Modélisation mathématique et physique macroscopique
mathematical modeling and macroscopic physics
(Parcours du M2 master M2 'Mathématiques Appliquées et THéoriques')

A total of 6 courses (6 x 6 ECTS) is required over the year. All the courses are optional (no mandatory course), but the program study must include at least two courses in maths (chosen from the fundamental mathematical courses listed below), and two courses in physics (from the 6-ECTS courses below).
Provided that timetables are compatible, students will also be able to choose from other M2 courses, for example from the ICFP master. Students will have to validate an internship in a research laboratory (24 ECTS).
For students who want to get up to date in mathematics, it is possible to take refresher courses in mid-September. We strongly advise students with a background in physics only to follow these short courses.


    Refresher courses (mid-september)


    A review of functional analysis tools for PDEs (D. Gontier)
    (15h)

    1. Lp spaces, Sobolev spaces
    2. Distributions, Fourier transform, Laplace, heat and Schrödinger equations in the whole space
    3. Self-adjoint compact operators, Laplace and Poisson equations in a domain.


    A review of numerical methods for PDEs (Guillaume Legendre)
    (15h)

    1. Review of classical numerical methods for PDEs
    2. Tests and Applications with Matlab


    A review of probability theory foundations [NEW] (Djalil Chafai)
    (15h)

    1. Random variables, expectations, laws, independence
    2. Inequalities and limit theorems, uniform integrability
    3. Conditioning, Gaussian random vectors
    4. Bounded variation and Lebegue-Stieltjes integral
    5. Stochastic processes, stopping times, martingales
    6. Brownian motion : martingales, trajectories, construction
    7. Wiener stochastic integral and Cameron-Martin formula


    A review of differential calculus for ODEs and PDEs [NEW] (Emeric Bouin)
    (15h)

    1. C^1 function, optimization, convex function, the inverse function theorem;
    2. Domain and its boundary, submanifold in R^n, The divergence theorem;Brouwer Theorem;Applications to PDE.
    3. ODE: Examples, Cauchy-Lipschitz theorem;Gronwall lemma;Smooth dependence by perturbations;
    4. Linear stability; Nonlinear stability and Lyapunov function;Volume preserving flow;Variations calculus and Euler-Lagrange equation.



    Semester 1



    Physics courses:


    Instabilities and nonlinear phenomena
    Laurette Tuckerman (ESPCI), Stephan Fauve (LPENS)
    ECTS credit: 6
    Details of the course here:


    Plasma Physics and advanced fluid dynamics
    Jean-Marcel Rax (Univ. Paris-Saclay, Ecole Polytechnique), Christophe Gissinger (LPENS)
    ECTS credit: 6
    Details of the course here:


    Systems out of equilibrium and non-linear dynamics
    Kirone Mallick (CEA, IPhT), Francois Petrelis (LPENS)
    ECTS credit: 6
    Details of the course here:



    Math courses:


    Introduction to non-linear PDEs (Eric Sere)
    (ECTS credit: 6 ; 30h+7,5h of tutorial)

    1. Bifurcation theory applied to nonlinear elliptic PDEs.
    2. Existence of weak solutions by variational methods
    3. Regularity of weak solutions to linear and nonlinear elliptic PDEs.
    4. Fixed point theorems applied to nonlinear elliptic PDEs
    5. Maximum principles and applications
    > more details here...


    Hamiltonian Dynamical Systems (Jacques Féjoz)
    (ECTS credit: 6 ; 30h)

    1. Reminder on differential equations
    2. Hamiltonian systems on R2n
    3. Smooth manifolds, tangent and cotangent bundles
    4. Differential forms
    5. Symplectic manifolds
    6. Hamiltonian systems on symplectic manifolds
    7. Integrability of Hamiltonian systems
    8. Hamiltonian perturbation theory
    9. The KAM theorem
    10. The Nekhoroshev theorem
    > more details here...


    Introduction to evolution PDEs (Stéphane Mischler)
    (ECTS credit: 6 ; 30h)

    1. Parabolic equations
    2. Transport equations
    3. Evolution equation and semigroup
    4. Heat equation
    5. Entropy and applications
    6. Markov semigroups and the Harris-Meyn-Tweedie theory
    7. The parabolic-elliptic Keller-Segel equation
    > more details here...


    Additional possible choices in the M2 MATH (PSL)
    > more details here...






    Semester 2


    Internship in a research laboratory
    (ECTS credit: 24)


    Physics courses:


    Numerical methods for fluid dynamics
    Emmanuel Dormy (DMA ENS)
    ECTS credit: 6
    Details of the course here:


    Turbulence (Alexandros Alexakis)
    (ICFP master)

    1. Introduction and examples
    2. Conserved quantities and symmetries
    3. Turbulence, dynamical systems and chaos
    4. Cascades and phenomenology of hydrodynamic turbulence
    5. Kolmogorov 1941 theory
    6. Vorticity dynamics and 2D turbulence
    7. Turbulence and statistical mechanics
    8. Intermittency and multifractal analysis
    > more details here...


    Non Linear Solid Mechanics (Matteo Ciccotti, Benoit Roman )


    This class is motivated by a current interest in research for non-linear mechanics for several reasons: i) instabilities are not anymore seen as failure of the structure, but as challenging phenomena that can be harnessed and useful; ii) the current development and better understanding of very soft materials (gels, polymers). The aim of the class is dealing with some selected instabilities in the deformation of soft elastic solids, such as wrinkling and stringing. The teaching will be organized as working in pairs on selected projects that regard instabilities which are still debated in the scientific community. The teaching will be in the form of a weekly guidance to acquire the relevant necessary concepts from continuum mechanics, energetic methods, dynamics, surface physics and nonlinear phenomena and to apply them into the personal investigation of the selected topic by combining several parallel approaches form mechanical physics research strategies such as 1) developing simple mechanical models by scaling methods, 2) realizing simple experiments at home and 3) identifying and critically analyzing the relevant existing papers.

    Math courses:


    Additional possible choices in the M2 MATH (PSL)
    > more details here...