1. CJM 2015 (vol 67 pp. 1109)
 Nohara, Yuichi; Ueda, Kazushi

Goldman Systems and Bending Systems
We show that the moduli space
of parabolic bundles on the projective line
and the polygon space are isomorphic,
both as complex manifolds
and symplectic manifolds equipped with structures of completely integrable systems,
if the stability parameters are
small.
Keywords:toric degeneration Categories:53D30, 14H60 

2. CJM 2013 (vol 66 pp. 31)
 Bailey, Michael

Symplectic Foliations and Generalized Complex Structures
We answer the natural question: when is a transversely holomorphic
symplectic foliation induced by a 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. As a consequence, under certain topological
hypotheses, we infer that we actually have a symplectic fibre bundle
over a complex base. We then show how to compute the full obstruction
via a spectral sequence. We give various concrete necessary and
sufficient conditions for the vanishing of the obstruction.
Throughout, we give examples to test the sharpness of these
conditions, including a symplectic fibre bundle over a complex base
which does not come from a generalized complex structure, and a
regular generalized complex structure which is very unlike a
symplectic fibre bundle, i.e., for which nearby leaves are not
symplectomorphic.
Keywords:differential geometry, symplectic geometry, mathematical physics Category:53D18 

3. CJM 2012 (vol 65 pp. 553)
4. CJM 2012 (vol 65 pp. 467)
 Wilson, Glen; Woodward, Christopher T.

Quasimap Floer Cohomology for Varying Symplectic Quotients
We show that quasimap Floer cohomology for varying symplectic
quotients resolves several puzzles regarding displaceability of toric
moment fibers. For example, we (i) present a compact Hamiltonian
torus action containing an open subset of nondisplaceable
orbits and a codimension four singular set, partly answering a
question of McDuff, and (ii) determine displaceability for most of the
moment fibers of a symplectic ellipsoid.
Keywords:Floer cohomology, Hamiltonian displaceability Category:53Dxx 

5. CJM 2011 (vol 64 pp. 991)
 Damianou, Pantelis A.; Petalidou, Fani

Poisson Brackets with Prescribed Casimirs
We consider the problem of constructing Poisson brackets on smooth
manifolds $M$ with prescribed Casimir functions. If $M$ is of even
dimension, we achieve our construction by considering a suitable
almost symplectic structure on $M$, while, in the case where $M$ is
of odd dimension, our objective is achieved by using a convenient
almost cosymplectic structure. Several examples and applications are
presented.
Keywords:Poisson bracket, Casimir function, almost symplectic structure, almost cosymplectic structure Categories:53D17, 53D15 

6. CJM 2011 (vol 63 pp. 878)
 Howard, Benjamin; Manon, Christopher; Millson, John

The Toric Geometry of Triangulated Polygons in Euclidean Spac
Speyer and Sturmfels associated GrÃ¶bner toric
degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$
of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each
trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations
induce toric
degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the
space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line.
Our goal in this paper is to give a
geometric (Euclidean polygon) description of the toric fibers
and describe the action of the
compact part of the torus
as "bendings of polygons".
We prove the conjecture of Foth and Hu that
the toric fibers are homeomorphic
to the spaces defined by Kamiyama and Yoshida.
Categories:14L24, 53D20 

7. CJM 2011 (vol 63 pp. 938)
 LiBland, David

AVCourant Algebroids and Generalized CR Structures
We construct a generalization of Courant algebroids that are
classified by the third cohomology group $H^3(A,V)$, where $A$ is a
Lie Algebroid, and $V$ is an $A$module. We see that both Courant
algebroids and $\mathcal{E}^1(M)$ structures are examples of
them. Finally we introduce generalized CR structures on a manifold,
which are a generalization of generalized complex structures, and show
that every CR structure and contact structure is an example of a
generalized CR structure.
Category:53D18 

