  We review some rigorous estimates for the ground state energy of dilute Bose
gases. We start with Dyson's upper bound, which provides the correct leading
order asymptotics for hard spheres. Afterwards, we discuss a rigorous version
of Bogoliubov theory, which recently led to an estimate for the ground state
energy in the Gross-Pitaevskii regime, valid up to second order, for particles
interacting through integrable potentials. Finally, we explain how these ideas
can be combined to establish a new upper bound, valid to second order, for the
energy of hard spheres in the Gross-Pitaeavskii limit. Here we only sketch the
main ideas, details will appear elsewhere.
