Let $\gamma$ be an $S^1$-subgroup in $Ham (M, \omega)$ generated by a Morse
Hamiltonian $H$. We give a simple proof of the conjecture stated in
\cite{virtmorse}, relating the Morse index of $ \gamma$, as a critical point of
the Hofer length functional, with the Conley Zehnder index of the extremizer
$x_{\max}$ of $ H$, considered as a periodic orbit of $H$.

We answer the natural question: when are a regular Poisson structure along
with a complex structure transverse to its leaves induced by generalized
complex structure? The leafwise symplectic form and transverse complex
structure determine an obstruction class in a certain cohomology, which
vanishes if and only if our question has an affirmative answer. We first study
a component of this obstruction, which gives the condition that the leafwise
cohomology class of the symplectic form must be transversely pluriharmonic.

We compute the Floer cohomology of monotone tori in the Stein surfaces
obtained by a linear plumbing of cotangent bundles of spheres, also known as
the Milnor fibre associated with the complex surface singularity of type A_n.
We next study some finite quotients of the A_n Milnor fibre which coincide with
the Stein surfaces that appear in Fintushel and Stern's rational blowdown
construction.

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a
Riemann surface \Sigma with boundary, which consists of symplectic vector
bundle on \Sigma and a Lagrangian subbundle on \partial{\Sigma} as well as its
generalization for transversely intersecting Lagrangian boundary conditions. We
discuss their properties and relations to the known topological definitions. As
a main application, we extend Maslov index to the case with orbifold interior
singularites, via curvature integral, and find also an analogous topological
definition in these cases.

Let $M$ be an exact symplectic manifold with contact type boundary such that
$c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya
category of $M$ has the structure of an involutive Lie bialgebra.Inspired by a
work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism
from the linearized contact homology of $M$ to the cyclic cohomology of the
Fukaya category. Our study is also motivated by string topology and
2-dimensional topological conformal field theory.

We give an h-principle type result for a class of Legendrian embeddings in
contact manifolds of dimension at least 5. These Legendrians, referred to as
loose, have trivial pseudo-holomorphic invariants. We demonstrate they are
classified up to ambient contact isotopy by smooth embedding class equipped
with an almost complex framing. This result is inherently high dimensional:
analogous results in dimension 3 are false.

We study symplectic embeddings of ellipsoids into balls. In the main
construction, we show that a given embedding of 2m-dimensional ellipsoids can
be suspended to embeddings of ellipsoids in any higher dimension. In dimension
6,s if the ratio of the areas of any two axes is sufficiently large then the
ellipsoid is flexible in the sense that it fully fills a ball. We also show
that the same property holds in all dimensions for sufficiently thin ellipsoids
E(1,..., a).

We prove Conjecture 1.1 in [Chan-Lau-Leung] for toric Calabi-Yau manifolds of
the form $K_Y$ where $Y$ is a toric Fano manifold. In particular, we show that
the coefficients of the Taylor series expansions of the inverse mirror map for
$K_Y$ can be expressed in terms of disk open Gromov-Witten invariants defined
by Fukaya-Oh-Ohta-Ono.

We prove that any closed connected exact Lagrangian manifold L in a connected
cotangent bundle T*N is up to a finite covering space lift a homology
equivalence. We prove this by constructing a fibrant parametrized family of
ring spectra FL parametrized by the manifold N. The homology of FL will be
(twisted) symplectic cohomology of T*L. The fibrancy property will imply that
there is a Serre spectral sequence converging to the homology of FL and the
product combined with intersection product on N induces a product on this
spectral sequence.

Consider a Stein manifold M obtained by plumbing cotangent bundles of
manifolds of dimension greater than or equal to 3 at points. We prove that the
Fukaya category of closed exact Lagrangians with vanishing Maslov class in M is
generated by the compact cores of the plumbing.

The restricted planar three-body problem has a rich history, yet many
unanswered questions still remain. In the present paper we prove the existence
of a global surface of section near the smaller body in a new range of energies
and mass ratios for which the Hill's region still has three connected
components. The approach relies on recent global methods in symplectic geometry
and contrasts sharply with the perturbative methods used until now.

