  This mostly expository paper centers on recently proved conjectures in two
areas:
  A) A conjecture of A. Oppenheim on the values of real indefinite quadratic
forms at integral points.
  B) Conjectures of Dani, Raghunathan, and Margulis on closures of orbits in
spaces of lattices such as ${\BSL}_n(\R)/{\BSL}_n(\Z)$.
  At first sight, A) belongs to analytic number theory and B) belongs to
ergodic and Lie theory, and they seem to be quite unrelated. They are discussed
together here because of a very interesting connection between the two pointed
out by M. S. Raghunathan, namely, a special case of B) yields a proof of A).
