  These are notes of the mini-course I gave during the CIMPA summer school at
Villa de Leyva, Colombia, in July $2014$. The subject was my joint work with
Damien Gayet on the topology of random real hypersurfaces, restricting myself
to the case of projective spaces and focusing on our lower estimates. Namely,
we estimate from (above and) below the mathematical expectation of all Betti
numbers of degree $d$ random real projective hypersurfaces. For any closed
connected hypersurface $\Sigma$ of $\mathbb{R}^n$, we actually estimate from
below the mathematical expectation of the number of connected components of
these degree $d$ random real projective hypersurfaces which are diffeomorphic
to $\Sigma$.
