Abstract: Symplectic maps appear as a natural generalization of area-preserving diffeomorphisms of surfaces. They play a central role in the mathematical model of classical mechanics.

We will focus on the following topics:

(a) growth rate of symplectic maps and the trichotomy hyperbolic/parabolic/elliptic in the context of diffeomorphisms;

(b) obstructions to symplectic actions of finitely generated groups, including a symplectic version of the Zimmer program on actions of lattices.

We describe recent progress in these directions based on modern methods of symplectic topology.