Noncommutative Differential Geometry of Deformations of Commutative Rings

TSYGAN，9970591本文研究光滑流形的形变量子化的各种问题。 最近，Kontsevich对所有这些量子化进行了分类。 这种分类是一个后果的一般形式定理李代数的Hochschild上链。 提出了这个定理的一个类比。 这个类比和它的各种推广是关于Hochschild和循环复形的定理。 在建议的一般形式定理的后果是：一般泊松流形的A^类的定义;一般等变指标定理;分类的变形与跟踪。 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形变量子化和三维流形的Rozansky-Witten不变量之间的联系。 一个来自量子力学。另一个动机来自微分方程。 本论文的主要工作是将Kontsevich和Tamarkin的方法与形变方法相结合，应用于非交换微分几何理论，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.