sentences:  He proved that it was possible to resolve singularities of varieties over fields of characteristic 0 by repeatedly blowing up along non-singular subvarieties, using a very complicated argument by induction on the dimension.Simplified versions of
  his formidable proof were given by several people, including , , , , , .Some of the recent proofs are about a tenth of the length of Hironaka's original proof, and are easy enough to give in an introductory graduate course.