In 1932 von Neumann proposed classifying the statistical behavior of measure preserving diffeomorphisms of the torus, In joint work with B. Weiss I prove this is impossible (in a convincing and rigorous sense) even in the simplest and most concrete case: the 2-torus. 

By luck our work has accidental, but far-reaching consequences inside ergodic theory.  It establishes a “global structure theorem” for ergodic measure preserving transformations that gives heretofore unknown and surprising examples of diffeomorphisms of the torus.