The talk will present some results about Laplace-Beltrami operators on complete manifolds with corner of codimension two obtained by adapting methods coming from the spectral analysis of Schroedinger operators. We will outline also possible research projects that exploit in similar ways analogies between Schroedinger and Laplace-Beltrami operators.

In recent years, low dimensional topologists have become interested in the study of "generic" smooth maps to surfaces. The approach is similar to Morse theory, only with two dimensional target. In this talk, I will discuss a specific problem in the 4-dimensional context which is analogous to the (uniqueness of) cancellation of critical points of Morse functions. I will also indicate applications to certain pictorial descritpions of 4-manifolds in terms of curve configurations on surfaces. This is joint work with Kenta Hayano.

La torsion de Frobenius dans la catégorie P des foncteurs polynomiaux stricts induit les monomorphismes entre les groupes d’extensions : Ext (F, G) → Ext (F^(1),G^(1)). Comme la catégorie P est une sous-catégorie pleine de la catégorie U des modules instables, ceci amène à étudier les morphismes Ext(M, N) → Ext(ΦM, ΦN). La recherche sur la résolution injective minimale du module F(1) a donc pour but de comprendre le cas particulier des morphismes Ext (Φ^n(F(1)), Φ^n(F (1))) → Ext(Φ^n+1(F (1)), Φ^n+1(F (1))) . Dans cet exposé, on donne de l’information partielle sur la partie nilpotente de la résolution injective minimale de F (1) qui confirme l’injectivité des morphismes des groupes d’extensions induits par la torsion de Frobenius dans plusieurs cas.

On s'intéresse à la rigidité spectrale de l’opérateur de Laplace-Beltrami dans le cas où la dynamique classique est du type KAM (intégrable ou proche d'un système intégrable). On montre que les tores invariants KAM (de Kolmogorov-Arnold-Moser) fournissent des invariants iso-spectraux. Les invariants sont donnés par la fonction $\beta$ de Mather.

We investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively.

The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers [1, 2, 3] for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.

In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure [4, 5]. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme.
A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency.

A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.

If time permits we will present the extension of the a posteriori treatment to construct a subcell limiter for Discontinuous Galerkin methods of high accuracy (polynomial degree 9).

References:
[1] S. Clain, S. Diot, and R. Loubère, A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD). Journal of Computational Physics, 230(10):4028 – 4050, 2011.
[2] S. Diot, S. Clain, and R. Loubère, Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Computers and Fluids, 64:43 – 63, 2012.
[3] S. Diot, R. Loubère, and S. Clain, The MOOD method in the three-dimensional case: Very-high-order finite volume method for hyperbolic systems. International Journal of Numerical Methods in Fluids, 73:362–392, 2013.
[4] M. Dumbser, Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier- Stokes equations. Computers and Fluids, 39:60–76, 2010.
[5] M. Dumbser, M. Castro, C. Parés, and E.F. Toro, ADER schemes on unstructured meshes for non-conservative hyperbolic systems: Applications to geophysical flows. Computers and Fluids, 38:1731 – 1748, 2009.

Afin de développer de nouveaux schémas numériques préservant l’asymptotique pour des équations cinétiques en régimes hyperbolique ou diffusif, il est intéressant d’appliquer la technique d’intégration projective introduite par Gear et Kevrekidis pour des grands systèmes différentiels apparaissant en chimie. Après avoir introduit l’intégration projective pour les équations cinétiques, j’exposerai notamment une méthode pour la montée en ordre de ces schémas.