Low-dimensional symplectic geometry from embedded contact homology computations

Embedded contact homology (ECH) is a 3D Floer-type homology theory due to Hutchings. Its access to both symplectic and topological information means that it is well-suited to solving problems in low-dimensional symplectic and contact geometry. We will explain recent applications to surface dynamics and 4D symplectic embeddings, which require understanding ECH at the chain level in new topological and geometric settings, including its cobordism maps. In part based on several recent and upcoming joint works with Magill, McDuff, Nelson, and Pires.