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This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem.It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area .The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.