The Hubbard fermionic model : introduction and recent advances

A paradigm of the physics of interacting quantum systems, the Hubbard model has a status in this field similar to that of the Ising model in statistical physics. It is the simplest model to formulate, but one which we can hope will suffice to understand at least qualitatively certain collective phenomena such as magnetism, metal-insulator transitions or unconventional superconductivity. Despite its simplicity, this model represents a formidable theoretical challenge. After an introduction to the physical motivations - from materials with strong electronic correlations to cold atoms in optical lattices - this year's lecture will review the current state of our understanding of this model, particularly in two dimensions, and present the main methods that have led to recent progress or hold out hope for future advances.