Double doubles and maximal real quartics in RP^{^4}

Small variations of doubled real algebraic varieties is a classical construction technique which has led to many interesting results, for instance regarding the topology of real algebraic surfaces in the three-dimensional real projective space. We highlight a slight variation of this technique, which roughly consists in applying it two times in a row, and discuss some of its applications. In particular, we explain how this method can be used to obtain different topological types of maximal quartic hypersurfaces in the four-dimensional real projective space.