We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing $1$-parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal...

Nous étudions les aspects infinitésimaux du problème suivant. Soit $H$ un hamiltonien de ${ℝ}^{2n}$ dont la surface d’énergie $\{H=1\}$ borde un domaine compact et étoilé de volume identique à celui de la boule unité de ${ℝ}^{2n}$. La surface d’énergie $\{H=1\}$ contient-elle une orbite périodique du système hamiltonien$$\left\{\begin{array}{cc}\dot{q}\hfill & =\frac{\partial H}{\partial p}\hfill \\ \dot{p}\hfill & =-\frac{\partial H}{\partial q}\hfill \end{array}\right.$$dont l’action soit au plus $\pi$ ?

Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified...