The notion of renormalization is at the core of several spectacular achievements of contemporary physics, and in the last years powerful techniques have been developed allowing to put renormalization on a firm mathematical basis. This book provides a self-consistent and accessible introduction to the sophisticated tools used in the modern theory of non-perturbative renormalization, allowing an unified and rigorous treatment of Quantum Field Theory, Statistical Physics and Condensed Matter models. In particular the first part of this book is devoted to Constructive Quantum Field Theory, providing a mathematical construction of models at low dimensions and discussing the removal of the ultraviolet and infrared cut-off, the verification of the axioms and the validity of Ward Identities with the relative anomalies. The second part is devoted to lattice 2D Statistical Physics, analyzing in particular the theory of universality in perturbed Ising models and the computation of the non-universal critical indices in Vertex or Ashkin-Teller models. Finally the third part is devoted to the analysis of complex quantum fluids showing Luttinger of Fermi liquid behavior, like the 1D or 2D Hubbard model. Key Features • Serves as a self-consistent and accessible introduction to the powerful and sophisticated tools used in the modern theory of renormalization • Includes a unique implementation of exactly modified Ward identities in an approach based on multiscale analysis, which is built upon a recently developed technique allowing the rigorous analysis of models with non trivial fixed points and anomalous behavior • Focuses mainly on fermionic rather than bosonic integrals, unlike most books on renormalization; this has the effect of cleaning up the structure of renormalization • Presents a unified treatment of QFT, lattice spin models and quantum liquids • Presents a number of recent and important results, appearing up to now only in specialized journals, in an accessible manner • Is devoted to mathematicians or physicists aiming to understand this important field; prerequisites are thus limited to a minimum

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