Unlike Riemannian geometry, symplectic topology is locally flexible; by Darboux's and Moser’s theorems, it carries no local invariants. However, despite this local flexibility, Mikhail Gromov revealed global rigidity phenomena in 1985 by introducing the study of J-holomorphic curves in symplectic manifolds. In this talk, we will explore J-holomorphic curves in S² × S² and see how they can be used to understand the topology of the group of symplectomorphisms.