RANDOM MATRICES AND THE AVERAGE TOPOLOGY OF THE INTERSECTION OF TWO QUADRICS.

Let X_R be the zero locus in RPn of one or two independently and Kostlan distributed random real quadratic forms (this is equivalent to the corresponding symmetric matrices being in the Gaussian Orthogonal Ensemble). Denoting by b(X_R) the sum of the Betti numbers of X_R, we prove that (1)...The methods we use combine random matrix theory, integral geometry and spectral sequences: for one quadric hypersurface it is simply a corollary of Wigner's semicircle law; for the intersection of two quadrics it is related to the (intrinsic) volume of the set of singular symmetric matrices of norm one. [ABSTRACT FROM AUTHOR]