This paper studies the impact of U.S. immigration barriers on global knowledge production. We present four key findings. First, among Nobel Prize winners and Fields Medalists, migrants to the U.S. play a central role in the global knowledge network— representing 20-33% of the frontier knowledge producers. Second, using novel survey data and hand-curated life-histories of International Math Olympiad (IMO) medalists, we show that migrants to the U.S. are up to six times more productive than migrants to other countries—even after accounting for talent during one's teenage years. Third, financing costs are a key factor preventing foreign talent from migrating abroad to pursue their dream careers, particularly talent from developing countries. Fourth, certain 'push' incentives that reduce immigration barriers – by addressing financing constraints for top foreign talent – could increase the global scientific output of future cohorts by 42% percent. We conclude by discussing policy options for the U.S. and the global scientific community.