Studies in Noncommutative Geometry

摘要莫斯科维奇提出的研究涉及使用非对易几何的概念和方法的统一力量来解决几个影响数学的广泛领域的问题，从微分拓扑和几何到算子代数和数学物理。在与A.Connes的合作中，H.Moscovici将把他们的非对易局部指数形式应用于微分同胚等变特征类的研究，特别是将Gelfand-Fuchs上同调与Atiyah-Singer指数定理的微分同变扩张联系起来。第二个主题是将直陈特征结构应用到伪流形的特征类理论中，以便发展一种综合的方法，既包含光滑情形下的经典的Chern-Weil理论，又包含组合Pontryagin类的Cheeger公式。第三个项目集中于对Connes的非对易流形概念的研究，最终应该通过对普通流形上的几个基本的非对易运算来对这类对象进行完整的分类。非对易几何是一个新兴的数学领域，植根于久经考验的量子力学算符形式主义，在法国著名数学家阿兰·康纳斯的愿景中，该理论提供了一个统一的几何框架，能够与连续统平等地对待离散，以及与普朗克尺度的时空结构同等对待爱因斯坦的广义相对论。

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.