In 1932 von Neumann proposed classifying the statistical behavior of physical systems. The idea was to take a diffeomorphism of a compact manifold and describe what one might observe as random (as in coin flipping) or predictable (as in a translation on a compact group), or even better have a dictionary  in which one could look up the precise behavior.

Remarkable progress was made on this problem; benchmarks include the Halmos-von Neumann theorem on discrete spectrum and the work of Kolmogorov on Entropy that culminated in the Ornstein classification of Bernoulli shifts. One genre of applications of this theory were the results of Furstenberg on Szemeredi’s theorem and eventually the work of  Green and Tao.

Still the problem resisted a complete solution. Strange examples of completely determined systems that showed completely random statistical behavior began to surface. Starting in the 1990’s anti-classification theorems began to appear. These results showed, in a rigorous way, that complete invariants for measure preserving systems cannot exist. Moreover the isomorphism relation itself is completely intractable. Very recently these results were extended to measure preserving diffeomorphisms of the 2-torus.