Description:

Title: Surjectivity of random integral matrices on integral vectors

Abstract: A random nxm matrix gives a random linear transformation
from Z^m to Z^n (between vectors with integral coordinates).  Asking
for the probability that such a map is injective is a question of the
non-vanishing of determinants.  In this talk, we discuss the
probability that such a map is surjective, which is a more subtle
integral question.  We show that when m=n+u, for u at least 1, as n
goes to infinity, the surjectivity probability is a non-zero product
of inverse values of the Riemann zeta function.  This probability is
universal, i.e. we prove that it does not depend on the distribution
from which you choose independent entries of the matrix, and this probability
 also arises in the Cohen-Lenstra heuristics predicting the
distribution of class groups of real quadratic fields.  This talk is on joint work with Hoi Nguyen.