A "square root law" for the expected number of singularities of Kostlan polynomials.

I will consider the problem of counting the number of points on the m-dimensional Sphere at which a homogenous polynomial attains a given nondegenerate singularity, i.e. points where the r-jet prolongation of the polynomial meets transversally some given semialgebraic submanifold of the space of r-jets. I will present a recent result, obtained jointly with Antonio Lerario, which establishes that in the Kostlan case, under reasonable hypothesis on the singularity, the expected value of this number grows like the square root of the corresponding deterministic upper bound as the degree of the polynomial tends to infinity.