Is every finite group isomorphic to the Galois group of some Galois extension of the rational numbers? Although this question remains open in general, powerful methods have led to an affirmative answer in some cases, including that of solvable groups, symmetric and alternating groups, and most of the sporadic groups. In this talk, we call upon seemingly disconnected areas of algebra, topology, and complex analysis to describe the rigidity method of inverse Galois theory.