An Introduction to Embedded Contact Homology

Embedded contact homology (ECH) is a powerful invariant of contact 3-manifolds due to Hutchings and Taubes. It is key to the proofs of the Weinstein conjecture (existence of Reeb orbits), Arnold's chord conjecture, Irie's closing lemma, characterizations of symplectic embeddings, and obstructions of symplectic fillings of contact manifolds. I give an introduction to ECH, provide some examples, and explain some of its applications to dynamics.