Automorphisms of the Weyl algebra
  We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic. 
Alexei Belov-Kanel, Maxim Kontsevich