In this course we will review recent work on the different upper bounds for the ground state energy of a gas of interacting bosons in the thermodynamic limit. We start by considering quasi-free states to realize that these reproduce the leading order asymptotics of the energy but not the expected first correction term. This part is based on work by Erdos-Schlein-Yau and Napiorkowski-Reuvers-Solovej. Then we pass to the more involved trial states by Yau-Yin and prove that such a state can indeed give the expected correction term.
The course will essentially be elementary. Only a basic knowledge of mathematical quantum mechanics will be needed.