Local and global invariants in Noncommutative Geometry

非交换几何中的局部和全局不变量

• 批准号: 1300548
• 负责人: Henri Moscovici
• 金额: $ 18万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300548&HistoricalAwards=false
• 关键词: Local; global; invariants; Noncommutative; Geometry

The comprehensive goal of this project is to achieve a conceptual understanding of the local invariants for noncommutative spaces and of the way the local and global properties interact and condition each other. Of primary interest is the concept of intrinsic curvature, which lies at the very core of the classical geometry but remained for a long time intangible in noncommutative geometry. One of the main thrusts of the present project is to build on the latest advances in this direction, which led to the uncovering of the nonunimodular character of the conformal geometry of the noncommutative torus. Other themes concern the continuation of the development of new techniques for computing Hopf algebra cyclic cohomology, and applying them to the determination of explicit expressions for the transverse characteristic classes of a large class of noncommutative spaces occurring in geometry and number theory; the investigation of the invariants called higher indices for manifolds with boundary in order to elucidate their role in detecting geometric and topological properties of these spaces; the exploration of the potential arithmetic implications of the obtained results in connection with several open problems at the interface between noncommutative geometry and number theory.In noncommutative geometry, the paradigm of space as a manifold formed of points labeled by numerical coordinates is replaced by one of a much more general nature, in which the coordinates are operator-valued and may no longer commute, as in quantum physics. This new foundational principle allows for a substantial enrichment of the class of objects that can be treated as geometric spaces and has already found spectacular applications in both mathematics and theoretical physics. While the most basic algebraic, topological, and analytical tools for the global treatment of such spaces have been already successfully developed, the local treatment, and even the meaning of locality in the absence of the deeply ingrained spatial conceptualization, is still very much under construction. Its truly conceptual understanding, which makes the primary object of this project, engenders new applications to several central fields of contemporary mathematics and is quite likely of considerable relevance for the latest developments in particle and high-energy physics.

这个项目的全面目标是实现对非对易空间的局部不变量以及局部和全局性质相互作用和条件的方式的概念性理解。最令人感兴趣的是本征曲率的概念，它位于经典几何的核心，但在非对易几何中很长一段时间是无形的。本项目的主要目的之一是在这一方向的最新进展的基础上，揭示非对易环面的共形几何的非么正特征。其他主题涉及计算Hopf代数循环上同调的新技术的继续发展，并将它们应用于确定几何和数论中出现的一大类非交换空间的横向特征类的显式表达式；研究有边界流形的称为高指数的不变量，以阐明它们在检测这些空间的几何和拓扑性质中的作用；在非对易几何和数论的交界处，对所得到的结果的潜在算术含义的探索与几个公开问题有关。在非对易几何中，空间作为由数值坐标标示的点形成的流形的范例被一种更一般的性质所取代，其中坐标是算符价值的，并且可能不再像在量子物理中那样可交换。这一新的基本原理允许大量丰富可以被视为几何空间的对象的类别，并且已经在数学和理论物理中找到了壮观的应用。虽然已经成功地开发了对这类空间进行全局处理的最基本的代数、拓扑和分析工具，但在没有根深蒂固的空间概念化的情况下，局部处理，甚至局部性的意义，仍然在很大程度上处于构建之中。它的真正概念性理解是本项目的主要目标，它在当代数学的几个中心领域产生了新的应用，并很可能与粒子和高能物理的最新发展有相当大的相关性。