Abstract: In the last decades, since the first experimental realizations of Bose- Einstein condensates, the study of large bosonic systems has been a very active field of research both in physics and in mathematics. In experiments Bose gases are often very dilute and can be well described in the Gross-Pitaevskii limit, i.e. as quantum systems of N confined particles, interacting through a potential with scattering length of order 1/N where N tends to infinity. In this talk we present a recent result on a hard-sphere Bose gas in this regime. Namely, we prove a second order upper bound on the ground state energy matching the known expression of the energy for integrable potentials. We also discuss a new upper bound for hard-spheres in the thermodynamic limit, where the number of particles and the size of the box are sent to infinity keeping the density fixed. Our result resolves the ground state energy up to an error of the order of the so-called Lee-Huang-Yang correction. Based on joint works with S. Cenatiempo, A. Giuliani, A. Olgiati, G. Pasqualetti, B. Schlein. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.]