Embedded contact homology and applications
Here is a talk from a few years ago which gives an overview of ECH and some of its older applications:
 Applications of ECH, Freedman birthday conference, June 8,
2011. slides video.
To get started reading about ECH, see the lecture notes [15] below.

[1] An index inequality for embedded pseudoholomorphic curves in
symplectizations
Journal of the European Math. Soc.
4 (2002), 313361. PDF
PS
This paper defines the index, or grading, in PFH. When Y is the
mapping torus of a surface symplectomorphism, this index associates to
each pseudoholomorphic curve C in R x Y an integer I(C), which
depends only on the relative homology class of C. One can think of
I(C) as a kind of measure of the complexity of C, which is quite
different from usual measures of complexity such as genus or number of
ends. In particular, we show that for a generic almost complex
structure, there are no curves in R x Y with I<0; a curve has
I=0 if and only if it is a cover of a union of Rinvariant cylinders;
I=1 curves are (mostly) embedded, have Fredholm index 1, and satisfy
additional topological constraints; and there are good compactness
results for moduli spaces of I=1 and I=2 curves. An analogous story
holds for ECH; see paper [3] below. The differential in PFH or ECH
counts curves with I=1. The index I is the key, nontrivial part of
the definition of PFH and ECH.

[2] The periodic Floer homology of a Dehn twist (with M. Sullivan),
Algebraic and Geometric
Topology 5 (2005), 301354. download.
This paper reviews the definition of PFH and computes the PFH of some
Dehn twists. Some of the geometric techniques introduced here play an
important role in paper [3] below.

