This is a report on joint work with Kyler Siegel that develops new ways to count $J$-holomorphic curves in $4$-dimensions, both in the projective plane with multi-branched tangency constraints, and in noncompact cobordisms between ellipsoids. These curves stabilize, i.e. if they exist in a given four dimensional target manifold $X$ they still exist in the product $X \times {\mathbb R}^{2k}$. This allows us to establish new cases of the stabilized embedding conjecture for symplectic embeddings of an ellipsoid into a ball (or ellipsoid).