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，首席研究员：Markus J. Pflaumthe拟议的工作将通过非交通性几何形状来推动奇异性的研究。在数学的各个领域，具有奇异性的空间显得很自然。开发用于研究平滑歧管或平滑品种的标准方法通常不扩展到单数环境，因此必须开发新方法。 最有希望的新提案将为奇异理论提供进步的最初和原始提案是确定在具有奇异性空间的功能代数的循环同源性理论的想法。 这是从非交通性几何形状的观点，它可以追溯到A. Connes的作品，事实证明，它不仅提供了更深入的数学见解，不仅是非交通性的结构理论，而且还提供了交换代数的结构理论。除了在奇异空间上的函数代数的循环同源性外，PI计划将奇异空间分层理论的最新结果与非交通性几何形状结合起来，以打开检查奇异性的新路径。通过这种方法，新的奇异性拓扑不变的构建也有望为索引理论提供奇异性空间的进步。 特别是，它旨在定义与适当的li子类固醇相关的惯性空间，并研究其奇异性结构，目的是构建一种数学设备，以跟踪奇异性对卷积代数对适当lie群体的循环同源性的贡献。 最后，相对循环的共同体学理论将用于在奇异情况下构建和描述几何算子的次级不变性。单一性理论是数学学科，其中描述和研究包含所谓奇异性的几何对象，例如corners，edges或dertices。除了这些相当基本的奇点外，不仅在数学本身中，而且在许多物理或技术应用中都出现了相当复杂的奇异点，例如水力动力学，弦乐理论，机器人技术或灾难理论，在“灾难性”现象中在Laser物理学或种群动力学中对“灾难性”现象的理论理解在理论上起着基本作用。 因此，对奇点的更好数学理解不仅会导致数学内的进步，而且还将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（即非共同的几何形状）联系起来来改善对奇异性的基础知识。可以预见的是，可以构建这种新的数学不变性。这将提供进一步的至关重要的步骤，以在数学，科学或工程中出现奇异性分类。 为了加强项目的更广泛影响，PI计划通过专门设计用于传播数学知识的网站来呈现奇异性的可视化。