$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero   Center Lyapunov Exponents
  We prove that if $f$ is a $C^1$-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if $f$ is not Anosov then all the exponents in the center bundle vanish. This establishes in full a result announced by R. Ma\~{n}\'{e} in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks. 
Jairo Bochi