Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds, - the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds, - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

We study a system of pseudodifferential equations which is elliptic in the Petrovskii sense on a closed smooth manifold. We prove that the operator generated by the system is a Fredholm operator in a refined two-sided scale of Hilbert function spaces. Elements of this scale are special isotropic spaces of Hörmander-Volevich-Paneah.

For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic...

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure $\mu$ for the geodesic flow $g^t$ is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the ${37}^{\text{èmes}}$ Journées EDP (Port d’Albret-June, 7-11 2010))

We prove the existence of solutions to nonlinear parabolic problems of the following type: $$\left\{\begin{array}{cc}{\displaystyle \frac{\partial b\left(u\right)}{\partial t}}+A\left(u\right)=f+\mathrm{div}\left(\Theta (x;t;u)\right)\hfill & \text{in}Q,\hfill \\ u(x;t)=0\hfill & \text{on}\partial \Omega \times [0;T],\hfill \\ b\left(u\right)(t=0)=b\left(u_0\right)\hfill & \text{on}\Omega ,\hfill \end{array}\right.$$ where $b:ℝ\to ℝ$ is a strictly increasing function of class ${𝒞}^1$, the term $$A\left(u\right)=-\mathrm{div}\left(a\right(x,t,u,\nabla u\left)\right)$$ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta :\Omega \times [0;T]\times ℝ\to ℝ$ is a Carathéodory, noncoercive function which satisfies the following condition: ${sup}_{\left|s\right|\le k}\left|\Theta (·,·,s)\right|\in E_{\psi }\left(Q\right)$ for all $k>0$, where $\psi$ is the Musielak complementary function of $\Theta$, and the second term $f$ belongs to $L^1\left(Q\right)$.

Nous donnons des résultats analytiques sur les propriétés de régularité du laplacien hypoelliptique de Jean-Michel Bismut et plus généralement sur les opérateurs $P$ de type Fokker-Planck géométrique agissant sur le fibré cotangent $\Sigma =T^{*}X$ d’une variété riemannienne compacte $X$. En particulier, nous prouvons un résultat d’hypoellipticité maximale pour $P$, et nous en déduisons des bornes sur la localisation de ses valeurs spectrales.

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also...