In the spectral theory of positive elliptic operators, an important role is
played by certain smoothing kernels, related to the Fourier transform of the
trace of a wave operator, which may be heuristically interpreted as smoothed
spectral projectors asymptotically drifting to the right of the spectrum. In
the setting of Toeplitz quantization, we consider analogues of these, where the
wave operator is replaced by the Hardy space compression of a linearized
Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz
operators.

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$
into $J^1(R^2)=R^5$ which lie in pairwise distinct Legendrian isotopy classes
and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally
minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the
Legendrian contact homology DGA does not admit any augmentation over $Z/2Z$,
and hence cannot be linearized. We also investigate these surfaces from the
point of view of the theory of generating families.

We reduce the computation of Poisson traces on quotients of symplectic vector
spaces by finite subgroups of symplectic automorphisms to a finite one, by
proving several results which bound the degrees of such traces as well as the
dimension in each degree. This applies more generally to traces on all
polynomial functions which are invariant under invariant Hamiltonian flow.

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold
$(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two
components, $X$ and $Y$. We first show that, if $\dim(X) + \dim(Y) +2 =
\dim(M)$, then $M$ is simply connected. Using this result and the results in
\cite{LT} on the integral cohomology ring and the Chern classes of $M$, we
obtain further classification results for such manifolds using techniques from
surgery theory. In particular, we prove that up to diffeomorphism there are at
most finitely many such manifolds in each dimension.

We extend the Local-to-Global-Principle used in the proof of convexity
theorems for momentum maps to not necessarily closed maps whose target space
carries a convexity structure which need not be based on a metric. Using a new
factorization of the momentum map, convexity of its image is proved without
local fiber connectedness, and for almost arbitrary spaces of definition.

In this short note we give a complete characterization of a certain class of
compact corank one Poisson manifolds, those equipped with a closed one-form
defining the symplectic foliation and a closed two-form extending the
symplectic form on each leaf. If such a manifold has a compact leaf, then all
the leaves are compact, and furthermore the manifold is a mapping torus of a
compact leaf. These manifolds and their regular Poisson structures admit an
extension as the critical hypersurface of a b-Poisson manifold as we consider
in another paper.

In this note we discuss the effect of the symplectic sum along spheres in
symplectic four-manifolds on the Kodaira dimension of the underlying symplectic
manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we
are able to obtain precise results on the structure of the manifold obtained
from the blow down of an embedded symplectic -4-sphere.

We study the class of norms on the space of smooth functions on a closed
symplectic manifold, which are invariant under the action of the group of
Hamiltonian diffeomorphisms. Our main result shows that any such norm that is
continuous with respect to the $C^{\infty}$-topology, is dominated from above
by the $L_{\infty}$-norm.

We use symplectic cohomology to study the non-uniqueness of symplectic
structures on the smooth manifolds underlying affine varieties. Starting with a
Lefschetz fibration on such a variety and a finite set of primes, the main new
tool is a method, which we call homologous recombination, for constructing a
Lefschetz fibration whose total space is smoothly equivalent to the original
variety, but for which symplectic cohomology with coefficients in the given set
of primes vanishes (there is also a simpler version that kills symplectic
cohomology completely).

The Runge approximation theorem for holomorphic maps (U -> C) is a
fundamental result in complex analysis. The aim of this article is to prove
such a result for (pseudo-)holomorphic maps from a compact Riemann surface to a
compact (almost-)complex manifold M under certain assumptions. Though the
setting is definitively that of pseudo-holomorphic maps it also covers some
complex varieties.

Rabinowitz Floer homology has been investigated on a submanifold of contact
type. The contact condition, however, is quite restrictive. For example, a
product of contact hypersurfaces is rarely of contact type. In this article, we
study Rabinowitz Floer homology for a class of non-contact submanifolds. We
show for this example that there are infinitely many leafwise intersection
points by proving a K\"unneth formula for Rabinowitz Floer homology.

In this paper we discuss the change in contact structures as their supporting
open book decompositions have their binding components cabled. To facilitate
this and applications we define the notion of a rational open book
decomposition that generalizes the standard notion of open book decomposition
and allows one to more easily study surgeries on transverse knots. As a
corollary to our investigation we are able to show there are Stein fillable
contact structures supported by open books whose monodromies cannot be written
as a product of positive Dehn twists.

One constructs lagrangian fibrations on the flag variety $F^3$ and proves
that the fibrations are special.

Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which
has been cut into several pieces along vertical lines, we assign a differential
graded algebra to each piece and prove a van Kampen theorem describing the
Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then
use this theorem to construct maps between the invariants of Legendrian knots
related by certain tangle replacements, and to describe the linearized contact
homology of Legendrian Whitehead doubles.

This thesis studies the pre-quantization of quasi-Hamiltonian group actions
from a cohomological viewpoint. The compatibility of pre-quantization with
symplectic reduction and the fusion product are established, and are used to
understand the sufficient conditions for the pre-quantization of $M_G(\Sigma)$,
the moduli space of flat $G$-bundles over a closed surface $\Sigma$. For a
simply connected, compact, simple Lie group $G$, $M_G(\Sigma)$ is known to be
pre-quantizable at integer levels.

We prove homological mirror symmetry for Lefschetz fibrations obtained as
disconnected sums of polynomials of types A or D. The proof is based on the
behavior of the Fukaya category under the addition of a polynomial of type D.

The study of fixed points is a classical subject in geometry and dynamics. If
the circle acts in a Hamiltonian fashion on a compact symplectic manifold M,
then it is classically known that there are at least 1 + dim(M)/2 fixed points;
this follows from Morse theory for the momentum map of the action.

This paper deals with multiplicity of rotation type solutions for Hamiltonian
systems on $T^\ell\times \mathbb{R}^{2n-\ell}$. It is proved that, for every
spatially periodic Hamiltonian system, i.e., the case $\ell=n$, there exist at
least $n+1$ geometrically distinct rotation type solutions with given energy
rotation vector.

We review first Azumaya geometry and D-branes in the realm of algebraic
geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work
and then use it as background to introduce Azumaya $C^{\infty}$-manifolds with
a fundamental module and morphisms therefrom to a projective complex manifold.
This gives us a description of D-branes of A-type. Donaldson's picture of
Lagrangian and special Lagrangian submanifolds as selected from the zero-locus
of a moment map on a related space of maps can be merged into the setting.

In this paper we generalize the Rabinowitz Floer theory which has been
established in the hypersurfaces case. We apply it to the coisotropic
intersection problem which interpolates between the Lagrangian intersection
problem and the closed orbit problem. More specifically, we study leafwise
intersections on a contact submanifold and the displacement energy of a stable
submanifold. Moreover we prove that the Rabinowitz action functional is
generically Morse, so that Rabinowitz Floer homology is well-defined.

This survey wants to give a short introduction to the transversality problem
in symplectic field theory and motivate to approach it using the new Fredholm
theory by Hofer, Wysocki and Zehnder. With this it should serve as a lead-in
for the user's guide to polyfolds, which will appear soon and is the result of
a working group organized by J. Fish, R. Golovko and the author at MSRI
Berkeley in fall 2009.

In this article we prove that for a smooth fiberwise convex Hamiltonian, the
asymptotic Hofer distance from the identity gives a strict upper bound to the
value at 0 of Mather's $\beta$ function, thus providing a negative answer to a
question asked by K. Siburg in \cite{Siburg1998}. However, we show that
equality holds if one considers the asymptotic distance defined in
\cite{Viterbo1992}.

Schwarz showed that when a closed symplectic manifold (M,\om) is
symplectically aspherical (i.e. the symplectic form and the first Chern class
vanish on \pi_2(M)) then the spectral invariants, which are initially defined
on the universal cover of the Hamiltonian group, descend to the Hamiltonian
group Ham (M,\om). In this note we describe less stringent conditions on the
Chern class and quantum homology of M under which the (asymptotic) spectral
invariants descend to Ham (M,\om).

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds
in a symplectic manifold M. Suppose that M carries a symplectic involution,
which preserves both submanifolds. Under various topological hypotheses, we
prove a localization theorem for Floer cohomology, which implies a Smith-type
inequality for the Floer cohomology groups in M and its fixed point set. Two
applications to symplectic Khovanov cohomology are included.

In this article, using the idea of toric degeneration and the computation of
the full potential function of Hirzebruch surface $F_2$, which is \emph{not}
Fano, we produce a continuum of Lagrangian tori in $S^2 \times S^2$ which are
non-displaceable under the Hamiltonian isotopy.

