This talk will be a review of open questions and some recent results on the
(classical and quantum) Uniform Electron Gas (UEG), which is a fundamental
model in quantum chemistry. The UEG is a gas of electrons placed in such a way
that their density is constant everywhere in space.
We will in particular compare this model with the one-component plasma
(Jellium), which has a constant positive background, but no special constraint
on the electronic density. Jellium is believed to play a central role for
random matrices and random Schrödinger operators. In some cases Jellium is the
same as the UEG, but in some other cases, these could be different.
Results in collaboration with Elliott H. Lieb (Princeton) and Robert Seiringer
(Vienna).