Noncommutative Invariants of Singularities and Application to Index Theory

摘要奖：DMS 1105670，首席研究员：Markus J.Pflum这项工作将通过非对易几何推动奇点的研究。具有奇点的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推广到奇异情形，因此必须开发新的方法。确定奇点空间上函数代数的循环同调理论是最有希望的新的和原创性的提议之一，它将为奇点理论提供进展。这是非对易几何的观点，这可以追溯到A.Connes的工作，它不仅为非对易代数的结构理论提供了更深刻的数学见解，也为对易代数的结构理论提供了更深刻的数学见解。除了计算奇异空间上函数代数的循环同调外，PI还计划将奇异空间分层理论的最新结果与非对易几何相结合，以开辟研究奇点的新途径。用这种方法构造新的奇点拓扑不变量也有望为奇点空间上的指数理论提供新的进展。特别地，它的目的是定义与真李群胚相关的惯性空间，并研究它们的奇点结构，目的是构造一种跟踪奇点对真李群胚上卷积代数的循环同调的贡献的数学工具。最后，利用相对循环上同调理论来构造和描述奇异情形下几何算子的二次不变量。奇点理论是描述和研究包含所谓奇点的几何对象(如角、边或顶点)的数学学科。除了这些相当初级的奇点之外，更复杂的奇点不仅出现在数学本身，而且还出现在许多物理或技术应用中，例如流体动力学、弦理论、机器人学或突变理论，后者在从理论上理解激光物理学或种群动力学中的“灾难性”现象方面发挥了基础性作用。因此，对奇点的更好的数学理解不仅会导致数学上的进步，而且会在出现奇点现象的情况下对理论物理或工程产生影响。该项目的目的是通过将奇点理论与另一种现代数学理论--非对易几何--联系起来，提高奇点的基础知识。可以预料的是，这样可以构造新的奇点的数学不变量。这将为对数学、科学或工程中出现的奇点进行分类提供进一步的关键步骤。为了加强该项目的更广泛影响，PI计划通过一个专门设计的网站来展示奇点的可视化，以传播数学知识。

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.