Heisuke Hironaka: “Resolution of Singularities in Algebraic Geometry”
Algebraic geometry in general has three fundamental types in terms of its base ground: (I) Q (and its fields extensions), (II) F(p) with a prime number p > 0 (and every finite field), and at last (III) Z in the case of the arithmetic geometry. In those three cases I will talk about resolution of singularities by means of blowups with permissible centers in smooth ambient spaces. (I) is done in 1964, (II) is proven recently with new concept and technique, while (III) is by combination of (I) and (II). Technically elaborate but conceptually interesting is the case of (II).