**Preface**
The electron liquid paradigm is at the basis of most of our current understanding of the physical properties of electronic systems. Quite remarkably, the latter are nowadays at the intersection of the most exciting areas of science: materials science, quantum chemistry, nano-electronics, biology, and quantum computation. Accordingly, its importance can hardly be overestimated. This course of the “Enrico Fermi” School of Physics was designed as a follow-up to a previous event in Varenna (Highlights of Condensed Matter Physics, 1983) in which Bardeen, Anderson, Schrieffer, Kohn, Pines, Peierls, Overhauser and others “went at it” about the electron gas in front of an audience of awed students, among which were these two directors. During the past 20 years the field has witnessed momentous developments, which are partly covered in this new volume. Advances in semiconductor technology have allowed the realizations of ultra-pure electron liquids whose density, unlike that of the ones spontaneously occurring in nature, can be tuned by electrical means, allowing a systematic exploration of both strongly and weakly correlated regimes. Most of these system are two or even one dimensional, and can be coupled together in the form of multi-layers or multi-wires, opening vast observational possibilities. The contributions by Nitin Samarth, Vladimir Pudalov, Sheena Murphy, Jun Zhu, Bilal Tanatar, and Francois Peeters in this volume will give the reader an up-to-date description of the wonderful progress that has been made in this direction. On the theoretical side, quantum Monte Carlo methods (authoritatively reviewed in the chapter by David Ceperley) have allowed an essentially exact determination of the ground-state energy of the electron liquid, and have provided partial answers to the still open question of the structure of its phase diagram. The Landau theory of Fermi liquid has been fully vindicated by detailed and often painstaking microscopic calculations and measurements, and has been found to hold far beyond its original naive field of validity, e.g. in complex systems such as heavy-fermion materials (see articles by Hans-Rudolf Ott and Peter Fulde). Renormalization group, as described in the chapters by Maurice Rice, and by Carlo Di Castro and Roberto Raimondi, has been developed into a powerful method for the exploration of novel states of matter. The interplay of disorder and correlation in the electron liquid (including the possibility of a metal-insulator transition in two dimensions) poses a formidable challenge to the theory: the status of this exceptionally difficult problem is reviewed in the chapter by Carlo Di Castro and Roberto Raimondi, which guides the reader through the intricacies of the diagrammatic renormalization group approach.

The emergence of the ground-state density functional theory (here reviewed in the chapter by Andrzej Holas) as the standard tool for the calculation of the electronic structure of matter has anointed the electron liquid as the holder of the prototypical correlations in electronic systems. In a more recent development, the time-dependent density functional theory (chapter by Miguel Marques and Eberhard Gross) coupled with recent advances in our understanding of dynamical correlations in the electron liquid (chapters by Mario Tosi and Robert van Leeuwen) has emerged as one of the most promising approaches to the calculation of excitation energies. Of particular interest, in this area, is the application of density functional theory to superconductors, which allows for the first time a genuine inclusion of Coulomb interaction effects, as opposed to a merely phenomenological treatment of the latter.

Starting from the 1980s some truly revolutionary concepts have emerged, which are well represented in these proceedings: for example, the notion of fractionally charged excitations in one-dimensional systems and in the quantum Hall liquid, the Luttinger liquid model for one-dimensional systems and for the edges of a quantum Hall liquid (chapters by Matthew Grayson and Jainendra Jain), and the beautiful composite-fermion theory of the quantum Hall effect (Jainendra Jain and John Quinn). These concepts transcend the traditional Landau picture of the interacting electron liquid as the continuation of the noninteracting one. What makes these developments particularly significant is the fact that the new scenarios have been found to emerge in the low-energy and low-temperature limit, subverting a traditional wisdom which saw in the high-energy limit the true frontier of physics.

This course would not have taken place without the early stimulus and the steady encouragement from the President of the Italian Physical Society (SIF), our former advisor Prof. Franco Bassani: we are indebted to him in more than one way. To the whole staff of the SIF, Barbara Alzani, Giovanna Bianchi Bazzi, Ramona Brigatti, Guglielmo Comini, and Marcella Missiroli, a special thank for making this course a wonderful experience, quite often making up for our organizational weakness with their expertise, kindness, and solicitude. In the end, the success of this course will be measured by the enthusiasm it has instilled in the young attendees: we expect them to come back to Varenna with their own story to tell sometime in the Summer of 2023.

G. F. Giuliani and G. Vignale