In this talk, I will give an overview over recent results and open problems related to key properties of fractional Hall systems. The ground state properties of such systems are famously described by Laughlin wavefunctions. Curiously, the latter are related to a 2D classical electrostatic problem known as jellium. Although this classical system already features one key property of fractional Hall systems, namely their incompressibility, a microscopic explanation of this phenomenon requires to establish a spectral gap in associated many-particle operators such as the Haldane pseudopotentials, which I will discuss. In particular, I will survey the recent mathematical results and methods concerning Haldane pseudopotentials in a thin cylinder geometry.