Gap probabilities and applications to geometry and random topology

We give an exact formula for the value of the derivative at zero of the gap probability in finite n x n Gaussian ensembles. As n goes to infinity our computation provides an asymptotic (with an explicit constant) of the order n^(1/2). As a first application, we consider the set of n x n (Real, Complex or Quaternionic) Hermitian matrices with Frobenius norm one and determinant zero. We give an exact formula for the intrinsic volume of this set and as n goes to infinity its asymptotic (with an explicit constant) is of the order n^(1/2). As a second application we consider the problem of computing Betti numbers of an intersection of k random Kostlan quadrics in RP^n. We show that the i-th Betti number is asymptotically expected to be one (for i sufficiently away from n/2). In the case k=2 the the sum of all Betti numbers was recently shown by the first author to equal n+o(n). Here we sharpen this asymptotic proving an asymptotic with two orders of precision and explicit constants.