Global and Local Noncommutative Geometry

现代物理学的一个根本挑战是在宏观层面上将时空视为一个连续体，而在亚原子尺度上观察到离散的量子特征。寻找一种能够以统一方式处理这些对偶问题的数学工具，激发了非对易几何的基础。在这个年轻的数学领域，由数值坐标标记的点组成的空间的经典概念被一个定性的更一般的范例所取代，在这个范例中，坐标的代数由有界的算子(可观测)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成一个无界算子，它的逆在经典几何中扮演着线元素的角色，它以有界的方式与可观察到的东西相互作用。由于先天的空间直觉由于丧失了交换性而变得不起作用，即使是最基本的几何概念，如局部性、对称性和曲率，也必须经历剧烈的概念蜕变。有用的线索来自物理学，前两者在量子场论中分别以高能极限和规范对称性的形式出现，而后两者在爱因斯坦的广义相对论中正是引力的表现。这个项目旨在加深对非对易框架中上述三个概念的概念性理解，并为其定量测量开发有效的数学工具。该项目的第一部分将在主要研究人员和合作者最近的工作基础上，将C*动力系统的增强伪微积分应用于一系列涉及具体非对易空间的曲率计算的问题，例如任意维的非对易环面。第二个子项目将改进闭流形上椭圆算子和Alexander-span ier余圈之间的高指数配对，并将其推广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，用于计算n元Toeplitz算子的完全反对称交换子的迹，从而将其推广到一个完全成熟的指数定理。推广了Perrot关于Radul上循环的周期循环上同调类的公式，以及Wodzicki剩余的周期循环上同调类的消失。第三个项目涉及一个新的Hopf代数K(N)，它的优点是直接作用于任何叶化的非交换叶空间，而不是作用于它的框架丛。它的Hopf循环上同调捕获了所有的横向陈氏类，但遗漏了次要类。我们的目标是构造一个增强的Hopf代数K(N)的拓扑版，它的Hopf循环上同调将作为所有几何特征类的叶的通用容器。此外，项目的这一部分试图用显式的Hopf循环余圈来表示这些类，并使用具体的表示来推导几何和拓扑结果。

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.