In this talk I will focus on the problem of understanding the topology of an intersection X of real quadrics. I will introduce a new notion of geometric complexity (inspired to fewnomials and related), using the discriminant in the space of quadratic forms. I will discuss a sort of duality'' between X and the set of singular quadrics in the linear system defining it; in the case of intersections of three quadrics this picture offers adual'' point of view on Hilbert's Sixteenth Problem.