8. CJM 2010 (vol 62 pp. 975)
 Bjorndahl, Christina; Karshon, Yael

Revisiting TietzeNakajima: Local and Global Convexity for Maps
A theorem of Tietze and Nakajima, from 1928, asserts that
if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex,
then it is convex.
We give an analogous ``local to global convexity" theorem
when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map
from a topological space $X$ to $\mathbb{R}^n$ that satisfies
certain local properties.
Our motivation comes from the CondevauxDazordMolino proof
of the AtiyahGuilleminSternberg convexity theorem in symplectic geometry.
Categories:53D20, 52B99 

9. CJM 2010 (vol 62 pp. 1082)
 Godinho, Leonor; SousaDias, M. E.

The Fundamental Group of $S^1$manifolds
We address the problem of computing the fundamental
group of a symplectic $S^1$manifold for nonHamiltonian actions on
compact manifolds, and for Hamiltonian actions on noncompact
manifolds with a proper moment map. We generalize known results for
compact manifolds equipped with a Hamiltonian $S^1$action. Several
examples are presented to illustrate our main results.
Categories:53D20, 37J10, 55Q05 

10. CJM 2008 (vol 60 pp. 572)
 Hitrik, Michael; Sj{östrand, Johannes

NonSelfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point
This is the third in a series of works devoted to spectral
asymptotics for nonselfadjoint
perturbations of selfadjoint $h$pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[1/C,1/C]+i\epsilon [F_01/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.
Keywords:nonselfadjoint, eigenvalue, periodic flow, branching singularity Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 

11. CJM 2007 (vol 59 pp. 981)
 Jiang, Yunfeng

The ChenRuan Cohomology of Weighted Projective Spaces
In this paper we study the ChenRuan cohomology ring of weighted
projective spaces. Given a weighted projective space ${\bf
P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted
sectors and the corresponding degree shifting numbers. The main
result of this paper is that the obstruction bundle over any
3\nobreakdashmulti\sector is a direct sum of line bundles which we use to
compute the orbifold cup product. Finally we compute the
ChenRuan cohomology ring of weighted projective space ${\bf
P}^{5}_{1,2,2,3,3,3}$.
Keywords:ChenRuan cohomology, twisted sectors, toric varieties, weighted projective space, localization Categories:14N35, 53D45 

12. CJM 2007 (vol 59 pp. 845)
 Schaffhauser, Florent

Representations of the Fundamental Group of an $L$Punctured Sphere Generated by Products of Lagrangian Involutions
In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixedpoint set of an
antisymplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixedpoint set is
nonempty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasiHamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasiHamiltonian $U$space
$(M, \w, \mu\from M \to U)$ for it to induce an antisymplectic involution on
the reduced space $M/\!/U := \mu^{1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, antisymplectic involutions, quasiHamiltonian Categories:53D20, 53D30 

13. CJM 2006 (vol 58 pp. 362)
 Goldin, R. F.; Martin, S.

Cohomology Pairings on the Symplectic Reduction of Products
Let $M$ be the product of two compact Hamiltonian
$T$spaces $X$ and $Y$. We present a formula for evaluating
integrals on the symplectic reduction of $M$ by the diagonal $T$
action. At every regular value of the moment map for $X\times Y$, the
integral is the convolution of two distributions associated to the
symplectic reductions of $X$ by $T$ and of $Y$ by $T$. Several
examples illustrate the computational strength of this relationship.
We also prove a linear analogue which can be used to find cohomology
pairings on toric orbifolds.
Category:53D20 

14. CJM 2005 (vol 57 pp. 1314)
 Zhitomirskii, M.

Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the DarbouxGivental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 

15. CJM 2005 (vol 57 pp. 750)
 Sabourin, Hervé

Sur la structure transverse Ã une orbite nilpotente adjointe
We are interested in Poisson structures to
transverse nilpotent adjoint orbits in a complex semisimple Lie algebra,
and we study their polynomial nature. Furthermore, in the case
of $sl_n$,
we construct some families of nilpotent orbits with quadratic
transverse structures.
Keywords:nilpotent adjoint orbits, conormal orbits, Poisson transverse structure Categories:22E, 53D 