Consider the cotangent bundle of a Riemannian manifold $(M,g)$ endowed with a
twisted symplectic structure defined by a closed weakly exact 2-form $\sigma$
on $M$ whose lift to the universal cover of $M$ admits a bounded primitive. We
compute the Rabinowitz Floer homology of energy hypersurfaces
$\Sigma_{k}=H^{-1}(k)$ of mechanical (kinetic energy + potential) Hamiltonians
$H$ for the case when the energy value k is greater than the Mane critical
value c.

Given a collection of exact Lagrangians in a Liouville manifold, we construct
a map from the Hochschild homology of the Fukaya category that they generate to
symplectic cohomology. Whenever the identity in symplectic cohomology lies in
the image of this map, we conclude that every Lagrangian lies in the idempotent
closure of the chosen collection. The main new ingredients are (1) the
construction of operations controlled by discs with two outputs on the Fukaya
category, and (2) the Cardy relation.

Rabinowitz Floer homology is the semi-infinite dimensional Morse homology
associated to the Rabinowitz action functional used in the pioneering work of
Rabinowitz. Gradient flow lines are solutions of a vortex-like equation. In
this survey article we describe the construction of Rabinowitz Floer homology
and its applications to symplectic and contact topology, global Hamiltonian
perturbations and the study of magnetic fields.

We present a discrete analog of the recently introduced Hamilton-Pontryagin
variational principle in Lagrangian mechanics. This unifies two, previously
disparate approaches to discrete Lagrangian mechanics: either using the
discrete Lagrangian to define a finite version of Hamilton's action principle,
or treating it as a symplectic generating function. This is demonstrated for a
discrete Lagrangian defined on an arbitrary Lie groupoid; the often encountered
special case of the pair groupoid (or Cartesian square) is also given as a
worked example.

In [EH89, Theorem 1] Ekeland-Hofer prove that for a centrally symmetric,
restricted contact type hypersurface in R^{2n} and for any global, centrally
symmetric Hamiltonian perturbation there exists a leaf-wise intersection point.
In this note we show that if we replace restricted contact type by star-shaped
there exists infinitely many leaf-wise intersection points or a leaf-wise
intersection point on a closed characteristic.

Spectral invariant were introduced in Hamiltonian Floer homology by Viterbo,
Oh, and Schwarz. We extend this concept to Rabinowitz Floer homology. As an
application we derive new quantitative existence results for leaf-wise
intersections. The importance of spectral invariants for the presented
application is that spectral invariants allow us to derive existence of
critical points of the Rabinowitz action functional even in degenerate
situations where the functional is not Morse.

We describe families of monotone symplectic manifolds constructed via the
symplectic cutting procedure of Lerman from the cotangent bundle of manifolds
endowed with a free circle action. We also give obstructions to the monotone
Lagrangian embedding of some compact manifolds in these symplectic manifolds.

We relate previously defined quantum characteristic classes to Morse
theoretic aspects of the Hofer length functional on $\ls$. As an application we
prove a theorem which can be interpreted as stating that this functional
behaves "virtually" as a perfect Morse-Bott functional with a flow. This can be
applied to study topology and Hofer geometry of $ \text {Ham}(M, \omega)$. We
also use this to give a prediction for the index of some geodesics for this
functional, which was recently partially verified by Yael Karshon and Jennifer
Slimowitz.

We prove that an arbitrary Poisson structure omega^{ij}(u) and an arbitrary
closed 3-form T_{ijk}(u) generate the local Poisson structure A^{ij}(u,u_x) =
M^i_s(u,u_x)omega^{sj}(u), where M^i_s(u,u_x)(delta^s_j +
omega^{sp}(u)T_{pjk}(u)u^k_x) = delta^i_j, on the corresponding loop space. We
obtain also a special graded epsilon-deformation of an arbitrary Poisson
structure omega^{ij}(u) by means of an arbitrary closed 3-form T_{ijk}(u).

Here we develop some basic analytic tools to study compactness properties of
$J$-curves (i.e.

This (partially expository) paper discusses Lagrangian Floer cohomology in
the context of Lefschetz fibrations, with emphasis on the algebraic structures
encountered there. In addition to the well-known directed A_infinity algebras
which appear in this situation, one has additional information encoded in a
certain bimodule homomorphism. There are two approaches to constructing this
homomorphism: in terms of the (noncompact) Lefschetz thimbles in the total
space, or else in terms of vanishing cycle in the fibre.

Starting from the work of Bhupal, we extend to the contact case the Viterbo
capacity and Traynor's construction of symplectic homology. As an application
we get a new proof of the Non-Squeezing Theorem of Eliashberg, Kim and
Polterovich.

