In this talk I will consider maps from the $m$-sphere to $R^k$, defined by $k$ homogeneous polynomials of degree $d$ in $m+1$ variables. I will investigate the topology of set $Z$ of points where those maps and their derivatives (up to order $r$) satisfy a given list of polynomial equalities and inequalities. I call such set of conditions a "singularity class" and $Z$ a ''singularity". Examples are: the zero set (here no derivative is involved, i.e. $r=0$); the set of critical points ($r=1$); the set of critical points with a prescribed signature of the hessian $(k=1,r=2)$. I will present the content of an homonymous paper (a joint work with with Antonio Lerario, on arxiv) concerning the asymptotic of the Betti numbers of $Z$ as $d$ grows, within a fixed "singularity class".

For what regards the maximal topology, I will try to convince you that it is possible to find maps for which $Z$ has arbitrarily large Betti numbers (this is subtle when derivatives are involved in the singularity class), however there is a constraint on the degree: $b(Z)=O(d^m)$. On the other hand, the typical behaviour of the Betti numbers is $\Theta (d^m/2)$ (the square root of the maximal). The word "typical" means that I am taking the expected value (i.e. the integral) with respect to a certain natural probability on the space of polynomials, called Kostlan measure.