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The outstanding open problem in the interface between smooth dynamics and ergodic theory is whether or not every finite entropy abstract ergodic transformation is isomorphic to a smooth diffeomorphism preserving volume element on a compact manifold. While the problem was essentially formulated by von Neumann in 1932 there has been very little progress and it is open even for very basic examples such as odometers. I will discuss some recent work on the problem (joint with Matt Foreman) of two kinds. On the one hand we provide a host of new examples that can be realized, while on the other hand we show that the isomorphism problem for smooth diffeomorphisms preserving Lebesgue measure on the torus is as complex as the general abstract isomorphism problem for ergodic transformations.