Sujets TER 2014-2015

1. [M2] J.-J. Alibert (IMATH)  Inégalités de Sobolev-Poincaré.
   Ces inégalités peuvent être utilisées pour valider certains résultats de Gamma-convergence de fonctionnelles sur l’espace des mesures boréliennes bornées du plan. Ces résultats de Gamma-convergence valident certains modèles continus en mécanique du solide par exemple.
2. [M1/2] J. Asch (CPT)  Occurrence of conical eigenvalue crossings.
   The object is to study the occurrence of conical eigenvalue intersections in a basic model of solid state quantum theory. In reference [1] these are shown to occur for the family of operators (-id_x-k)^2+V(x) on L^2(T) where T is a torus, V belongs to a certain class of functions, k varies in the dual torus which is related to a honeycomb lattice. The work to be done is to study reference [1] and to exhibit a way to prove stability of the result under deformations of the honeycomb structure.
   Prerequisites: Basic Functional Analysis, Operator and Spectral Perturbation theory.
   References:
   [1] C.L. Fefferman, M.L. Weinstein, Honeycomb lattice potentials and Dirac points, Journal of the American Mathematical Society, 25 (4), 2012, 1160-1220
3. [M2] W. Aschbacher (CPT)  The fermionic Federbush model.
   Although the standard model of particle physics is the fundamental conceptual framework for the description of the elementary particles and the strong and electroweak forces between them, it does not yet have the status of a physical theory in the sense that a rigorous mathematical construction and a proof of its consistency are still lacking (see the millenium problem « Yang-Mills and mass gap »). Important contributions to this question were made in the program of constructive quantum field theory initiated by Glimm and Jaffe in the middle of the 60ies. Unfortunately, this program could not reach its goal of constructing an interacting theory in the physically relevant four spacetime dimensions, but, based on Wightman’s axioms, existence and consistency of interacting quantum field theoretical models in two and three spacetime dimensions have been rigorously established. I propose to study the so-called fermionic Federbush model which describes two species of interacting fermions in two spacetime dimensions. The Federbush model was the first relativistic field theory model for which not only the Wightman axioms but also asymptotic completeness have been rigorously proved.
   References:
   [1] Ruijsenaars S N M, The Wightman axioms for the fermionic Federbush model, Commun. Math. Phys. 87 (1982) 181-228
   [2] Ruijsenaars S N M, Integrable quantum field theories and Bogoliubov transformations, Ann. Phys. 132 (1981) 328-382
   [3] Summers S J, A perspective on constructive quantum field theory, http://arxiv.org/abs/1203.3991 (2012)
4. [M2] Y. Aubry (IMATH)  Du théorème de l’indice de Hodge aux nombres de points des courbes sur les corps finis.
   Cf. ici
5. [M2] Y. Aubry (IMATH)  Corps de fonctions de nombre de classes 1.
   Il s’agit d’étudier l’article Function fields of class number one de Qibin Shen, Shuhui Shi publié sur arXiv le 6 février 2015.
   Abstract:
   In 1975, [LMQ] listed 7 function fields over finite felds (up to isomorphism) with positive genus and class number (i.e., the size of the divisor class group of degree zero) one and claimed to prove that these were the only ones such. In [S1], Claude Strirpe found 8th one! In this paper, we fix the argument in [LMQ] to show that this 8th example could have been found by [LMQ] method and is the only one, so that the list is now complete.
   References:
   [1] arXiv:1412.3505
6. [M1/2] J.-M. Barbaroux (CPT)  Régularité des solutions de l’équation de Boltzmann homogène.
   Cf. ici
7. [M1/2] G. Bouchitté (IMATH)  Distances entre probabilités: quelques variantes de la distance de Wasserstein et applications en statistiques.
8. [M1/2] F. Chittaro (LSIS)  Control of bilinear Schrödinger equations.
