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.

Contents: Introduction to Renormalization: Basic Notions; Fermionic Functional Integrals; Quantum Field Theory: The Ultraviolet Problem in Massive QED2; Infrared Problem and Anomalous Behavior; Ward Identities and Vanishing of the Beta Function; Thirring and Gross-Neveu Models; Axioms Verification and Wilson Fermions; Infrared QED4 with Large Photon Mass; Lattice Statistical Mechanics: Universality in Generalized Ising Models; Nonuniversality in Vertex or Isotropic Ashkin-Teller Models; Universality-Nonuniversality Crossover in the Ashkin-Teller Model; Quantum Liquids: Spinless Luttinger Liquids; The 1d Hubbard Model; Fermi Liquids in Two Dimensions; BCS Model with Long Range Interaction.