We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a
Floer-theoretic quantity called its "boundary depth," and establish basic
results about how the boundary depths of different Hamiltonians are related. As
applications, we prove that certain Hamiltonian symplectomorphisms supported in
displaceable subsets have infinitely many nontrivial geometrically distinct
periodic points, and we also significantly expand the class of coisotropic
submanifolds which are known to have positive displacement energy.

The main purpose of the present paper is a study of orientations of the
moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real}
Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic
involutions $\tau$ on a symplectic manifold. We introduce the notion of
$\tau$-relatively spin structure for an anti-symplectic involution $\tau$, and
study how the orientations on the moduli space behave under the involution
$\tau$. We also apply this to the study of Lagrangian Floer theory of real
Lagrangian submanifolds.

A presymplectic structure on odd dimensional manifold is given by a closed
2-form which is nondegenerate, i.e., of maximal rank. We investigate geometry
of presymplectic manifolds. Some basic theorems analogous to those in
symplectic and contact topology are given and applied to study constructions of
presymplectic manifolds. In particular, we describe how to glue presymplectic
manifolds along a presymplectic submanifold, including surgery along a
presymplectic circles.

We associate an A-infinity category with a dimer model, and show that it is
derived-equivalent to the category of representations of the quiver with
relations associated with the dimer model if the dimer model is consistent. We
also associate an exact Lefschetz fibration with a pair of a dimer model and an
internal perfect matching on it, and use it to prove a version of homological
mirror symmetry for two-dimensional toric Fano stacks.

We show that the Hamiltonian Lagrangian monodromy group, in its homological
version, is trivial for any weakly exact Lagrangian submanifold of a symplectic
manifold. The proof relies on a sheaf approach to Floer homology given by a
relative Seidel morphism.

This paper calculates the function $c(a)$ whose value at $a$ is the infimum
of the size of a ball that contains a symplectic image of the ellipsoid
$E(1,a)$. (Here $a \ge 1$ is the ratio of the area of the large axis to that of
the smaller axis.) The structure of the graph of $c(a)$ is surprisingly rich.
The volume constraint implies that $c(a)$ is always greater than or equal to
the square root of $a$, and it is not hard to see that this is equality for
large $a$.

We define relative Ruan invariants that count embedded connected symplectic
submanifolds which contact a fixed stable symplectic hypersurface V in a
symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders
(in addition to insertions on X\V) for stable V. We obtain invariants of the
deformation class of (X,V,w). Two large issues must be tackled to define such
invariants: (1) Curves lying in the hypersurface V and (2) genericity results
for almost complex structures constrained to make V pseudo-holomorphic (or
almost complex).

The theorem of Chekhanov asserts that a Lagrangian submanifold L has positive
displacement energy under natural assumptions on the symplectic topology at
infinity. It is greater than or equal to the minimal area of holomorphic disks
bounded by L. This estimate was obtained by Y.V. Chekhanov in 1998. Section 1
presents a direct proof based on the use of holomorphic curves and their
Hamiltonian perturbations. In section 2, we define a filtered version of the
Lagrangian Floer homology, without any assumption on L.

The aim of this paper is to address the following question: given a contact
manifold $(\Sigma, \xi)$, what can be said about the aspherical symplectic
manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$ ? We first extend a theorem
of Eliashberg, Floer and McDuff to prove that under suitable assumptions the
map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We
then apply this method in the case of contact manifolds having a contact
embedding in $ {\mathbb R}^{2n}$ or in a subcritical Stein manifold.

This is the first of a series of papers on foundations of Floer theory. We
give an axiomatic treatment of the geometric notion of a semi-infinite cycle.
Using this notion, we introduce a bordism version of Floer theory for the
cotangent bundle of a compact manifold M. Our construction is geometric and
does not require the compactness and gluing results traditionally used to setup
Floer theory. Finally, we prove a bordism version of Viterbo's theorem relating
Floer bordism of the cotangent bundle to the ordinary bordism groups of the
free loop space of M.

The aim of this paper is to explain a link between symplectic isotopies of
open objects such as balls and flexibility properties of symplectic
hypersurfaces. We get connectedness results for spaces of symplectic ellipsoids
or maximal packings of $P^2$.

