Oscar Zariski solves the problem of resolution of singularities for algebraic varieties of dimension 2 (1939) and dimension 3 (1944) over fields of characteristic zero, introducing valuation theory to locally approach the singularities. His rigorous algebraic approach, inherited from the Castelnuovo-Enriques-Severi school but formalized with strict commutative algebra, lays the methodological groundwork that Hironaka — later a doctoral student of Zariski's at Harvard — would extend twenty years later to arbitrary dimension.