Anti-classification results in smooth dynamics

Squaring the circle, trisecting of an angle, finding an explicit formula for roots of a polynomial of fifth degree—none of those problems has a solution. Let us now ask one more question—can one classify all dynamical systems? Specifically, is it possible to classify all diffeomorphisms of a given manifold up to a topological conjugacy? To show that such classification exists (as, for example, in the case of diffeomorphisms of the circle) it is enough to explicitly present it. But what if it doesn't? What exactly does it mean? And how can one prove it?
In our joint work with Matt Foreman we prove that for smooth diffeomorphisms of a two-dimensional manifold there is no reasonably defined numerical invariant such that it would take the same values exactly on diffeomorphisms that are topologically conjugate. For diffeomorphisms of manifolds of dimension five and higher such classification is impossible in another, much stronger sense. In the talk we will explain the details of those statements and discuss some open problems.