Global and Local Noncommutative Geometry

全局和局部非交换几何

• 批准号: 1600541
• 负责人: Henri Moscovici
• 金额: $ 33.17万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600541&HistoricalAwards=false
• 关键词: Global; Local; Noncommutative; Geometry

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.

现代物理学的一个基本挑战是调和在宏观层面上作为连续体的时空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对偶方面的数学装置激发了非交换几何的基础。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种定性上更普遍的范式所取代，在这种范式中，坐标的代数由有界算子（可观察值）组成，这些算子不一定在自然乘法下交换。非交换空间的几何被编码为无界算子，其逆在经典几何中扮演着线素的角色，它以有界的方式与可观测对象相互作用。由于交换性的丧失，天生的空间直觉变得无效，即使是最基本的几何概念，如局部性、对称性和曲率，也必须经历一个剧烈的概念变形。物理学提供了一些有用的提示，前两个在量子场论中分别作为高能极限和规范对称出现，而最后一个是爱因斯坦广义相对论中引力的表现。本项目旨在加深对非交换框架中上述三联概念的概念理解，并开发有效的数学工具来定量测量它们。除了与物理学相关外，设想的发展在几何和拓扑学中也有应用的潜力。该项目的第一部分，建立在首席研究员和合作者最近的工作基础上，将把C*动力系统的增强伪微分微积分应用于一系列涉及具体非交换空间曲率计算的问题，如任意维的非交换环面。第二个子项目将改进封闭流形上椭圆算子与Alexander-Spanier环之间的高指数配对，并将其推广到在有界伪微分算子代数上定义的配对。特别地，这将产生一个升级版本的Helton-Howe积分公式，用于n元Toeplitz算子的完全反对称对易子的迹，将其提升为一个成熟的指标定理。推广了Radul环的周期环上同类的Perrot公式，以及Wodzicki残基的周期环上同类的消失性。第三个项目涉及一个新的Hopf代数K(n)，它表现出直接作用于任何叶的叶的非交换空间而不是作用于其框架束的优势。它的Hopf环上同调捕获了所有的横向陈氏类，但忽略了次要类。目标是构造一个增强的Hopf代数K(n)的拓扑版本，其Hopf循环上同调将作为所有叶形的几何特征类的通用容器。此外，项目的这一部分试图用显式的Hopf循环来表达这些类，并使用具体的表示来推导几何和拓扑结果。