1. CJM 2015 (vol 67 pp. 961)
 Abuaf, Roland; Boralevi, Ada

Orthogonal Bundles and SkewHamiltonian Matrices
Using properties of skewHamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r2)n\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n1$ and $n\ge 8$. We speculate that the result holds in
greater generality.
Keywords:orthogonal vector bundles, moduli spaces, skewHamiltonian matrices Categories:14J60, 15B99 

2. CJM 2015 (vol 67 pp. 1270)
 Carcamo, Cristian; Vidal, Claudio

Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems
In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.
Keywords:difference equations, Hamiltonian systems, stability in the Lyapunov sense Categories:34D20, 34E10 

3. CJM 2013 (vol 67 pp. 450)
 Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian

Motion in a Symmetric Potential on the Hyperbolic Plane
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
Keywords:Hamiltonian systems with symmetry, symmetries, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

4. CJM 2012 (vol 65 pp. 1164)
 Vitagliano, Luca

Partial Differential Hamiltonian Systems
We define partial differential (PD in the following), i.e., field
theoretic analogues of Hamiltonian systems on abstract symplectic
manifolds and study their main properties, namely, PD Hamilton
equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in
standard multisymplectic approach to Hamiltonian field theory, in our
formalism, the geometric structure (kinematics) and the dynamical
information on the ``phase space''
appear as just different components of one single geometric object.
Keywords:field theory, fiber bundles, multisymplectic geometry, Hamiltonian systems Categories:70S05, 70S10, 53C80 

5. 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 

6. 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 

7. CJM 2004 (vol 56 pp. 566)
 Ni, Yilong

Geodesics in a Manifold with Heisenberg Group as Boundary
The Heisenberg group is considered as the boundary of a manifold. A class
of hypersurfaces in this manifold can be regarded as copies of the Heisenberg
group. The properties of geodesics in the interior and on the hypersurfaces
are worked out in detail. These properties are strongly related to those of
the Heisenberg group.
Keywords:Heisenberg group, Hamiltonian mechanics, geodesic Categories:53C22, 53C17 
