The Mott metal-insulator transition is 60 years old. Despite major recent progress with the help of DMFT it retains mysteries. Ancient experiments on electron-hole droplets in germanium in silicon display a first order transition, similar to that found in the half filled fermion Hubbard model. What is the corresponding order parameter? How does one evolve from that first order regime to the simple 1949 Mott picture? The issue remains open. The recent discovery of supersolids extends the Mott transition to Bose liquids. A Bose Hubbard model with integer filling has a choice between superfluidity or localization at zero temperature, with a second order transition: as usual bosons are simpler than fermions! In practice we want a real crystal where weak crystallization is due to freezing of soft phonon modes: the challenge is to introduce localization into such a spontaneous solid. The seminar will try to put these questions in perspective. If time allows the issue of supersolidity in He4 will be discussed in some detail. It seems to occur only in defects , but it opens a whole new field.
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