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The notion of renormalization is at the core of several spectacular achievements of contemporary physics. This book provides an introduction to the sophisticated tools used in the theory of non-perturbative renormalization, allowing a unified and rigorous treatment of Quantum Field Theory, Statistical Physics and Condensed Matter models.

This text is based on the lectures given at the CIME Summer Centre 'Quantum Many Body Systems' held in the summer of 2010. It provides a tutorial introduction to advances in the mathematics of interacting systems, written by four leading experts in the field.