This paper establishes compactness results for the moduli stack of
holomorphic curves in suitable exploded manifolds. This result together with
the analysis in arXiv:0902.0087 allows the definition of Gromov Witten
invariants of these exploded manifolds.

In this paper, we prove that the "quantization commutes with reduction"
phenomenon of Guillemin-Sternberg applies in the context of the metaplectic
correction.

Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed
almost-K\"ahler manifolds, we extend to arbitrary dimension a non-existence
result proved in complex dimension 2.

The paper is a summary of the results of the authors concerning computation
of symplectic invariants of Weinstein manifolds. We also sketch the proofs and
consider some examples and applications.

This note studies the geometric structure of monotone moment polytopes (the
duals of smooth Fano polytopes) using probes. The latter are line segments that
enter the polytope at an interior point of a facet and whose direction is
integrally transverse to this facet. A point inside the polytope is
displaceable by a probe if it lies less than half way along it. Using a
construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope
has a central point that is not displaceable by probes.

In his 1992 article on generating functions Viterbo constructed a
bi-invariant metric on the group of compactly supported Hamiltonian
symplectomorphisms of R^2n. Using the set-up of 0901.3112 we extend the Viterbo
metric to the group of compactly supported contactomorphisms of R^2n x S^1
isotopic to the identity. We also prove that the contactomorphism group is
unbounded with respect to this metric.

The purpose of this paper is to describe a method for computing homotopy
groups of the space of $\alpha$-stable representations of a quiver with fixed
dimension vector and stability parameter $\alpha$. The main result is that the
homotopy groups of this space are trivial up to a certain dimension, which
depends on the quiver, the choice of dimension vector, and the choice of
parameter.

Fix a compact 4-dimensional manifold with self-dual 2nd Betti number one and
with a given symplectic form. This article proves the following: The Frechet
space of tamed almost complex structures as defined by the given symplectic
form has an open and dense subset whose complex structures are compatible with
respect to a symplectic form that is cohomologous to the given one. The theorem
is proved by constructing the new symplectic form by integrating over a space
of currents that are defined by pseudo-holomorphic curves.

We define the Maslov index of a loop tangent to the characteristic foliation
of a coisotropic submanifold as the mean Conley--Zehnder index of a path in the
group of linear symplectic transformations, incorporating the "rotation" of the
tangent space of the leaf -- this is the standard Lagrangian counterpart -- and
the holonomy of the characteristic foliation. Furthermore, we show that, with
this definition, the Maslov class rigidity extends to the class of the
so-called stable coisotropic submanifolds including Lagrangian tori and stable
hypersurfaces.

This paper provides an introduction to exploded manifolds. The category of
exploded manifolds is an extension of the category of smooth manifolds with an
excellent holomorphic curve theory. Each exploded manifold has a tropical part
which is a union of convex polytopes glued along faces. Exploded manifolds are
useful for defining and computing Gromov Witten invariants relative to normal
crossing divisors, and using tropical curve counts to compute Gromov Witten
invariants.

In the early 1970's, Gelfand, Kalinin and Fuks found an exotic characteristic
class of degree 7 in the Gelfand-Fuks cohomology of the Lie algebra of formal
Hamiltonian vector fields on the plane. We prove that this cohomology class can
be decomposed as a product of a certain leaf cohomology class of degree 5 and
the transverse symplectic class. This is similar to the well known
factorization of the Godbillon-Vey class for codimension n foliations.

We prove that the algebra of singular cochains on a smooth manifold, equipped
with the cup product, is equivalent to the A-infinity structure on the
Lagrangian Floer cochain group associated to the zero section in the cotangent
bundle.

In this article, we study the question of existence of leafwise intersection
points for contact manifolds which are not necessarily of restricted contact
type. Moreover we can find a leafwise intersection point on the symplectization
for special Hamiltonian functions.

This paper is a fusion of a survey and a research article. We focus on
certain rigidity phenomena in function spaces associated to a symplectic
manifold. Our starting point is a lower bound obtained in an earlier paper with
Zapolsky for the uniform norm of the Poisson bracket of a pair of functions in
terms of symplectic quasi-states. After a short review of the theory of
symplectic quasi-states, we extend this bound to the case of iterated Poisson
brackets. A new technical ingredient is the use of symplectic integrators.

We estimate the Hamiltonian displacement energy of a bidisk inside a
cylinder.

