Interacting bosons are unique quantum systems, whose low temperature phases exhibit fascinating quantum mechanics effects at a macroscopic scale. In the past two decades, the mathematical understanding of these systems improved tremendously. However, their behavior in the thermodynamic limit is still poorly understood, although this is the appropriate large scale limit to prove the emergence of scaling laws and universality, as well as to investigate the occurrence of phase transitions. MaTCh aims at investigating the low energy properties of interacting bosons in the thermodynamic limit, and at gaining a mathematical understanding of the emergence of correlated phases, in the form of Bose-Einstein condensation and quasi-long range order, as well as of their instabilities, due to thermal fluctuations or three-body recombination effects of Efimov type. Our plan is to exploit scaling limits as a framework to identify and overcome, one at a time, the mathematical obstructions that currently prevent us to control the system at finite density in the thermodynamic limit. In order to make progress on this program, MaTCh will introduce novel mathematical methods, inspired by renormalization group approaches and grounded in the second quantization techniques developed by the P.I. and collaborators, valid on an increasing sequence of scales. Ultimately, the research led by MaTCh will lay the foundation for the rigorous description of several phenomena which are at the frontiers of present theoretical and experimental research, where collective excitations of quantum systems are described in terms of emergent Bose gases, such as in the BCS theory for superconductivity, the molecular description of strongly interacting Fermi gases, and the spin-wave theory for quantum magnetism.