Symplectic packing problem  is one of the central theme in symplectic geometry, after Gromov’s celebrating non-squeezing theorem.  Many works have been done to study whether a symplectic region can be packed into a given symplectic manifold.  

We take a different perspective and study the topology of the space of symplectic packing when the packing exists.  It turns out to be related to the symplectic automorphism of certain symplectic manifolds called the “ball-swapping”.  Furthermore, we relate the story to monodromy problem in algebraic geometry, and another more classical symplectic automorphism called “Dehn twists”.  This leads to applications to uniqueness theorems of Lagrangian embeddings, classifications of homotopy types of symplectic automorphism groups and classification of finite group actions.