In this talk I will give an overview of recent progress regarding mathematical questions and proofs of the spectral properties of Haldane pseudo-potentials. These are short range interactions projected onto the lowest Landau level and are tailored to model properties of fractional Hall fluids and their excitations as well as rotating Bose gases. I will sketch a strategy which establishes the spectral gap in a truncated system which describes a thin-cylinder limit.  (Based on joint works with B. Nachtergaele and A. Young)