The ability to fabricate increasingly smaller and cheaper sensing and actuation devices portends a future in which swarms of robots provide critical services, including searchand-rescue at disaster sites, environmental monitoring, and border security. But as individual robots and sensors shrink in size and cost, they multiply in number, requiring methods of coordination. One of the most fundamental and challenging problems is moving from local information (at the level of individual robots) to a global understanding of an environment (at the level of the full swarm). A century ago, mathematicians invented a new field — “topology,” the study of abstract spaces — to handle very similar issues of passing from local to global. A century of subsequent work has yielded a dizzying array of elegant algebraic tools which have remained largely hidden within Mathematics. This talk will illustrate several surprising applications of this once-esoteric mathematical subject to the understanding and control of robot and sensor swarms.