Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are key to defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will present ongoing work on the symplectic topology of these spaces using Liouville sectorial techniques, along with examples and applications of these decompositions in the context of homological mirror symmetry