Noncommutative Differential Geometry of Deformations of Commutative Rings

Northwestern University, Evanston IL

Investigators

Abstract

TSYGAN, 9970591 The 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.