  We describe a Schubert induction theorem, a tool for analyzing intersections
on a Grassmannian over an arbitrary base ring. The key ingredient in the proof
is the Geometric Littlewood-Richardson rule, described in a companion paper.
  Schubert problems are among the most classical problems in enumerative
geometry of continuing interest. As an application of Schubert induction, we
address several long-standing natural questions related to Schubert problems,
including: the "reality" of solutions; effective numerical methods; solutions
over algebraically closed fields of positive characteristic; solutions over
finite fields; a generic smoothness (Kleiman-Bertini) theorem; and monodromy
groups of Schubert problems. These methods conjecturally extend to the flag
variety.
