Abundance of projective subspaces on real algebraic hypersurfaces

In a joint work with V.Kharlamov, we gave an estimate for the numbers of lines on real algebraic n-dimensional hypersurfaces of degree (2n-1) using evaluation of a certain signed count of these lines. The signs involved can be viewed as a generalized version of the Welschinger indices.  Our approach allows also a generalization to obtain a similar estimate for the numbers of projective subspaces on real algebraic  hypersurfaces (of certain dimensions and degrees).   I will present our method in a concrete example of counting the  3-subspaces on a 7-dimensional real cubic.  A relation to some problems of the real Schubert calculus will be disclosed.   "Abundance" in the title of talk means logarithmic proportionality for the asymptotics of the numbers of the projective subspaces over R and C.