[3] Rounding corners of polygons and the
embedded contact homology of T^3 (with M. Sullivan),
Geometry and Topology 10 (2006), 169266.
download
This paper gives an introduction to ECH, and computes the ECH of T^3.
The latter is described by a combinatorial chain complex which is
generated by labeled convex polygons in the plane with vertices at
lattice points, and whose differential involves "rounding corners".
Eli Lebow has extended the methods of this paper to compute the ECH of
(most) T^2bundles over S^1. Some more recent applications of this paper
are described in this blog post.
 [4] Gluing pseudoholomorphic curves along
branched covered cylinders I (with C. H. Taubes.)
Journal of Symplectic Geometry 5 (2007), 43137.
ps
This paper and its sequel [5] contain the
gluing analysis to prove that d^2=0 in embedded contact homology and
periodic Floer homology. This requires an extension of the standard
gluing machinery, in order to glue together two Jholomorphic curves
in
R x Y whose ends are at the same Reeb orbits, but with
different covering multiplicities. To glue two such curves together,
we need to insert a branched cover of an
Rinvariant cylinder between them. The number of gluings is
given by a count of zeroes of a certain section of an obstruction
bundle over a noncompact moduli space of branched covered cylinders.
We obtain a general combinatorial formula for the number of gluings as
a certain sum over labeled forests. In the case of interest for ECH
and PFH, the number of gluings comes out to be 1. This is just what
is needed to show that d^2 counts the number of boundary points of a
compact 1manifold, so that d^2=0. Paper [4] explains the more algebraic
aspects of this story, using an analytic result from [5].
 [5] Gluing pseudoholomorphic curves along branched covered
cylinders II (with C. H. Taubes.)
Journal of Symplectic Geometry 7
(2009), 29133. pdf
This paper completes
all the analysis that was needed in [4]. The gluing technique
explained here is in principle applicable to more gluing problems. We
also prove some lemmas concerning the generic behavior of
pseudoholomorphic curves in symplectizations, which may be of
independent interest.
 [6] The embedded contact homology index revisited
New perspectives and challenges in symplectic field
theory ("Yashafest"), 263297, CRM Proc. Lecture Notes, 49. AMS, 2009,
nearly final version available at
arXiv:0805.1240. Erratum.
In this paper we refine the relative grading on ECH to an absolute
grading, which associates to each union of Reeb orbits a homotopy
class of oriented 2plane fields. This is obtained by
modifying the contact plane field in a canonical (up to homotopy)
manner in a neighborhood of each Reeb orbit in the union.
We also simplify the proof of the ECH index inequality, and extend it to
symplectic cobordisms between threemanifolds with Hamiltonian
structures. Included are new inequalities on the ECH index
of unions and multiple covers of holomorphic curves in cobordisms.
Finally, we define a new relative filtration on ECH, or any other
kind of contact homology of a contact 3manifold, which is similar
to the ECH index and related to the Euler characteristic of
holomorphic curves. This does not give new topological invariants
except possibly in special situations, but it is a useful
computational tool.
 [7] The Weinstein conjecture for stable Hamiltonian
structures (with C. H. Taubes.)
Geometry and Topology 13 (2009), 901941. download
We use the
equivalence between embedded contact homology and SeibergWitten Floer
homology to obtain the following improvements on the Weinstein
conjecture. Let Y be a closed oriented connected 3manifold with a
stable Hamiltonian structure, and let R denote the associated Reeb
vector field on Y. We prove that if Y is not a T^2bundle over S^1,
then R has a closed orbit. Along the way we prove that if Y is a
closed oriented connected 3manifold with a contact form such that all
Reeb orbits are nondegenerate and ellitpic, then Y is a lens space.
Related arguments show that if Y is a closed oriented 3manifold with
a contact form such that all Reeb orbits are nondegenerate, and if Y
is not a lens space, then there exist at least three distinct embedded
Reeb orbits.
 [8] Embedded contact homology and its applications
Proceedings of the 2010 ICM, vol. II, 10221041,
nearly final version available at
arXiv:1003.3209.
In this
article we first give an overview of the definition of embedded
contact homology. We then outline its applications to
generalizations of the Weinstein conjecture, the Arnold chord
conjecture, and obstructions to symplectic embeddings in four
dimensions.
Here are the slides from the associated talk.
 [9] Sutures and contact homology I (with V. Colin,
P. Ghiggini, and K. Honda)
Geometry and Topology 15 (2011), 17491842. download
Among other things, this
paper defines a version of ECH for sutured 3manifolds, which is
conjecturally isomorphic to the sutured (Heegaard) Floer homology of Juhasz.
 [10] Proof of the Arnold chord conjecture in three dimensions
I (with C. H. Taubes)
arXiv:1004.4319,
Math. Res. Lett. 18 (2011), 295313.
We prove that every Legendrian knot in a closed
threemanifold with a contact form has a Reeb chord. The proof uses another theorem, asserting that an exact
symplectic cobordism between contact 3manifolds induces a map on
(filtered) embedded contact homology satisfying certain axioms. The
latter theorem is proved in [12] using SeibergWitten
theory.
 [11] Quantitative embedded contact homology
J. Differential Geometry 88 (2011) 231266, close to final version at arXiv:1005.2260 (Conjecture 8.5 in the arxiv version is false and does not appear in the published version)
Define a "Liouville domain" to be a compact exact symplectic manifold
with contacttype boundary. We use embedded contact homology to assign
to each fourdimensional Liouville domain (or subset thereof) a
sequence of real numbers, which we call "ECH capacities". The ECH
capacities of a Liouville domain are defined in terms of the "ECH
spectrum" of its boundary, which measures the amount of symplectic
action needed to represent certain classes in embedded contact
homology. Using cobordism maps on embedded contact homology (defined
in joint work with Taubes), we show that the ECH capacities are
monotone with respect to symplectic embeddings. We compute the ECH
capacities of ellipsoids, polydisks, certain subsets of the cotangent
bundle of T2, and disjoint unions of examples for which the ECH
capacities are known. The resulting symplectic embedding obstructions
are sharp in some interesting cases, for example for the problem of
embedding an ellipsoid into a ball (as shown by work of
McDuffSchlenk) or embedding a disjoint union of balls into a ball. We
also state and present evidence for a conjecture under which the
asymptotics of the ECH capacities of a Liouville domain recover its
symplectic volume. There is some followup discussion in this blog post.
 [12] Proof of the Arnold chord conjecture in three dimensions II (with C. H. Taubes)
Geometry and Topology 17 (2013), 26012688. download.
We complete the proof of the threedimensional chord conjecture from [10] by showing that an exact symplectic cobordism
between contact threemanifolds induces a map on (filtered) embedded
contact homology satisfying certain axioms.
We also prove that filtered embedded contact
homology does not depend on the choice of almost complex structure
used to define it. For more discussion, see this blog
post.
 [13] From one Reeb orbit to two (with D. CristofaroGardiner)
J. Diff. Geom. 102 (2016), 2536.
arXiv version, blog post.
 [14] The asymptotics of ECH capacities (with
D. CristofaroGardiner and V.G.B. Ramos)
Invent. Math. 199 (2015), 187214, arXiv:1210.2167. blog post
 [15] Lecture notes on embedded contact homology
Contact and symplectic topology, Bolyai Soc. Math. Stud. 26 (2014), 389484. Nearly final version at arXiv:1303.5789. blog post
 [16] Symplectic embeddings into fourdimensional concave toric domains (with K. Choi, D. CristofaroGardiner, D. Frenkel and V.G.B. Ramos)
Journal of Topology 7 (2014), 10541076. arXiv:1310.6647, blog post.
 [17] Beyond ECH capacities
Geometry and Topology 20 (2016), 10851126. Almost final version at arXiv:1409.1352. blog post.
 [18] Mean action and the Calabi invariant
Journal of Modern Dynamics 10 (2016), 511539. Almost final version at arXiv:1509.02183. blog post.
 [19] Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (with D. CristofaroGardiner and D. Pomerleano)
arXiv:1701.02262.
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