  We address two longstanding open problems, one originating in PL topology,
another in birational geometry.
  First, we prove the weighted version of Oda's \emph{strong factorization
conjecture} (1978), and prove that every two birational toric varieties are
related by a common iterated blowup (at rationally smooth points).
  Second, we prove that every two PL homeomorphic polyhedra have a common
stellar subdivisions, as conjectured by Alexander in~1930.
