Define a “Liouville domain” to be a compact exact symplectic manifold with contact-type
boundary. We use embedded contact homology to assign to each four-dimensional Liouville …

These notes give an introduction to embedded contact homology (ECH) of contact three-
manifolds, gathering together many basic notions which are scattered across a number of …

We show that every (possibly degenerate) contact form on a closed three-manifold has at
least two embedded Reeb orbits. We also show that if there are only finitely many …

In a previous paper, the second author used embedded contact homology (ECH) of contact
three-manifolds to define “ECH capacities” of four-dimensional symplectic manifolds. In the …

We survey some recent progress on understanding when one four-dimensional symplectic
manifold can be symplectically embedded into another. In 2010, McDuff established a …

We prove that every nondegenerate contact form on a closed connected three-manifold
such that the associated contact structure has torsion first Chern class has either two or …

ECH (embedded contact homology) capacities give obstructions to symplectically
embedding one symplectic four-manifold with boundary into another. We compute the ECH …

We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of
twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the …

In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving
homeomorphisms of the $ n $-ball is simple for $ n\ge 3$, asked if the same statement holds …

We prove that every Reeb flow on a closed connected three-manifold has either two or
infinitely many simple periodic orbits, assuming that the associated contact structure has …