Mathematical Conversations

In 1850, Liouville proved a rather surprising fact: any $C^{3}$ conformal map in a three-dimensional domain is a Möbius transformation; this is in stark contrast with the two-dimensional case, where conformal maps abound. Since then, Liouville's theorem has been generalized to higher dimensions and the smoothness assumptions weakened, but the optimal statement is still not known. In this talk, I will recount the story of this fascinating open problem.