  Now that the low temperature properties of quantum-mechanical many-body
systems (bosons) at low density, $\rho$, can be examined experimentally it is
appropriate to revisit some of the formulas deduced by many authors 4-5 decades
ago. For systems with repulsive (i.e. positive) interaction potentials the
experimental low temperature state and the ground state are effectively
synonymous -- and this fact is used in all modeling. In such cases, the leading
term in the energy/particle is $2\pi\hbar^2 a \rho/m$ where $a$ is the
scattering length of the two-body potential. Owing to the delicate and peculiar
nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged
bosons), four decades of research failed to establish this plausible formula
rigorously. The only previous lower bound for the energy was found by Dyson in
1957, but it was 14 times too small. The correct asymptotic formula has
recently been obtained by us and this work will be presented. The reason behind
the mathematical difficulties will be emphasized. A different formula,
postulated as late as 1971 by Schick, holds in two-dimensions and this, too,
will be shown to be correct. With the aid of the methodology developed to prove
the lower bound for the homogeneous gas, two other problems have been
successfully addressed. One is the proof by us that the Gross-Pitaevskii
equation correctly describes the ground state in the `traps' actually used in
the experiments. For this system it is also possible to prove complete Bose
condensation, as we have shown. Another topic is a proof that Foldy's 1961
theory of a high density Bose gas of charged particles correctly describes its
ground state energy.
