Symplectic geometry provides the mathematical framework of classical mechanics in its Hamiltonian formulation, and also underlies modern approaches to quantum theories. About twenty years ago, the introduction of new ideas and techniques has led to the discovery of "symplectic rigidity" phenomena that have no counterparts in classical differential geometry. Since then, symplectic geometry evolved into an independent field of research at the crossroads of geometric topology, algebraic geometry, dynamical systems, and gauge theory. The study of symplectic diffeomorphisms is at the heart of understanding symplectic rigidity phenomena. In this introductory talk, I will give an (somewhat biased) overview of symplectic rigidity, focussing mainly on homotopy-theoretic, geometric, and algebraic properties of symplectomorphism groups, as well as on properties of embedded symplectic balls and Lagrangian submanifolds.

I will discuss some recent results of a joint work with professor Rasul Shafikov, concerning a CR geometry problem, with a complex (yet simple) local analytic solution.

Our plan is to give an overview of equivariant cohomology for torus actions as it is understood after Goresky, Kottwitz and MacPherson.