16. CJM 2005 (vol 57 pp. 114)
 Flaschka, Hermann; Millson, John

Bending Flows for Sums of Rank One Matrices
We study certain symplectic quotients of $n$fold products of
complex projective $m$space by the unitary group acting
diagonally. After studying nonemptiness and smoothness of these
quotients we construct the actionangle variables, defined on an open
dense subset, of an integrable Hamiltonian system. The semiclassical
quantization of this system reporduces formulas from the
representation theory of the unitary group.
Category:53D20 

17. CJM 2004 (vol 56 pp. 553)
 Mohammadalikhani, Ramin

Cohomology Ring of Symplectic Quotients by Circle Actions
In this article we are concerned with how to compute the cohomology ring
of a symplectic quotient by a circle action using the information we have
about the cohomology of the original manifold and some data at the fixed
point set of the action. Our method is based on the TolmanWeitsman theorem
which gives a characterization of the kernel of the Kirwan map. First we
compute a generating set for the kernel of the Kirwan map for the case of
product of compact connected manifolds such that the cohomology ring of each
of them is generated by a degree two class. We assume the fixed point set is
isolated; however the circle action only needs to be ``formally Hamiltonian''.
By identifying the kernel, we obtain the cohomology ring of the symplectic
quotient. Next we apply this result to some special cases and in particular
to the case of products of two dimensional spheres. We show that the results
of Kalkman and HausmannKnutson are special cases of our result.
Categories:53D20, 53D30, 37J10, 37J15, 53D05 

18. CJM 2002 (vol 54 pp. 3)
 Alekseev, A.; KosmannSchwarzbach, Y.; Meinrenken, E.

QuasiPoisson Manifolds
A quasiPoisson manifold is a $G$manifold equipped with an invariant
bivector field whose Schouten bracket is the trivector field generated
by the invariant element in $\wedge^3 \g$ associated to an invariant
inner product. We introduce the concept of the fusion of such
manifolds, and we relate the quasiPoisson manifolds to the previously
introduced quasiHamiltonian manifolds with groupvalued moment maps.
Category:53D 

19. CJM 2002 (vol 54 pp. 30)
 Treloar, Thomas

The Symplectic Geometry of Polygons in the $3$Sphere
We study the symplectic geometry of the moduli spaces
$M_r=M_r(\s^3)$ of closed $n$gons with fixed sidelengths in the
$3$sphere. We prove that these moduli spaces have symplectic
structures obtained by reduction of the fusion product of $n$
conjugacy classes in $\SU(2)$ by the diagonal conjugation action of
$\SU(2)$. Here the fusion product of $n$ conjugacy classes is a
Hamiltonian quasiPoisson $\SU(2)$manifold in the sense of
\cite{AKSM}. An integrable Hamiltonian system is constructed on
$M_r$ in which the Hamiltonian flows are given by bending polygons
along a maximal collection of nonintersecting diagonals. Finally,
we show the symplectic structure on $M_r$ relates to the
symplectic structure obtained from gaugetheoretic description of
$M_r$. The results of this paper are analogues for the $3$sphere of
results obtained for $M_r(\h^3)$, the moduli space of $n$gons with
fixed sidelengths in hyperbolic $3$space \cite{KMT}, and for
$M_r(\E^3)$, the moduli space of $n$gons with fixed sidelengths in
$\E^3$ \cite{KM1}.
Category:53D 

20. CJM 1999 (vol 51 pp. 1123)
 Arnold, V. I.

First Steps of Local Contact Algebra
We consider germs of mappings of a line to contact space and
classify the first simple singularities up to the action of
contactomorphisms in the target space and diffeomorphisms of the
line. Even in these first cases there arises a new interesting
interaction of local commutative algebra with contact structure.
Keywords:contact manifolds, local contact algebra, Diracian, contactian Categories:53D10, 14B05 
