Bose-Einstein condensation for two dimensional interacting bosons: mean field and Gross-Pitaevskii scalings

This thesis is concerned with static properties of large bosonic systems in two dimensions. These systems at very low-temperatures are expected to exhibit emph{Bose-Einstein condensation}. From a mathematical and physical, point of view it is interesting to provide conditions for the occurrence of Bose-Einstein condensation. Obviously, studying a system of N particles, where N is large, is very challenging. However, to overcome this problem we can rely on effective theories, which describe the collective behaviour of the particles. The aim of the manuscript is to present new results regarding the occurrence of Bose-Einstein condensation in two-dimensional bosonic systems in suitable scaling limits. Our first result consists of the rigorous derivation of complete Bose-Einstein condensation of low-energy states in a regime where the interaction potential scales as N^2bV(N^b ), for b >0 such that log (N^b) <