Semi-classical spectral estimates for magnetic Laplacians.

We consider semi-classical estimates for moments of eigenvalues of
magnetic Laplacians on bounded domains. We prove that, in the magnetic case, Polya's conjecture is violated even for tiling domains. We present some new bounds which go beyond semi-classics and which allow to prove stability of relativistic matter with magnetic fields for nuclear charges up to the critical value.

This talk is based on joint works with M. Loss and T. Weidl and with E. H. Lieb and R. Seiringer.