In a series of works with Laszlo Szekelyhidi Jr. we pointed out an unusual analogy between two problems in rather distant areas: a long standing conjecture of Onsager in the theory of turbulence and a (less known) critical regularity problem in classical differential geometry. In both cases the issue is to find the critical regularity above which the solutions to a certain system of partial differential equations are ``well-behaved'' and below which they are instead rather ``wild''. The circle of ideas which, thanks to the contributions of several people, have solved both problems leads also to other surprising facts and raises many further questions.