We consider pairs of Lagrangian submanifolds $(L_0,L), (L_1, L)$ belonging to
the class of Lagrangian submanifolds with \emph{conic} ends on \emph{Weinstein
manifolds}. The main purpose of the present paper is to define a canonical
chain map $h_\CL: CF(L_0,L) \to CF(L_1,L)$ of Lagrangian Floer complex inducing
an isomorphism in homology, under the Hamiltonian isotopy $\CL=\{L_s\}_{0 \leq
s\leq 1}$ generated by \emph{conic} Hamiltonian functions such that the
intersections $L \cap L_s$ do not escape to infinity.

Complete integrability in a symplectic setting means the existence of a
Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we
describe complete integrability in a contact set-up as a more subtle structure:
a flag of two foliations, Legendrian and co-Legendrian, and a
holonomy-invariant transverse measure of the former in the latter. This turns
out to be equivalent to the existence of a canonical $\R\ltimes \R^{n-1}$
structure on the leaves of the co-Legendrian foliation.

Let X(\Sigma) be a smooth projective toric variety for a complex torus T_\C.
In this paper, a real T_\C-invariant Poisson structure \Pi_\Sigma is
constructed on the complex manifold X(\Sigma), the symplectic leaves of which
are the T_\C-orbits in X(\Sigma). It is shown that each leaf admits a
Hamiltonian action by a sub-torus of the compact torus T\subset T_\C. However,
the global action of T_\C on (X(\Sigma),\Pi_\Sigma) is Poisson but not
Hamiltonian. The main result of the paper is a lower bound for the first
Poisson cohomology of these structures.

Let $(V, \Om)$ be a symplectic vector space and let $\phi: M \ra V$ be a
symplectic immersion. We show that $\phi(M) \subset V$ is (locally) an
extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if
and only if the second fundamental form of $\phi$ is parallel.

Furthermore, we show that any symmetric space which admits an immersion as an
e.s.s.s. also admits a {\em full} such immersion, i.e., such that $\phi(M)$ is
not contained in a proper affine subspace of $V$, and this immersion is unique
up to affine equivalence.

We state Bennequin inequalities in the relative case, and show that the
relative invariants are additive under relative connected sums. We show they
exhibit similar limitations as their classical analogues. We study relatively
Legendrian simple knots and give some classification results.

We define $S^1$-equivariant symplectic homology for symplectically aspherical
manifolds with contact boundary, using a Floer-type construction first proposed
by Viterbo. We show that it is related to the usual symplectic homology by a
Gysin exact sequence. As an important ingredient of the proof, we define a
parametrized version of symplectic homology, corresponding to families of
Hamiltonian functions indexed by a finite dimensional smooth parameter space.
We define a parametrized version of the Robbin-Salamon index, which gives the
grading for these new versions of symplectic homology.

An origami manifold is a manifold equipped with a closed 2-form which is
symplectic except on a hypersurface where it is like the pullback of a
symplectic form by a folding map and its kernel defines a circle fibration. We
can move back and forth between origami and symplectic manifolds using cutting
(unfolding) and radial blow-up (folding), modulo compatibility conditions.

This paper explains an application of Gromov's h-principle to prove the
existence, on any orientable 4-manifold, of a folded symplectic form.

We construct a smooth codimension-one foliation on the five-sphere in which
every leaf is a symplectic four-manifold and such that the symplectic structure
varies smoothly. Our construction implies the existence of a complete regular
Poisson structure on the five-sphere.

Recently, Gaiotto-Moore-Neitzke \cite{GMN08} proposed a new construction of
hyperk\"{a}hler metrics. In particular, they give a new construction of the
Ooguri-Vafa metric, from which one comes across certain formulas, which we
interpret as wall-crossing formulas that appear in Auroux's construction
\cite{Auroux07}, \cite{Auroux09} of instanton-corrected mirror manifolds. This
shows that the Ooguri-Vafa metric is closely related to the counting of
nontrivial holomorphic discs with boundary on special Lagrangian torus fibers.

