Noncommutative Differential Geometry of Deformations of Commutative Rings

本文主要研究光滑流形的变形量化问题。最近，Kontsevich对所有这些量子化进行了分类。这种分类是关于Hochschild协链代数的一般形式定理的一个结果。本文提出了这个定理的一个类比。这个类比，以及它的各种推广，是关于Hochschild和循环复合体的定理。所提出的一般形式性定理的结果包括：一般泊松流形的A^类的定义；一般等变指标定理；带痕迹的变形分类。另一个相关的研究方向是研究具有全纯辛结构的复流形的变形量子化与3-流形的Rozansky-Witten不变量之间的联系。研究量化有两个相关的动机。一个来自量子力学。微分方程的另一个动机。将Kontsevich和Tamarkin方法结合变形方法应用于理论非交换微分几何，得到新的几何和拓扑不变量。

TSYGAN, 9970591The propsed research is devoted to various questions of deformation quantization of smooth manifolds. Recently, Kontsevich classified all such quantizations. This classification is a consequence of a general formality theorem about Lie the algebra of Hochschild cochains. An analog of this theorem is proposed. This analog, and its various generalizations, are theorems about Hochschild and cyclic complexes. Among the consequences of the proposed general formality theorems are: definition of an A^ class of a general Poisson manifold; general equivariant index theorems; classification of deformations with trace. Another, related line of research studies the links between deformation quantization of complex manifolds with holomorphic symplectic structures and Rozansky-Witten invariants of 3-manifolds.There are two related motives for studying quantization. One come from quantum mechanics. The other motive from differential equations. The proposed research will be devoted to the applications of methods of Kontsevich and Tamarkin combined with deformation methods to the theory noncommutative differential geometry, to obtain new geometrical and topological invariants.