All triangulations have a common stellar subdivision

We address two longstanding open problems, one originating in PL topology, another in birational geometry. First, we prove the weighted version of Oda’s 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.