  I discuss results bearing on a variational problem of a new type, inspired by
fractional quantum Hall physics. In the latter context, the main result
reviewed herein can be spelled as "the phase of independent quasi-holes
generated from Laughlin's wave-function is stable against external potentials
and weak long-range interactions". The main ingredient of the proof is a
connection between fractional quantum Hall wave-functions and statistical
mechanics problems that generalize the 2D one-component plasma (jellium model).
Universal bounds on the density of such systems, coined "Incompressibility
estimates" are obtained via the construction of screening regions for any
configuration of points with positive electric charges. The latter regions are
patches of constant, negative electric charge density, whose shape is optimized
for the total system (points plus patch) not to generate any electric potential
in its exterior.
