The asymptotics of ECH capacities and absolute gradings on Floer homologies

Embedded contact homology (ECH) capacities were defined by Hutchings and provide a family of obstructions to embeddings of four-dimensional symplectic manifolds. In Part I of this thesis, we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This was joint work with Daniel Cristofaro-Gardiner and Michael Hutchings. In Part II of this thesis, we construct topological absolute gradings in Heegaard Floer homology and bordered Floer homology that satify all of the expected properties. This was joint work with Yang Huang. We also show that the isomorphism between Heegaard Floer homology and ECH preserves the absolute gradings