von Neumann’s Classification Program is provably impossible

von Neumann can be viewed as the founder of Ergodic Theory, the study of the quantitative (probabilistic) behavior of systems that evolve through time. His foundational Mean Ergodic Theorem set the stage for later developments. Ergodic theorems rigorously justify estimating functions by sampling their values at regular intervals. In his 1932 paper he proposed classifying the behavior of differentiable systems up to measure preserving transformations (the “Isomorphism Problem”). Work of von Neumann and Halmos was an early success–it classified ergodic translations on Abelian groups, using spectral methods. Their results inspired an explosion of results, including major successes such as classifying Bernoulli systems. This talk will present a negative final outcome: von Neumann’s program of classification of C∞ measure preserving diffeomorphisms of the 2-torus is impossible in a rigorous sense. Very recent results show that Smale’s Program (from 1964) of classifying the qualitative behaviour of C∞-diffeomorphisms is similarly impossible. These will be briefly discussed at the end of the talk.