1. CJM 2002 (vol 54 pp. 1187)
 Cobo, Milton; Gutierrez, Carlos; Llibre, Jaume

On the Injectivity of $C^1$ Maps of the Real Plane
Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of
(complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists
$\epsilon >0$ such that $\Spec(X)\cap(\epsilon,\epsilon)=\emptyset$, then
$X$ is injective. Some applications of this result to the real Keller Jacobian
conjecture are discussed.
Categories:34D05, 54H20, 58F10, 58F21 

2. CJM 2000 (vol 52 pp. 1235)
 Hurtubise, J. C.; Jeffrey, L. C.

Representations with Weighted Frames and Framed Parabolic Bundles
There is a wellknown correspondence (due to Mehta and Seshadri in
the unitary case, and extended by Bhosle and Ramanathan to other
groups), between the symplectic variety $M_h$ of representations of
the fundamental group of a punctured Riemann surface into a compact
connected Lie group~$G$, with fixed conjugacy classes $h$ at the
punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles
on the unpunctured surface with a parabolic structure at the puncture
points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs
(representations of the fundamental group into $G$, ``weighted frame''
at the puncture points), and a corresponding complex variety ${\cal
P}$ of moduli of ``framed parabolic bundles'', which encompass
respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that
one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal
M}_h$ from ${\cal P}$ by a complex quotient. This allows us to
explain certain features of the toric geometry of the $\SU(2)$ moduli
spaces discussed by Jeffrey and Weitsman, by giving the actual toric
variety associated with their integrable system.
Categories:58F05, 14D20 

3. CJM 2000 (vol 52 pp. 582)
 Jeffrey, Lisa C.; Weitsman, Jonathan

Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems
This paper treats the moduli space ${\cal M}_{g,1}(\Lambda)$ of
representations of the fundamental group of a Riemann surface of
genus $g$ with one boundary component which send the loop around
the boundary to an element conjugate to $\exp \Lambda$, where
$\Lambda$ is in the fundamental alcove of a Lie algebra. We
construct natural line bundles over ${\cal M}_{g,1} (\Lambda)$ and
exhibit natural homology cycles representing the Poincar\'e dual of
the first Chern class. We use these cycles to prove differential
equations satisfied by the symplectic volumes of these spaces.
Finally we give a bound on the degree of a nonvanishing element of
a particular subring of the cohomology of the moduli space of
stable bundles of coprime rank $k$ and degree $d$.
Category:58F05 

4. CJM 1998 (vol 50 pp. 134)
 Médan, Christine

On critical level sets of some two degrees of freedom integrable Hamiltonian systems
We prove that all Liouville's tori generic bifurcations of a
large class of two degrees of freedom integrable Hamiltonian
systems (the so called JacobiMoserMumford systems) are
nondegenerate in the sense of Bott. Thus, for such systems,
Fomenko's theory~\cite{fom} can be applied (we give the example
of Gel'fandDikii's system). We also check the Bott property
for two interesting systems: the Lagrange top and the geodesic
flow on an ellipsoid.
Categories:70H05, 70H10, 58F14, 58F07 

5. CJM 1997 (vol 49 pp. 583)
 Pal, Janos; Schlomiuk, Dana

Summing up the dynamics of quadratic Hamiltonian systems with a center
In this work we study the global geometry of planar quadratic
Hamiltonian systems with a center and we sum up the dynamics of
these systems in geometrical terms. For this we use the
algebrogeometric concept of multiplicity of intersection
$I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$,
$Q(x,y,z) = 0$ at a point $p$ of the plane. This is a
convenient concept when studying polynomial systems and it
could be applied for the analysis of other classes of nonlinear
systems.
Categories:34C, 58F 

6. CJM 1997 (vol 49 pp. 212)
 Coll, B.; Gasull, A.; Prohens, R.

Differential equations defined by the sum of two quasihomogeneous vector fields
In this paper we prove, that under certain hypotheses,
the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$,
$\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are
quasihomogeneous vector fields, has at most two limit cycles.
The main tools used in the proof are the generalized polar
coordinates, introduced by Lyapunov to study the stability of degenerate
critical points, and the analysis of the derivatives of the Poincar\'e
return map. Our results generalize those obtained for polynomial
systems with homogeneous nonlinearities.
Categories:34C05, 58F21 

7. CJM 1997 (vol 49 pp. 338)
 Rousseau, C.; Toni, B.

Local bifurcations of critical periods in the reduced Kukles system
In this paper, we study the local bifurcations of critical periods
in the neighborhood of a nondegenerate centre of the reduced Kukles
system. We find at the same time the isochronous systems. We show
that at most three local critical periods bifurcate from the
ChristopherLloyd centres of finite order, at most
two from the linear isochrone and at most one critical period from the
nonlinear isochrone. Moreover, in all cases, there exist
perturbations which lead to the maximum number of critical
periods. We determine the isochrones, using the method of Darboux:
the linearizing transformation of an isochrone is derived from the
expression of the first integral.
Our approach is a combination of computational algebraic techniques
(Gr\"obner bases, theory of the resultant, Sturm's algorithm), the
theory of ideals of noetherian rings and the transversality theory
of algebraic curves.
Categories:34C25, 58F14 
