Studies in Noncommutative Geometry

提出的研究涉及利用非交换几何的概念和方法的统一能力来解决影响数学广泛领域的几个问题，从微分拓扑和几何到算子代数和数学物理。与A. Connes、H. Moscovici合作，将他们的非交换局部指标形式论应用于微分同态等变特征类的研究，特别是将Gelfand-Fuchs上同调与Atiyah-Singer指标定理的微分同态等变推广联系起来。第二个主题是将直陈氏特征构造应用到伪泛形的特征类理论中，以便发展一种综合的方法，既包括光滑情况下的经典陈氏-魏尔理论，也包括组合庞特里亚金类的Cheeger公式。第三个项目的重点是调查Connes的非交换流形的概念，并最终导致根据在普通流形上执行的一些基本非交换操作对这些对象进行完整的分类。非对易几何是一个新兴的数学领域，植根于经过时间考验的量子力学算子形式，在著名的法国数学家阿兰·康内斯的愿景中，它提供了一个统一的几何框架，能够将离散与连续同等对待，并将爱因斯坦的广义相对论与普朗克尺度的时空结构等同对待。

ABSTRACT Moscovici The proposed research involves employing the unifying power of the concepts and methods of noncommutative geometry to address several problems which affect broad areas of mathematics, ranging from differential topology and geometry to operator algebras and mathematical physics. In collaboration with A. Connes, H. Moscovici will apply their noncommutative local index formalism to the study of diffeomorphism-equivariant characteristic classes, in particular connecting the Gelfand-Fuchs cohomology with the diffeomorphism-equivariant extension of the Atiyah-Singer index theorem. A second theme concerns the application of the straight Chern character construction to the theory of characteristic classes of pseudomanifolds, in order to develop a comprehensive approach encompassing both the classical Chern-Weil theory, in the smooth case, as well as Cheeger's formulae for the combinatorial Pontryagin classes. The third project focuses on the investigation of Connes' notion of noncommutative manifold and should ultimately lead to a complete classification of such objects in terms of a few basic noncommutative operations performed on ordinary manifolds. Noncommutative geometry is a newly emerging field of mathematics, rooted in the time-tested operator formalism of quantum mechanics, which -- in the vision of the renowned French mathematician Alain Connes -- provides a unified geometric framework able to treat the discrete on equal footing with the continuum as well as Einstein's general relativity on a par with the Planck-scale structure of space-time.