Sujets TER 2013-2014

1. [M1] R. Aimino & S. Vaienti (CPT)  Inégalités de concentration.
   Cf. ici
2. [M1] 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. Ces résultats de Gamma-convergence valident certains modèles continus en mécanique du solide par exemple.
3. [M1/2] W. Aschbacher (CPT)  Quantum field theory without fields.
   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. Important contributions to this problem were made in the program of constructive quantum field theory initiated by Glimm and Jaffe in the middle of the 60ies. Based on Wightman’s axioms, existence and consistency of interacting quantum field theoretical models in two and three spacetime dimensions could be rigorously established. Unfortunately, the physically relevant four spacetime dimensions were out of reach, and the program came to rest. To this day, not a single example of a relativistic model of interacting particles in physical spacetime could be constructed.
   Recently, a change of paradigm within the framework of algebraic quantum field theory has opened up a new and promising perspective on this construction problem. I propose to study this new approach and its connection to Wightman quantum field theory.
   References:
   [1] D. Buchholz, S.J. Summers, Warped convolutions: A novel tool in the construction of quantum field theories, in: Quantum Field Theory and Beyond, edited by E. Seiler and K. Sibold (World Scientific, Singapore), pp. 107-121, 2008.
   [2] H. Grosse, G. Lechner, Wedge-local quantum fields and noncommutative Minkowski space, JHEP, 0711, 012 (2007).
   [3] H. Grosse, G. Lechner, Noncommutative deformations of Wightman quantum field theories, JHEP, 0809, 131 (2008).
4. [M2] Y. Aubry (IMATH)  Sur une application de la descente du corps de définition d’une tour de corps de fonctions.
   Cf. ici
5. [M1] J.-M. Barbaroux (CPT)  Dynamical properties of solutions of the Schrödinger equation for a model of Graphene.
   Cf. ici
6. [M1/2] G. Bouchitté (IMATH)  Spectral analysis of sign changing diffusion operators.
   Cf. ici
7. [M1/2] G. Bouchitté (IMATH)  Subwavelength transmission of the light (plasmonic wave guides).
   Cf. ici
8. [M1/2] G. Bouchitté (IMATH)  Calculus of variations: duality approach for non convex variational problems.
   Cf. ici
9. [M1/2] T. Champion (IMATH)  Displacement convexity on the set of probabilities.
   Cf. ici
10. [M1/2] F. Chittaro (LSIS)  Passage adiabatique au travers des intersections coniques.
    Cf. ici
11. [M2] C. Galusinski (IMATH)  Comparison of numerical schemes based on MAC grid discretization to solve the incompressible Navier-Stokes equations.
    One scheme has to be implemented to be compare to the already implemented code.
12. J.P. Gauthier & F. Chittaro (LSIS)  Quantum Control.
    We study the controllability, and the effective control/ stabilization of the solution of the Schrödinger equation, with external control fields. Several techniques, mostly from optimal control theory were developed in our research group [1], [2]. Also, adiabatic methods were considered [3]. The project is to continue to exploit the geometry of the finite dimensional Galerkin approximation to design control and ensemble-control of such quantum systems. Potential applications are in the fields of quantum information, or also nuclear magnetic resonance.
    References:
    [1] U. Boscain, G. Charlot, J.P. Gauthier, S. Guérin, H. Jauslin, Optimal Control in laser-induced populati transfer for two and three level quantum systems, Journal of Mathematical Physics, Vol. 43, pp. 2107-2132, 200.
    [2] U. Boscain, T. Chambrion, J.P. Gauthier, On the K+P problem for a 3-level quantum system: optimality implies resonance, Journal of dynamical and control systems, Vol. 8, No.4, pp. 547-572, Oct 2002.
    [3] U. V. Boscain, F. Chittaro, P. Mason, and M. Sigalotti, Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues, IEEE Trans. Automat. Control, 57 (2012), pp. 1970-1983.
    [4] T. Chambrion, P. Mason, M. Sigalotti, and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), pp. 329-349.
    [5] K. Beauchard, J.-M. Coron, P. Rouchon, Controllability Issues for Continuous Spectrum Systems and ensemble controllability of Bloch equations, Comm. Math. Phys. 296, 525-557 (2010).
13. J.P. Gauthier & N. Boizot (LSIS)  Motion Planning for Kinematic systems.
    The motion planning problem is one of the fundamental problems in robotics. It consists of evaluating the complexity of the realization of a nonadmissible path for a kinematic object, subject to nonholonomic constraints. See [1] for a survey. Following F. Jean, and other references there in, our methodology is based upon techniques from subriemannian geometry [8]. The paper [2] gives rise to a natural conjecture on the topological nature of optimal paths in nonholonomic interpolation. The purpose of this work is to extend the results of [2] to a nonholonomy degree larger than 4.
    References:
    [1] N. Boizot, J.P. Gauthier, Motion planning for Kinematic Systems, IEEE TAC Vol. 58, No. 6, June 2013, pp. 1430-1442.
    [2] N. Boizot, J.P. Gauthier, On the Motion Planning of the Ball with a Trailer, to appear in Mathematical Control and Related Fields, 2013.
    [3] J.P. Gauthier, B. Jakubczyk, V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM Journ. On Control and Opt., 48 (5), pp. 3433-3448, 2010.
    [4] J.P. Gauthier, V. Zakalyukin, On the one-step-bracket-generating motion planning problem, Jounal of dynamical and control systems, Vol. 11 No. 2, pp. 215-235, April 2005.
