Topics 2016-2017

  1. [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
  2. [M1] N. Boizot, A. Feddaoui (LSIS) Observateurs pour les systèmes linéaires continu-discrets : observabilité et preuve de convergence du filtre de Kalman déterministe.
    Ce sujet propose une introduction à la théorie du contrôle à travers l’un de ses algorithmes classiques, le filtre de Kalman vu dans un cadre déterministe. On se propose de présenter les grands principes de cette discipline ainsi que quelque résultats classiques : systèmes linéaires continus-discrets, observabilité, Grammien, filtre de Kalman. L’étudiant pourra alors étudier quelque théorèmes classiques ainsi que leurs preuves, et en fonction de sa sensibilité ainsi que du temps disponible, de programmer un filtre de Kalman sur un exemple.
  3. [M1] G. Bouchitté (IMATH) Calculus of variations : optimal shape of a rod in torsion.
    See here
  4. [M2] G. Bouchitté (IMATH) Study of some variants of the Monge-Kantorovich optimal transport problem.
      See here.
  5. [M2] E.Busvelle (LSIS) Optimal syntheses and observers.
    In linear control theory with quadratic cost, the separation principle is a theorem which is applied in order to split the output feedback control problem into two more simpler problems : state feedback control and observer. In more complex cases (nonlinear, time-optimal control), this separation principle does not apply. However, it is usual to split the problem in control and observation problems. We want to study whats happen when an exponential observer is used with an optimal synthesis (from Pontryagin maximum principle) with bang-bang trajectories. Filippov solutions are introduced to study the sub-optimality of the closed-loop control.
  6. [M1/2] T. Champion, M. Ersoy (IMATH) La méthode de Nesterov et son interprétation en terme d’équations différentielles / Nesterov’s method and an differential equation modelling.
    – Le méthode de Nesterov (1983) est une méthode de gradient accéléré qui a des propriétés optimales de convergence (en termes de rapidité) pour la minimisation d’une fonction convexe. Récemment, l’efficacité de cette méthode a été expliquée par divers auteurs via une interprétation comme différenciation d’une EDO d’ordre 2. On étudiera cette interprétation et on programmera cette méthode sur un problème de modélisation.
    – Nesterov’s method (1983) is an accelerated gradient method whose convergence is proven to be optimal for the minimization of a convex function. This optimality has recently been justified by several authors via a second order differential equations approach. We shall study these works, as well as apply the method on some modelisation problem.
  7. [M1] F. Chittaro (LSIS) Integrable systems, Liouville’s Theorem, and action-angle variables.
    The object of this stage is to study the main points of the theory of Integrable systems, from basics definitions of integrability of vector fields to symplectic geometry, Hamiltonian systems and the Liouville-Arnold Theorem. These notions are very useful in many fields of Mathematics and Mathematical Physics (Dynamical Systems, Hamiltonian Mechanics, Control Theory, Quantum Theory). The knowledge of them could be a first step for further development in a M2 project on Control Theory.
  8. [M2] F. Chittaro (LSIS) Ensemble controllability: swing-up of a collection of pendula.
    See here
  9. [M2] F. Chittaro (LSIS) Schrieffer-Wolff transformation in Quantum Mechanics.
    See here
  10. [M1] M. Ersoy, L. Yuschenko (IMATH) Schéma implicite MAC pour Navier-Stokes Compressible Barotrope/  Implicit Mac scheme for Barotropic Compressible Navier-Stokes.
    – Dans un premier temps, il s’agit d’étudier un schéma MAC (Marker And Cell) implicite pour les équations bi-dimensionnelles de Navier-Stokes barotrope. Dans un deuxième temps, un code écrit en language fortran ou language C/C++ sera développé. Enfin, les résultats numériques seront validés via des résultats théoriques et expérimentaux. 
    – In this work, we first study an implicit MAC (Marker And Cell) scheme for the two-dimensional Barotropic Compressible Navier-Stokes equations. 
    Then, a numerical code (written in Fortran or C/C++ language) will be developed. The numerical results will be validated through theoretical and experimental results. 
  11. [M2] Y. Kian (CPT) Problèmes inverses spectraux et théorème de Borg-Levinson.
    See here
  12. [M1/2] S. Meradji (IMATH) Implementation of periodic boundary conditions in the discrete ordinates method : Application to grassland fires using FireStar3D.
    A 3D physics-based model referred to as “FireStar3D” has been developed in order to predict fire propagation in natural environment. It consists briefly in solving the conservation equations of the coupled system consisting of the vegetation and the surrounding gaseous medium. The model takes into account the phenomena of vegetation degradation (drying, pyrolysis, combustion), the interaction between an atmospheric boundary layer and a canopy (aerodynamic drag, heat transfer by convection and radiation, and mass transfer), and the transport within the fluid phase (convection, turbulence, gas-phase combustion).
    The objective of this project is to evaluate using the FireStar3D source code (written in Fortran90/95 and parallelized with OpenMP directives) the rates of spread of grassland fires for different wind speeds, and compare these rates to those obtained during experimental fires. An infinite fire front will be considered by assuming periodic boundary conditions in the horizontal direction perpendicular to the wind direction. Conducting these simulations requires first the implementation of periodic boundary conditions in the discrete ordinates method used to solve the radiative transfer equation.
  13. [M1] A. Panati (CPT) Stone-von Neumann theorem in Quantum Mechanics.
    In this stage the student will be asked to study and write a pedagogical introduction to Stone-von Neumann theorem about representation unicity of canonical commutation’s relations, one of the cornerstone theorem in quantum mechanics.
  14. [M1] A. Panati (CPT) Entropy in Classical and Quantum Mechanics.
    In this stage, the student will be asked to study and write a pedagogical introduction
    to the concept of entropy and its mathematical formulations, first in classic mechanics, then in quantum mechanics in the simplified framework of confined (i.e. finite dimensional) systems.
  15. [M1] C-A. Pillet (CPT) La géométrie Riemanienne.
    See here
  16. [M1] M. Rouleux (CPT) Modèle de Hubbard et spins à sym\’etrie continue.
    L’objet de ce stage est l’étude de la décroissance des corrélations (fonctions à 2 points) pour le modèle de Hubbard sur les réseaux Z ou Z^2, et sa généralisation possible aux systèmes de spins à symétrie continue. Le modèle de Hubbard décrit le Hamiltonien (quantique) d’un système d’électrons itinérant d’un site à l’autre du réseau, éventuellement soumis à un champ magnétique extérieur. Le sujet comprend une étude analytique et éventuellement des simulations numériques.
  17. [M1] S. Vaienti (CPT) Central limit theorem via martingale techniques.
    We propose to prove the central limit theorem using martingale theory. This is an alternative technique to the Levy’s approach. The student should first give an overview of martingale’s theory (not developed in the course but easily accessible with the given knowledge), and then follow the article by S. Lalley here :