Global and Local Noncommutative Geometry

现代物理学的一个基本挑战是协调在宏观水平上将时空作为连续体的处理与在亚原子尺度上观察到的离散量子特征。寻找一种能够以统一的方式处理这些双重方面的数学装置，激发了非对易几何的基础。在这个年轻的数学领域中，由数值坐标标记的点形成的空间的经典概念被一个定性的更一般的范式所取代，其中坐标代数由有界算子（可观量）组成，这些算子在自然乘法下不一定是可交换的。非对易空间的几何被编码在一个无界算子中，其逆算子扮演着经典几何中的线元的角色，它以有界的方式与观测量相互作用。由于天生的空间直觉由于交换性的丧失而变得模糊，甚至最基本的几何概念，如局部性、对称性和曲率，也不得不经历剧烈的概念变形。有用的提示来自物理学，其中前两个出现在量子场论中，分别作为高能极限和规范对称，而最后一个是爱因斯坦广义相对论中引力的表现。本项目旨在加深对上述三个概念在非对易框架中的概念理解，并开发有效的数学工具对其进行定量测量。该项目的第一部分将在首席研究员和合作者最近工作的基础上，将C*-动力系统的增强伪微分应用于一系列涉及具体非交换空间（如任意维的非交换环面）曲率计算的问题。第二个子项目将改进椭圆算子和Alexander-Spanier上循环之间的高指数配对，并将其扩展到定义在有界伪微分算子代数上的配对。特别是，这将产生一个升级版本的Helton-Howe积分公式的轨迹的全反对称交换子的n元组的Toeplitz运营商，促进它的一个成熟的指标定理。它还将扩展Perrot的公式的周期循环上同调类的Radul上循环，以及消失的周期循环上同调类的Wodzicki剩余。第三个项目涉及一个新的霍普夫代数K（n），提出了直接作用于非交换空间的叶的任何叶状，而不是其框架丛的优势。它的Hopf循环上同调捕获所有横陈类，但错过了次要类。我们的目标是构建一个增强的，拓扑版本的霍普夫代数K（n），其霍普夫循环上同调将作为一个通用的容器的所有几何特征类的叶理。此外，该项目的这一部分旨在表达这些类的显式的霍普夫循环上圈和采用具体的表示，以获得几何和拓扑的后果。

A fundamental challenge of modern physics is to reconcile the treatment of space-time as a continuum at the macroscopic level with the discrete, quantum features observed at the subatomic scale. The search for a mathematical apparatus capable of handling these dual aspects in a unified manner motivated the foundation of noncommutative geometry. In this young field of mathematics, the classical concept of a space formed of points labeled by numerical coordinates is replaced by a qualitatively more general paradigm, in which the algebra of coordinates consists of bounded operators (observables) that do not necessarily commute under the natural multiplication. The geometry of a noncommutative space is encoded in an unbounded operator, whose inverse plays the role of the line element in classical geometry, which interacts with the observables in a bounded fashion. With innate spatial intuition rendered inoperative by the forfeit of commutativity, even the most basic geometric notions, such as locality, symmetry, and curvature, have to undergo a drastic conceptual metamorphosis. Helpful hints come from physics, where the first two appear in quantum field theory as the high-energy limit and gauge symmetry, respectively, while the last is the very manifestation of gravity in Einstein's general relativity. This project aims to deepen the conceptual understanding of the above triad of notions in the noncommutative framework, as well as to develop effective mathematical tools for their quantitative measurement. Besides relevance for physics, the envisaged developments hold a proven potential of having applications in geometry and topology.The first part of the project, building on recent work of the principal investigator and collaborators, will apply the enhanced pseudodifferential calculus for C*-dynamical systems to a host of problems involving curvature calculations for concrete noncommutative spaces, such as the noncommutative tori of arbitrary dimension. The second subproject will refine the higher index pairing between elliptic operators and Alexander-Spanier cocycles on a closed manifold and extend it to a pairing defined over the algebra of bounded pseudo-differential operators. In particular, this will yield an upgraded version of the Helton-Howe integral formula for the trace of the totally antisymmetric commutator of an n-tuple of Toeplitz operators, promoting it to a full-fledged index theorem. It will also extend Perrot's formula for the periodic cyclic cohomology class of the Radul cocycle, as well as the vanishing of the periodic cyclic cohomology class of the Wodzicki residue. The third project concerns a new Hopf algebra K(n) that presents the advantage of acting directly on the noncommutative space of leaves of any foliation rather than on its frame bundle. Its Hopf cyclic cohomology captures all transverse Chern classes, but misses the secondary classes. The goal is to construct an enhanced, topological version of the Hopf algebra K(n), whose Hopf cyclic cohomology will serve as a universal receptacle for all the geometric characteristic classes of foliations. In addition, this part of the project seeks to express these classes in terms of explicit Hopf cyclic cocycles and employ the concrete representations to derive geometric and topological consequences.