Between 1947 and 1950, Laurent Schwartz (1915-2002) went from being almost unknown outside of France to being an international mathematical celebrity. This paper accounts for Schwartz's rapid ascent by focusing on the social, institutional, and mathematical contexts of his crucial trajectory from Nancy, via Copenhagen, to the world stage, culminating in his 1950 Fields Medal awarded in Cambridge, Massachusetts. We identify, based on new archival findings, the pivotal role of Danish mathematician Harald Bohr along this trajectory. Our analysis reveals the emerging dynamics of early postwar international mathematics, and explains how certain individuals and theories could rise to prominence in this period.