Many economists believe knowledge production generates positive spillovers among knowledge producers. The available evidence, however, is mixed. We argue that spillovers can exist along three dimensions: idea, geographic, and collaboration space. To isolate the key channel through which knowledge spills over, we use a unique data set to examine the impact of a large post-1992 exodus of Soviet mathematicians on the output of the nonémigrés. Although the data reveal strong competitive effects in idea space, there is evidence of knowledge spillovers in collaboration space, when high-quality researchers directly engage with other researchers in the joint production of new knowledge.