In 1947 Bogoliubov proposed a heuristic theory to compute the low energy spectrum of weakly interacting bosons, predicting a linear dispersion relation for small momenta. A derivation of this property - which has important consequences from the point of view of physics - has been for long beyond the reach of rigorous analysis. In this talk I will present recent tools allowing to establish the validity of Bogoliubov predictions for a system of interacting bosons in the Gross-Pitaevskii limit, a scaling regime which is particularly relevant for the description of quantum fluids. Remarkably, similar tools are being successfully applied to different scaling regimes and dimensionalities.

Based on joint works with G. Basti, C. Boccato, C.Brennecke, C. Caraci and B.Schlein.