The Shape Invariant for Toric Domains

We discuss the shape invariant, a sort of set valued symplectic capacity defined by the Lagrangian tori inside a domain of R4. Partial computations for convex toric domains are sometimes enough to give sharp obstructions to symplectic embeddings, but in general the shape is far from a complete invariant. We then consider continuous families of Lagrangian embeddings, and describe a seemingly close relation to stabilized symplectic embeddings. This is ongoing work with Ely Kerman and Jun Zhang.