   Cf. ici
9. [M1/2] F. Chittaro (LSIS)  How rare are multiple eigenvalues?
   Cf. ici
10. [M1/2] F. Chittaro (LSIS)  Géométrie sous-Riemannienne.
    Cf. ici
11. [M1/2] M. Ersoy (IMATH) 
    The first topic is to construct a robust and accurate kinetic scheme. Up to now, there are no well-balanced and entropic numerical scheme constructed from the kinetic approach when the source term is complex. Thus, we mainly focus on how to construct such a scheme which are designeds to solve hyperbolic systems of equations, e.g. Saint-Venant.
12. [M1/2] M. Ersoy (IMATH) 
    This topic deals with the rigorous justification of the Saint-Venant-Exner equations for sediment transport for which it is well-known that Exner equation governs the morphodynamic part of the flow and Saint-Venant the hydrodynamic one. These equations are coupled through the topography term. The idea is to use the Vlasov equation for the sediment transport modeling and the Euler equation for the fluid. The main task will be to define correctly the modeling of the kinetic boundary conditions (which describes incoming and outgoing sediment particles). Hydrodynamic limit and a thin layer asymptotic analysis will be used to get a sediment transport model.
13. [M1/2] M. Ersoy (IMATH) 
    This topic concerns an exact Riemann solver for a general coupled hyperbolic systems trough a discontinuous flux gradient. This arise, for instance, in the modeling of unsteady mixed flows in closed water pipes for which the sound speed is necessary continuous with a discontinuous gradient leading to a discontinuous flux gradient. The main task is then to define rigorously the solution through the discontinuity which is not necessary fixed.
14. [M1/2] C. Galusinski (IMATH)  Condition limite en entrée et sortie pour un écoulement incompressible à Reynolds élevé.
    Cf. ici
15. [M1/2] C. Galusinski (IMATH)  Suivi d’interfaces par maillage cartésien mouvant.
    Cf. ici
16. [M1] H. Jaber (CPT)  Principe du maximum.
    Cf. ici
17. [M1/2] S. Meradji (CPT/IMATH)  Modeling of flame spread in engineered cardboard fuelbeds.
    Cf. ici
18. [M1/2] C. Pidéri (IMATH)  Comportement effectif d’un matériau homogène avec inclusions élastiques.
    On se propose d’étudier le comportement d’un matériau élastique Ω qui contient des inclusions élastiques réparties périodiquement au voisinage d’un segment Γ. En dehors de cette zone, le matériau est homogène. En première approximation, les inclusions n’ont aucune influence sur le comportement global du matériau et nous cherchons à comprendre leur effet au second ordre. Pour cela nous utiliserons une méthode de double échelle basée sur les développements asymptotiques raccordés. Une fois cette étape franchie, on pourrait s’intéresser au cas où les inclusions sont rigides.
    Cf. ici
19. [M1/2] C.-A. Pillet (CPT)  Riemannian geometry, statistical mechanics, and thermodynamics.
    Cf. ici
20. [M1/2] M. Rouleux (CPT)  Ionization properties of an atom in a periodic electric field: the semi-classical approach.
    Cf. ici
21. [M2] P. Seppecher (IMATH)  Effet du contraste sur la pertinence des modèles homogénéisés en élasticité.
    Cf. ici
22. [M1] S. Vaienti (CPT)  Principe d’invariance et thèoreme de Donsker.
    Le principe d’invariance est un raffinement du theorème central limite. En introduisant de manière convenable un temps continu, on montre la convergence de la somme de variables aléatoires vers un mouvement brownien.

 

Organismes de recherche extérieurs

1. [M2] IMFT Implémentation et validation dans un code de DNS d’écoulements particulaires du déplacement et des collisions pour des particules solides et non-sphériques.
   Cf. ici
2. [M2] INRA Etude numérique d’un modèle de réaction-advection-diffusion à advection hétérogène définie à partir de noyaux de perception.
   Cf. ici

Entreprises

1. [M2] Dassault Systèmes Simulation comportementale massivement distribuée.
   Cf. ici