We consider cohomology defined by a system of local Lagrangian and investigate under which conditions the variational Lie derivative of associated local currents is a system of conserved currents. The answer to such a question involves Jacobi equations for the local system. Furthermore, we recall that it was shown by Krupka et al. that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form; we remark that...

We give an introduction into and exposition of Seiberg-Witten theory.

We prove that the space of orientation preserving homeomorphisms of the 2-sphere which fix pointwise a nontrivial nonseparating continuum is a contractible absolute neighborhood retract homeomorphic to the separable Hilbert space $l_2$.

For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related $L^p$ boundedness properties of these operators and show in particular that, although they are not bounded on $L^p$ in general, they are always bounded on suitable weighted $L^p$ spaces.

Following joint work with Dyatlov [DyGu], we describe the semi-classical measures associated with generalized plane waves for metric perturbation of ${ℝ}^d$, under the condition that the geodesic flow has trapped set $K$ of Liouville measure $0$.

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $\chi$ microlocally supported away from the trapping: $\parallel \chi R_h(E+i0)\chi \parallel =𝒪\left(h^{-1}\right)$, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde{\chi }$, to be supported at the trapped set, giving $\parallel \chi R_h(E+i0)\tilde{\chi }\parallel =𝒪\left(\sqrt{a\left(h\right)}{h}^{-1}\right)$ when the a priori bound is $\parallel \tilde{\chi }{R}_h(E+i0)\tilde{\chi }\parallel =𝒪\left(a\left(h\right)h^{-1}\right)$.

Let $X$ be a compact Kähler manifold with integral Kähler class and $L\to X$ a holomorphic Hermitian line bundle whose curvature is the symplectic form of $X$. Let $H\in C^{\infty }(X,ℝ)$ be a Hamiltonian, and let $T_k$ be the Toeplitz operator with multiplier $H$ acting on the space ${\mathscr{H}}_k=H^0(X,L^{\otimes k})$. We obtain estimates on the eigenvalues and eigensections of $T_k$ as $k\to \infty$, in terms of the classical Hamilton flow of $H$. We study in some detail the case when $X$ is an integral coadjoint orbit of a Lie group.

En nous inspirant d’articles de Beardon, nous donnons des résultats concernant les points fixes et les orbites d’auto-applications contractantes et semi-contractantes des espaces connexes localement compacts. Des résultats plus précis sont obtenus dans le cas des variétés complexes Kobayashi hyperboliques.