Local and global invariants in Noncommutative Geometry

该项目的综合目标是实现对非交换空间的局部不变量以及局部和全局属性相互作用和相互制约的方式的概念性理解。主要的兴趣是内禀曲率的概念，它是经典几何的核心，但在很长一段时间内在非对易几何中仍然是无形的。本项目的主要目标之一是建立在这一方向的最新进展，这导致了非对易环面的共形几何的非么模特征的发现。其他主题涉及继续发展新的技术计算霍普夫代数循环上同调，并将其应用于确定明确的表达式的横向特征类的一大类非交换空间发生在几何和数论;调查的不变量称为高指数流形的边界，以阐明其作用，在检测几何和拓扑这些空间的性质;在非对易几何和数论之间的界面上，与几个公开问题有关的所得结果的潜在算术含义的探索。在非对易几何中，空间作为由数值坐标标记的点形成的流形的范例被更一般的性质所取代，其中坐标是算子值的，并且可能不再对易，就像量子物理学一样。这一新的基本原理大大丰富了可被视为几何空间的对象类别，并已在数学和理论物理中得到了惊人的应用。虽然最基本的代数，拓扑和分析工具的全球治疗这样的空间已经成功地开发，当地的治疗，甚至在没有根深蒂固的空间概念化的地方的意义，仍然是非常正在建设中。它真正的概念性理解，这使得这个项目的主要目标，产生新的应用到当代数学的几个中心领域，并很可能在粒子和高能物理学的最新发展相当相关。

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.