Noncommutative Invariants of Singularities and Application to Index Theory

奇点的非交换不变量及其在指数理论中的应用

• 批准号: 1105670
• 负责人: Markus Pflaum
• 金额: $ 14.45万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105670&HistoricalAwards=false
• 关键词: Noncommutative; Invariants; Singularities; Application; Index

AbstractAward: DMS 1105670, Principal Investigator: Markus J. PflaumThe proposed work will advance the study of singularities by means of noncommutative geometry. Spaces with singularities appear abundantly and naturally in various areas of mathematics. Standard methods developed to study smooth manifolds or smooth varieties can in general not be extended to the singular setting, so one has to develop new approaches. Among the most promising new and original proposals which will provide progress for singularity theory is the idea to determine the cyclic homology theory of function algebras over spaces with singularities. This is the viewpoint from noncommutative geometry which goes back to the work of A. Connes and which has turned out to provide deeper mathematical insight not only into the structure theory of noncommutative but also of commutative algebras. In addition to the computation of cyclic homologies of function algebras over singular spaces, the PI plans to combine recent results from the stratification theory of singular spaces with noncommutative geometry to open up new paths to examine singularities. The construction of new topological invariants of singularities by this approach also promises to provide progress for index theory over spaces with singularities. In particular, it is intended to define inertia spaces associated to proper Lie groupoids and study their singularity structure with the goal of constructing a mathematical device which keeps track of the contribution of singularities to the cyclic homology of convolution algebras over proper Lie groupoids. Finally, relative cyclic cohomology theory will be used to construct and describe secondary invariants of geometric operators in singular situations.Singularity theory is the mathematical discipline in which one describes and studies geometrical objects containing so-called singularities such as corners, edges or vertices. Besides these rather elementary singularities, considerably more complicated ones appear not only in mathematics itself but also in many physical or technical applications like for example hydro dynamics, string theory, robotics or catastrophe theory, which plays a fundamental role in the theoretical understanding of "catastrophic" phenomena in laser physics or population dynamics. A better mathematical understanding of singularities therefore will not only lead to progress within mathematics but also will have its impact for theoretical physics or engineering in situations where singular phenomena appear. The proposed project aims at improving the foundational knowledge on singularities by connecting singularity theory to another modern mathematical theory, namely noncommutative geometry. It is to be expected that this way new mathematical invariants for singularities can be constructed. This will provide further crucial steps towards a classification of singularities as they appear in mathematics, the sciences or engineering. To strengthen the broader impact of the project, the PI plans to present visualizations of singularities via a website specifically designed to disseminate mathematical knowledge.

AbstractAward：DMS 1105670，主要研究者：Markus J. Pflaum拟议的工作将通过非对易几何推进奇点的研究。奇点空间在数学的各个领域中大量而自然地出现。研究光滑流形或光滑簇的标准方法一般不能扩展到奇异设置，因此必须开发新的方法。 其中最有前途的新的和原来的建议，这将提供进展的奇异性理论的想法，以确定循环同调理论的功能代数空间的奇异性。 这是非对易几何的观点，它可以追溯到A. Connes和这已成为提供更深层次的数学洞察力，不仅结构理论的非交换，但也交换代数。除了奇异空间上函数代数的循环同调的计算外，PI计划将奇异空间的分层理论与非交换几何的最新结果联合收割机结合起来，以开辟新的途径来研究奇异性。通过这种方法构造的新的拓扑不变量的奇异性也有望提供的进展指标理论的空间与奇异性。 特别是，它的目的是定义惯性空间相关联的正常李群胚和研究他们的奇点结构的目标是建立一个数学设备，跟踪贡献的奇点的卷积代数的循环同调正常李群胚。 最后，相对循环上同调理论将用于构造和描述几何算子在奇异情况下的次不变量。奇点理论是一门描述和研究包含所谓奇点（如角点、边或顶点）的几何对象的数学学科。除了这些相当基本的奇点之外，相当复杂的奇点不仅出现在数学本身中，而且出现在许多物理或技术应用中，例如流体力学，弦理论，机器人或突变理论，这在激光物理学或人口动力学中的“灾难”现象的理论理解中起着基础作用。 因此，对奇点的更好的数学理解不仅会导致数学的进步，而且会在出现奇异现象的情况下对理论物理或工程产生影响。 拟议的项目旨在通过将奇点理论与另一种现代数学理论，即非交换几何联系起来，提高奇点的基础知识。可以预期，通过这种方式，可以构造新的奇点数学不变量。这将为数学、科学或工程中出现的奇点分类提供进一步的关键步骤。 为了加强该项目的广泛影响，PI计划通过专门设计用于传播数学知识的网站来展示奇点的可视化。