  We describe an explicit geometric Littlewood-Richardson rule, interpreted as
deforming the intersection of two Schubert varieties so that they break into
Schubert varieties. There are no restrictions on the base field, and all
multiplicities arising are 1; this is important for applications. This rule
should be seen as a generalization of Pieri's rule to arbitrary Schubert
classes, by way of explicit homotopies. It has a straightforward bijection to
other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's
puzzles.
  This gives the first geometric proof and interpretation of the
Littlewood-Richardson rule. It has a host of geometric consequences, described
in the companion paper "Schubert induction". The rule also has an
interpretation in K-theory, suggested by Buch, which gives an extension of
puzzles to K-theory. The rule suggests a natural approach to the open question
of finding a Littlewood-Richardson rule for the flag variety, leading to a
conjecture, shown to be true up to dimension 5. Finally, the rule suggests
approaches to similar open problems, such as Littlewood-Richardson rules for
the symplectic Grassmannian and two-flag varieties.
