homogeneous structureslike Euclidean, spherical, and hyperbolic geometry. Of these,hyperbolic geometry is the most prevalent. On a closed3-manifold a hyperbolic structure, if it exists, is unique by theMostow Rigidity Theorem. Thus, any geometric invariant is atopological invariant and can be used to distinguish different3-manifolds. When the 3-manifold has boundary, it can have alarge parameter space of hyperbolic structures; in this case, thequalitativeq properties of the structures as one varies over theparameter space are of particular interest. Kerckhoff studieshyperbolic structures on 3-manifolds, trying to describe thebehavior of these geometric structures under a topologicalprocedure, called Dehn surgery, and in the presence ofsingularities modeled on a cone. R.L. Cohen is pursuing severalprojects that involve studying algebraic topological aspects ofquestons stemming from geometry. These projects lie under thefollowing general headings. 1. The topology of moduli spaces ofholomorphic curves in complex and symplectic manifolds. 2. Thehomotopy type of the stable mapping class group and the MumfordConjecture. 3. Properties and applications of holomorphicK-theory. R.J. Milgram will pursue questions involving thecohomology of finite simple groups, as well as applications ofthe topology of configuration spaces to questions in the designand behavior of robotic arms.

AbstractAward：DMS-0305712 Principal研究者：R.L。Cohen，S.P。Kerckhoff，R.J。米尔格拉姆这个项目由代数和几何拓扑的广泛涉及研究计划组成。 首席评估者，R.L。Cohen教授，S.P。Kerckhoff，Andr.J。米尔格拉姆（Milgram）是高级拓扑师，其研究兴趣景观各个领域的拓扑和相关领域。在这些项目中，包括研究核心拓扑问题的研究，例如研究3-维几何学的研究，复杂及相关结构的拓扑结构的托管，on manifolds的拓扑结构，以及有限基团的拓扑组群。 它不断不断地包括在机器人技术领域进行工程问题的拓扑方法的应用。三维歧管的拓扑与差异几何形状密切相关，尤其是均匀的结构类似于类似的欧几里得，球形和双曲线几何形状。 其中，双曲线几何形状最为普遍。 在封闭的3个manifold上，双曲线结构（如果存在）是由theSostow刚度定理唯一的。 因此，任何几何不变剂都是久经式不变的，可用于区分不同的3个manifolds。 当3个manifold具有边界时，它可以具有双曲线结构的Alarge参数空间。在这种情况下，由于参数空间的变化，结构的质量质量特性特别感兴趣。 Kerckhoff研究在三个manifolds上的灌注液结构，试图描述这些几何结构的行为，在拓扑过程中，称为Dehn手术，以及在锥体上建模的呈现的存在。 R.L. Cohen正在追求几个项目，其中涉及研究由几何形状造成的代数拓扑方面。 这些项目位于标题下。 1。复杂和符号歧管中的曲线曲线的模量空间的拓扑。 2。稳定映射类组和芒福福德注射器的荷势类型。 3。Holomorphick理论的属性和应用。 R.J.米尔格拉姆将提出涉及有限简单组的杂物学的问题，以及配置空间拓扑的应用到机器人臂的设计和行为中的问题。