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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.

Physics. --- Quantum Field Theories, String Theory. --- Mathematical Physics. --- Functional Analysis. --- Functional analysis. --- Physique --- Analyse fonctionnelle --- Geometric quantization. --- Poisson algebras. --- Poisson manifolds. --- Quantum groups. --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Lie algebras. --- Algebras, Lie --- Mathematical physics. --- Quantum field theory. --- String theory. --- Algebra, Abstract --- Algebras, Linear --- Lie groups --- Differentiable manifolds --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Physical mathematics --- Models, String --- String theory --- Nuclear reactions --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Mathematics