The famous Gromov-Eliashberg -rigidity theorem is a miracle in symplectic geometry. Roughly speaking, the theorem states that the group of symplectomorphisms of a symplectic manifold is topologically closed in its group of diffeomorphisms, in the sense of uniform norm. It turns out that this rigidity phenomenon has its roots deeply mined in the interaction between Hamiltonian dynamics and Poisson brackets under -limit. The somehow interesting thing is that this functional analytical style properties are related to Hom structures of certain triangulated (or dg) categories. In this talk I will give an introduction to Hamiltonian diffeomorphism groups and their Lie algebras, and explain how the homological algebra of triangulated categories can be used to answer these symplectic -rigidity problems.