1. CJM Online first
 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 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 

3. 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 
