All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants...

These notes summarize the papers [8, 9] on the analysis of resolvent, Eisenstein series and scattering operator for geometrically finite hyperbolic quotients with rational non-maximal rank cusps. They complete somehow the talk given at the PDE seminar of Ecole Polytechnique in october 2005.

The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set Ω with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that Ω is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer problem to an arbitrary Riemannian manifold and also...

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

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.