Let $M$ be a connected smooth $G$-manifold, where $G$ is a connected compact
Lie group. In this paper, we first study the relation between $\pi_1(M)$ and
$\pi_1(M/G)$. Then we particularly focus on the case when $M$ is a connected
Hamiltonian $G$-manifold with an equivariant moment map $\phi$. In \cite{L2},
for {\em compact} $M$, we proved that $\pi_1(M)\cong \pi_1(M/G)\cong\pi_1(M_a)$
for all $a\in {image}(\phi)$, where $M_a$ is the symplectic quotient at $a$. We
generalize and extend these results to Hamiltonian $G$-manifolds with {\em
proper} moment maps.

We introduce certain homology and cohomology subgroups for any almost complex
structure and study their pureness, fullness and duality properties. Motivated
by a question of Donaldson, we use these groups to relate J-tamed symplectic
cones and J-compatible symplectic cones over a large class of almost complex
manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and
complex surfaces.

We study the parametrized Hamiltonian action functional for
finite-dimensional families of Hamiltonians. We show that the linearized
operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic
choice of Hamiltonian and almost complex structure. We also establish the
Fredholm property and transversality for generic $S^1$-invariant families of
Hamiltonians and almost complex structures, parametrized by odd-dimensional
spheres. This is a foundational result used to define $S^1$-equivariant Floer
homology.

Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie
group $G$ acts on $M$ transverse to the contact distribution $E$. In
arxiv:0712.2431v4, we defined a $G$-transversally elliptic Dirac operator
$\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on
the symplectic vector bundle $E\to M$, whose equivariant index is well-defined
as a generalized function on $G$, and gave a formula for its index.

Lagrangian Floer homology in a general case has been constructed by Fukaya,
Oh, Ohta and Ono, where they construct an $\AI$-algebra or an $\AI$-bimodule
from Lagrangian submanifolds, and studied the obstructions and deformation
theories. But for obstructed Lagrangian submanifolds, standard Lagrangian Floer
homology can not be defined.

In lines 8-11 of \cite[pp.

In arXiv:math/0605587, the first two authors have constructed a
gauge-equivariant Morse stratification on the space of connections on a
principal U(n)-bundle over a connected, closed, nonorientable surface. This
space can be identified with the real locus of the space of connections on the
pullback of this bundle over the orientable double cover of this nonorientable
surface. In this context, the normal bundles to the Morse strata are real
vector bundles.

We study the dynamics and symplectic topology of energy hypersurfaces of
mechanical Hamiltonians on twisted cotangent bundles. We pay particular
attention to periodic orbits, displaceability, stability and the contact type
property, and the changes that occur at the Mane critical value c. Our main
tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces
that are either stable tame or virtually contact, and it is invariant under
under homotopies in these classes.

We relate the Gromov-Witten invariants of $X\times S^2$ to the Seiberg-Witten
invariants of $X$ where $X$ is a simply-connected symplectic 4-manifold. We
also give examples that expose the similarity between the classification of
smooth 4-manifolds and some classification problems regarding symplectic
6-manifolds.

We generalize a result of Kostant and Wallach concerning the algebraic
integrability of the Gelfand-Zeitlin vector fields to the full set of strongly
regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to
stratify the strongly regular set by subvarieties $X_{D}$. We construct an
\'{e}tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and
$\hat{\mathfrak{g}}$ are smooth and irreducible.

To every Poisson algebraic variety X over an algebraically closed field of
characteristic zero, we canonically attach a right D-module M(X) on X. If X is
affine, solutions of M(X) in the space of algebraic distributions on X are
Poisson traces on X, i.e., distributions invariant under Hamiltonian flows.
When X has finitely many symplectic leaves, we prove that M(X) is holonomic.
Thus, when X is affine and has finitely many symplectic leaves, the space of
Poisson traces on X is finite-dimensional.

We generalize a result of Kostant and Wallach concerning the algebraic
integrability of the Gelfand-Zeitlin vector fields to the full set of strongly
regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to
stratify the strongly regular set by subvarieties $X_{D}$. We construct an
\'{e}tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and
$\hat{\mathfrak{g}}$ are smooth and irreducible.

We prove several generic existence results for infinitely many periodic
orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that
a Hamiltonian diffeomorphism of a complex projective space or Grassmannian
generically has infinitely many periodic orbits. We also consider
symplectomorphisms of the two-torus with irrational flux. We show that such a
symplectomorphism necessarily has infinitely many periodic orbits whenever it
has one and all periodic points are non-degenerate.

We give an example of a symplectic manifold with a stable hypersurface such
that nearby hypersurfaces are typically unstable.