We survey one of the most exciting and ambitious current programs in smooth dynamics, which is the classification of partially hyperbolic diffeomorphisms in dimension 3 up to leaf conjugacy. The principal tool in this classification is Novikov’s theorem from the theory of codimension one foliations. Highlights include:
 The BrinBuragoIvanov proof that S^3 admits no partially hyperbolic diffeomorphisms
 The complete classification of partially hyperbolic diffeomorphisms on 3manifolds with solvable fundamental group by Hammerlindl and Potrie
 The FranksWilliams surgery construction of Anosov flows on all sorts of 3manifolds
 The new BonattiGogolevPotrie construction of partially hyperbolic diffeomorphisms of the unit tangent bundle of a surface of genus > 1 in the isotopy class of any element of the mapping class group.
No dynamics knowledge will be assumed from the audience in this talk so even if what I wrote above makes absolutely no sense to you, you will still learn something awesome!
