RUI: Noncommutative Geometry: Curvature and Rigidity of Noncompact Manifolds

RUI：非交换几何：非紧流形的曲率和刚度

• 批准号: 0405867
• 负责人: Stanley Chang
• 金额: $ 9.45万
• 依托单位: Wellesley College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405867&HistoricalAwards=false
• 关键词: RUI; Noncommutative; Geometry; Curvature; Rigidity

AbstractAward: DMS-0405867Principal Investigator: Stanley S. ChangThe main projects funded by this award explore the existenceproblem for Riemannian metrics of positive scalar curvature onnoncompact manifolds. The work of Gromov and Lawson on thecompact case of this problem has stimulated important parts ofthe development of noncommutative geometry and controlledtopology and these projects are formulated within the context ofthe stable Gromov-Lawson-Rosenberg conjecture. The principalinvestigator and collaborators aim to construct an appropriateassembly map in the noncompact case, to test that theory innatural examples, and to study with the generalized Roe algebrathe coarse quasi-isometry type of manifolds that do admitcomplete metrics of positive scalar curvature. Noncompactmanifolds obtained by deleting submanifolds from larger compactmanifolds and non-Galois or irregular covering spaces of compactmanifolds are promising sources of examples for study, andnoncompact manifolds of finite asymptotic dimension will receiveparticular attention. Other directions of investigation includethe zero-in-the-spectrum conjecture and expander graphs.Noncommutative geometry is an approach to the study of geometricobjects through algebras of natural functions and operators. Forexample, much of the geometry of a sphere is captured insolutions to equations that model vibration, and by thedifferential operators that appear in those equations. Roundspheres are examples of manifolds of positive curvature, the flatplane has curvature zero, and the saddle point in a mountain passis a model for space of negative curvature. Mathematicians havefound a number of tests that must be passed by any space thatwould be a candidate to carry a geometry of positive curvature,and these projects will advance that effort. Another line ofwork described in the proposal concerns expander graphs, whichwere originally introduced in computer science as models of largenetworks with good communication properties but have also turnedout to be a source of potential counterexamples for geometricquestions. These will be the subject of seminars involvingstudents and faculty from Wellesley College and nearbyinstitutions.

摘要奖：DMS-0405867主要研究人员：Stanley S.Chang该奖项资助的主要项目探索正数量曲率非紧流形的黎曼度量的存在问题。Gromov和Lawson关于这个问题的紧情形的工作刺激了非对易几何和受控拓扑的重要部分的发展，这些方案是在稳定的Gromov-Lawson-Rosenberg猜想的背景下提出的。主要研究者和合作者的目标是在非紧的情况下构造适当的集合映射，检验这一理论的非自然例子，并用广义Roe代数研究具有正标量曲率的容许完备度量的粗拟等距流形。从较大的紧致流形中删除子流形得到的非紧致流形和紧致流形的非伽罗瓦或不规则覆盖空间是很有希望的例子来源，而有限渐近维非紧致流形将受到特别的关注。其他研究方向包括谱中零猜想和扩张图。非交换几何是通过自然函数和算子的代数来研究几何对象的一种方法。例如，球体的大部分几何形状都包含在建立振动模型的方程的解中，以及这些方程中出现的微分算符。圆球是正曲率流形的例子，平面的曲率为零，山中的鞍点是负曲率空间的模型。数学家们已经发现了许多测试，这些测试必须通过任何可能携带正曲率几何的空间，这些项目将推动这一努力。提案中描述的另一项工作涉及扩展图，它最初是在计算机科学中引入的，作为具有良好通信性能的大型网络的模型，但也被证明是几何问题的潜在反例。这将是卫尔斯理学院和附近机构的学生和教职员工参加的研讨会的主题。