Birationally rigid hypersurfaces
  We prove that for N greater than or equal to 4, all smooth hypersurfaces of degree N in P^N are birationally superrigid. First discovered in the case N = 4 by Iskovskikh and Manin in a work that started this whole direction of research, this property was later conjectured to hold in general by Pukhlikov. The proof relies on the method of maximal singularities in combination with a delicate formula on restrictions of multiplier ideals. 
Tommaso de Fernex