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 …

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 …

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book
decomposition. As an application we obtain that on a closed 3-manifold, every …

This is a sequel to four earlier papers by the author that construct an isomorphism between
the embedded contact homology and Seiberg–Witten Floer cohomology of a compact 3 …

Abstract In “Proof of the Arnold chord conjecture in three dimensions, I”[Math. Res. Lett. 18
(2011) 295–313], we deduced the Arnold chord conjecture in three dimensions from another …

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 …

We resolve three longstanding questions related to the large scale geometry of the group of
Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely:(1) …