  The fractional quantum Hall effect in 2D electron gases submitted to large
magnetic fields remains one of the most striking phenomena in condensed matter
physics. Historically, the first observed signature is a Hall resistance
quantized to the value (2m+1) when the filling factor (electron density divided
by magnetic flux quantum density) of a 2D electron gas is in the vicinity of an
inverse odd integer 1/(2m +1). This was one of the first observation of
fractional quantum numbers. A large part of our basic theoretical understanding
of this effect (and descendants) originates from Laughlin's theory of 1983,
reviewed here from a mathematical physics perspective. We explain in which
sense Laughlin's proposed ground and excited states for the system are
rigid/incompressible liquids, and why this is crucial for the explanation of
the effect.
