Résume | One of the outstanding problems of mathematics today is the Riemann Hypothesis, on the location of the zeros of the Riemann zeta function. A crucial insight obtained in past few decades concerning these zeros is that their local statistics can be modeled by those of eigenvalues of certain Random Matrix ensembles,and similarly for all other automorphic L-functions. A parallel theory dealt with the zeta function of a varieties over a finite field, for which the Riemann Hypothesis was established by Weil and Deligne. A fundamental conjecture of Katz and Sarnak about the statistics of zeros of families of such zeta-functions is that in many cases these statistics converge to those of the eigenvalues of a suitable Random Matrix ensemble, dictated by the symmetries of the underlying objects. Until now, there was only a loose analogy between the two settings, going back to Weil, Grothendieck and others, but no real implications. Recently that has changed, as we have discovered how to establish combinatorial identities which are required to identify number field statistics with Random Matrix Theory, and which have so far been intractable, by using corresponding results for function fields which are known due to the results of Deligne and Katz about the monodromy for various moduli spaces, such as the moduli space of hyperelliptic curves. The lecture will give an overview of these matters, and is intended for a very general audience. |