The embedded contact homology of toric contact manifolds

Embedded contact homology (ECH) is an invariant of a contact three-manifold. In Part I of this thesis, we provide a combinatorial description of the ECH chain complex of certain ``toric'' contact manifolds. This is an extension of the combinatorial description appearing in Hutchings-Sullivan. ECH capacities are invariants of a symplectic four-manifold with boundary, which give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. In Part II of this thesis, we compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called ``concave toric domains''. Examples include the (nondisjoint) union of two ellipsoids in $\R^4$. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. This is a joint work with D. Cristofaro-Gardiner, D. Frenkel, M. Hutchings and V. G. B. Ramos.