Michael Hutchings

A symplectic embedding of one domain in R^(2n) into another is a smooth embedding which preserves the standard symplectic form. It is a fundamental problem to determine when one domain can be symplectically embedded into another. Even for simple domains such as ellipsoids and polydisks, this is a difficult question, and the known answers often involve subtle combinatorics. In general, one can obstruct the existence of symplectic embeddings using various “symplectic capacities”. We introduce a new series of symplectic capacities based on positive S^{^1}-equivariant symplectic homology. These capacities can be computed combinatorially for “toric domains”, and lead to some sharp obstructions to symplectic embeddings in which the domain is a cube. Joint work with Jean Gutt.