Let us consider a system of N electrons and M nuclei interacting by inverse-square forces. Under suitable temperature and pressure this system may become a crystal or a gas. The physical laws that govern such a system have been known for a long time. But why these states? Understanding why those laws lead to the observed behavior of matter is a mthematical challenge.
Embarked for a long time on a program to prove that electrons and nuclei can form atomic and molecular gases the author in this very interesting paper explains what he has succeeded in proving and what remains to be done.
His main theorem is that up to some accuracy there exists a temperature and density such that in a sufficiently large box the Gibbs measure describes a gas of hydrogen atoms. A very difficult result to prove.