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.

项目编号：dms -0305712项目负责人：R.L. Cohen， S.P. Kerckhoff， R.J. milgramm该项目包括代数和几何拓扑领域的广泛研究项目。主要研究者，R.L.科恩教授，S.P. Kerckhoff教授和r.j.。Milgram是高级拓扑学家，他们的研究兴趣涉及拓扑学和相关领域的大量领域。这些项目包括对核心拓扑问题的研究，如三维几何的研究，流形上复杂结构和相关结构的拓扑研究，以及有限群的拓扑不变量。此外，它还包括拓扑方法在机器人领域工程问题中的应用。三维流形的拓扑结构与微分几何密切相关，特别是与欧几里得几何、球面几何和双曲几何等齐次结构密切相关。其中，双曲几何是最普遍的。在一个封闭的3流形上，双曲结构，如果它存在，根据mostow刚性定理是唯一的。因此，任何几何不变量都是拓扑不变量，可以用来区分不同的3流形。当三维流形具有边界时，它可以具有较大的双曲结构参数空间；在这种情况下，结构的性质随参数空间的变化是特别有趣的。Kerckhoff研究了3-流形上的双曲结构，试图在一种叫做Dehn手术的拓扑程序下描述这些几何结构的行为，并且存在以锥体为模型的奇点。R.L. Cohen正在进行几个项目，涉及研究源自几何的问题的代数拓扑方面。这些项目可归入以下总标题。1. 复辛流形中全纯曲线模空间的拓扑。2. 稳定映射类群的同伦类型与mumford猜想。3. 全纯理论的性质及应用。R.J. Milgram将研究有限单群的同调问题，以及构型空间拓扑在机械臂设计和行为中的应用。