This is a Joint Columbia-IAS-Princeton Symplectic Seminar.   ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and the target is a "convex toric domain” (see previous talk). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. In this talk we explain how more refined information from ECH gives stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.