RUI: Noncommutative Geometry: Curvature and Rigidity of Noncompact Manifolds

摘要奖：DMS-0405867首席研究员：Stanley S.该奖项资助的主要项目是探索非紧流形上正数量曲率的黎曼度量的存在性问题。 Gromov和Lawson关于这个问题的紧致情形的工作刺激了非交换几何和受控拓扑的发展的重要部分，这些项目是在稳定的Gromov-Lawson-Rosenberg猜想的背景下制定的。 principalinvestigator和合作者的目标是在非紧的情况下构建一个适当的assembly映射，在自然的例子中测试该理论，并与广义Roe algebrathe粗拟等距型流形，不承认正标量曲率的完整度量。 从较大的紧流形中删除子流形得到的非紧流形和紧流形的非Galois或不规则覆盖空间是很有希望的研究实例来源，而有限渐近维数的非紧流形将受到特别的关注。 其他的研究方向包括零的频谱猜想和expander graphs.非交换几何是通过自然函数和算子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可以在模拟振动的方程的解中得到，并通过出现在这些方程中的微分算子得到。 圆球是正曲率流形的例子，平面的曲率为零，山路上的鞍点是负曲率空间的模型。 数学家们已经发现了一些测试，必须通过任何空间，这将是一个候选人进行正曲率几何，这些项目将推进这一努力。 该提案中描述的另一项工作涉及扩展图，它最初是作为具有良好通信特性的大型网络模型引入计算机科学的，但也被证明是几何问题的潜在反例的来源。 这些都将成为研讨会的主题，研讨会的参与者包括来自韦尔斯利学院和附近机构的学生和教师。

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.