The goal of this internship is to apply bosonization, together with the non-perturbative functional renormalization group (a modern implementation of the Wilsonian renormalization group), to study weakly coupled one-dimensional Bose gases (Luttinger liquids). These highly anisotropic systems are expected to display a rich phase diagram, featuring superfluid, density-wave, and possibly supersolid phases. The internship will involve both analytical calculations and numerical solutions of renormalization-group flow equations. The project could be extended into a PhD thesis, focusing on Bose gases that deviate from the Luttinger-liquid paradigm. In particular, we will explore the effects of a periodic potential (leading to a superfluid–Mott-insulator transition, described by the sine-Gordon model) and of a random potential (which may give rise to a localized phase known as “Bose-glass”). In both cases, the non-perturbative functional renormalization group offers a powerful framework to determine the phase diagram and low-energy properties in the one-dimensional limit. Studying weakly coupled one-dimensional Bose gases, starting from the one-dimensional limit, is not only interesting in its own right but also provides a promising route to understanding the physics of two- and three-dimensional Bose gases, where direct approaches based on isotropic models are often difficult. Similar studies may be considered for Fermi gases.