  In his famous undergraduate physics lectures, Richard Feynman remarked about
the problem of fluid turbulence: "Nobody in physics has really been able to
analyze it mathematically satisfactorily in spite of its importance to the
sister sciences". This statement was already false when Feynman made it.
Unbeknownst to him, Lars Onsager decades earlier had made an exact mathematical
analysis of the high Reynolds-number limit of incompressible fluid turbulence,
using a method that would now be described as a non-perturbative
renormalization group analysis and discovering the first "conservation-law
anomaly" in theoretical physics. Onsager's results were only cryptically
announced in 1949 and he never published any of his detailed calculations.
Onsager's analysis was finally rescued from oblivion and reproduced by this
author in 1992. The ideas have subsequently been intensively developed in the
mathematical PDE community, where deep connections emerged with John Nash's
work on isometric embeddings. Furthermore, Onsager's method has more recently
been successfully applied to new physics problems, such as compressible fluid
turbulence and relativistic fluid turbulence, yielding many novel testable
predictions. This note will explain Onsager's exact analysis of incompressible
turbulence using modern ideas on renormalization group and conservation-law
anomalies, and it will also very briefly review subsequent developments.
