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This book is a survey of the theory of formal deformation quantization of Poisson manifolds, in the formalism developed by Kontsevich. It is intended as an educational introduction for mathematical physicists who are dealing with the subject for the first time. The main topics covered are the theory of Poisson manifolds, star products and their classification, deformations of associative algebras and the formality theorem. Readers will also be familiarized with the relevant physical motivations underlying the purely mathematical construction.

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Introduction.- Classical mechanics and Poisson structures.- Deformation quantization and formality theory.- Kontsevich s Formula and Globalization.- Appendix.- Vector bundles.- Cohomology.- Index.

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"The formality theorem gave deep insight into the homological algebra of smooth functions on a manifold. Many applications have grown out of this investigation, too many to mention here. ... the author explains its implications together with its origins in the theory of quantization. This is a valuable contribution since the formulation of the statement requires some sophisticated preparation, which is carefully discussed in this booklet." (Stefan Waldmann, Mathematical Reviews, February, 2016)

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