    [5] J.P. Gauthier, V. Zakalyukin, On the motion planning problem, complexity, entropy and nonholonomic interpolation, Journal of Dynamical and Control Systems, Vol. 12, No. 3, July 2006.
    [6] F. Jean, Complexity of Nonholonomic motion planning, International Journal of Control, Vol. 74 (8), 2001.
    [7] F. Jean, Entropy and Complexity of a Path in Sub-Riemannian Geometry, ESAIM: Cont. Opt. Calc. Var, Vol. 9, pp. 485-506, 2003.
    [8] A. Agrachev, D. Barilari, U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, book, preprint, SISSA 2011.
14. F. Golay (IMATH)  Computational Fluid Mechanics.
    We intend to numerically analyse the influence of the wind on waves using a finite volume code based on compressible Euler equation. The aim of this study is to validate a new mesh refinement procedure.
    Skills expected: mechanics, scientific computing, finite volume, fortran,…
15. [M2] F. Golay & L. Yushchenko (IMATH)  Computational Fluid Mechanics.
    We intend to numerically analyze air/water flows, like wave breaking, using a finite volume code based on compressible Euler equation. The aim of this study is to validate a new mesh refinement procedure (AMR) and local time stepping scheme.
    Skills expected: mechanics, scientific computing, finite volume, fortran 90,…
16. [M1/2] F. Golay & L. Yushchenko (IMATH)  Photorealistic rendering in computational Fluid Mechanics.
    We perform three-dimensional numerical simulations of air/water flows in the framework of AMR techniques. The aim of this study is to develop a post processing tool to capture iso-surfaces and perform realistic animations.
    Skills expected: scientific computing, image rendering, fortran 90,…
17. [M1/2] A. Novotny (IMATH)  L’Analyse Fourier appliquée à l’équation des ondes.
    Dans ce stage nous proposons de construire les solutions de l’équation des ondes dans ℝ, ℝ² ou ℝ³ en utilisant la transformée de Fourier. Ensuite, nous étudierons quelques propriétés de ces solutions.
18. [M2] C.-A. Pillet (CPT)  Entropic fluctuations in harmonic systems.
    Using the Girsanov-Cameron-Martin formula, the fluctuations of entropic flows in a mechanical system driven out of equilibrium by stochastic forces can be studied through a detailed analysis of a deformation of the Fokker-Planck operator describing the Markovian dynamics of the system. The aim of this project is to analyse this operator in simple cases where the mechanical system is harmonic.
19. [M1] C.-A. Pillet (CPT)  Initiation aux probabilités non commutatives: une théorie quantique des tests d’hypothèses.
    Les lois de la mécanique classique ne sont plus valables à l’echelle microscopique. Cette constatation faite au début du siècle dernier a forcé les physiciens à developper une nouvelle théorie structurelle: la mécanique quantique. Du point de vue des mathematiques, cette théorie se distingue radicalement de la mécanique classique de deux façons:
    1. Ses objets fondamentaux (une composante de la position ou de la vitesse d’une particule, par exemple) ne sont plus des éléments d’un corps commutatif (ℝ en l’occurence), mais d’une algèbre non-commutative.
    2. Ses prédictions sont intrinsèquement probabilistes. On ne peut pas prédire avec certitude la position et la vitesse d’une particule, mais seulement leur distribution de probabilité. De plus, à cause du point 1., cette distribution n’est pas une mesure, mais une fonctionnelle lineaire positive sur l’algèbre non-commutative évoquée ci-dessus.
    La mecanique quantique est une source inépuisable de nouveaux problèmes mathematiques qui sont les thèmes de recherche principaux des enseignants toulonais effectuant leur recherche au sein du laboratoire CPT.
    Le but de ce sujet de stage est de vous initier au cadre mathématique de la mécanique quantique en étudiant un problème spécifique: le test d’hypothèses. Il s’agit de comprendre avec quelle précision il est possible de distinguer deux états d’un système quantique. Cette question est devenue importante car il est de nos jours possible de manufacturer et de manipuler des états quantiques dans le laboratoire (en vue d’élaborer, par exemple, des ordinateurs quantiques). J’ai choisi cette question pour deux raisons:
    a) parce qu’elle m’interesse,
    b) parce que les objets qui entre en jeu dans son étude sont très simples (des matrices 2×2) et ne nécessitent donc pas d’outils d’analyse fonctionnelle très sophistiqués.
    Pratiquement, il s’agit:
    1. De comprendre le probleme et sa solution en lisant quelques articles assez recents et d’en faire la synthèse.
    2. De tenter de reformuler cette solution dans un formalisme (que je vous expliquerai) permettant en principe d’étendre l’analyse au cas plus compliqué de matrices infinies.
20. [M1/2] S. Vaienti (CPT) 
    We study extreme value theory applied to dynamical systems. In particular the stage would turn around some recent results of extreme value theory applied to systems which are randomly perturbed. The extreme value theory is a very well understood branch of probability for i.i.d. processes. After a review of the classical results, the student should look at relatively easy applications to expanding dynamical systems. The literature and the articles will be provided by myself.
21. S. Vaienti (CPT) 
    We work on statistical properties of dynamical systems, in particular limit theorems. These subjects are at the interface between ergodic theory, measure theory and probability. There are possibilities of PhD thesis.
22. P. Véron (IMATH)  Développement sur GPU d’algorithmes d’attaques sur des systèmes cryptographiques à base de codes correcteurs d’erreurs.
    Pré-recquis: bonnes bases en algère linéaire et bonnes bases de programmation en C.