One-dimensional quantum systems occupy a special place in many-body physics because they can be studied using a wide range of powerful theoretical methods. In particular, bosonization provides a remarkably simple framework to describe a broad class of systems known as Luttinger liquids, which include interacting fermions, bosons, and quantum spin chains. At low energies, these systems are governed by a quadratic Hamiltonian characterized by only two parameters: the velocity of the low-energy mode (i.e., the sound mode velocity in a Bose gas) and the Luttinger parameter, which encodes the interaction strength. The asymptotic behavior of the correlation functions is entirely determined by the Luttinger parameter, which thus sets the nature of the dominant fluctuations. The goal of this internship is to apply bosonization, together with the non-perturbative functional renormalization group (a modern formulation of the Wilsonian renormalization group), to study weakly coupled one-dimensional Bose gases. 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.