## 1 Introduction  
Ultracold atomic gases have been the cornerstone of atomic physics since 1995 [1; 2], when Bose-Einstein condensation was experimentally achieved for the first time. This groundbreaking result was the starting point of an ongoing and intense research effort to improve atoms trapping and cooling procedures [3]. Nowadays, modern laboratories have reached an exquisite control on the relevant physical parameters for atomic gases, such as the number density or the strength and range of atom-atom interaction. At the same time, it is also possible to tune the coupling with the external environment, enabling the observation of quantum dynamics without spurious smearing effects [4]. It is then natural to think of ultracold atomic gases as an extremely effective platform to probe quantum theories up to the macroscopic scale. One of the most striking examples is given by the Bose-Hubbard model and its superfluid-insulator transition originally predicted in [5]. This quantum phase transition was first observed in 2002 by loading a repulsive condensate in an optical lattice confining potential [6]. Similar achievements have been reached for low-dimensional quantum gases and their remarkable phenomenology [7; 8]. Among a wide and interesting literature, it is worth mentioning the first direct observation of the Tonks-Girardeau regime in an optical lattice filled with alkali atoms [9; 10] or the vortex proliferation associated with the Berezinskii-Kosterlitz-Thouless physics in two-dimensional setups [11; 12]. More recently, a growing interest has aroused on cold atoms as a promising quantum simulation platform. Together with the refined experimental expertise, it has been shown that, within peculiar setups, the superfluid dynamics of atoms obeys equations that can be mapped to other ones appearing in high-energy physics or cosmology. For instance, Rabi-coupled bosonic mixtures seem to be a burgeoning candidate to explore inflationary dynamics and space-timeexpansion in table top experiments [13; 14; 15]. Other proposals involve cold-atoms analog of quantum gravity [16] or the implementation of system mimicking quantum chromodynamics features [17]. Restraining ourselves to the condensed matter domain, ultracold atoms offer the interesting possibility to investigate the intriguing issue of quantum transport in a disordered environment [18]. Indeed, from the dawn of solid state theory, it was clear that a well-founded investigation has to consider the multifaceted role played by microscopic impurities and disordered external potential. For instance, a common incandescent lamp would not exist without a certain degree of impurity scattering. At the same time, it is reasonable to expect a smearing of experimental data due to randomly wrinkled substrate or the destruction of supertransport properties. Examples of this _parasitic_ role can be traced back to dirty superconductors or \({}^{4}\)He diffusion in porous medium [19; 20]. The complexity of this problem lies in the interplay between superfluidity (i.e., the occuring of some kind of ordered phase) and the localization phenomenon first theorized by [PERSON] in [21]. Given the broad interest and the cross-disciplinary relevance of this problem, it is reasonable to turn towards cold atoms, since they offer reliable experimental protocols to engineer disordered external potential and control the corresponding relevant parameters. The _dirty bosons_ problem then gathers all the investigations addressing the properties of ultracold bosons subject to an external random potential. The interplay between superfluidity and localization may lead to the observation of a new phase, the so-called Bose glass [5]. Similarly to the Bose-Hubbard model in the absence of disorder, optical lattices have been proved again to be a reliable setup to explore these exotic features [22].