Applied and Theoretical Mathematics - Master’s Year 2

Lecturer :
DAVID GONTIER

Total hours : 15

Overview :

- Lp spaces, Sobolev spaces ;

- Distributions, Fourier transform, Laplace, heat and Schrödinger equations in the whole space ;

- Self-adjoint compact operators ;

- Laplace and Poisson equations in a domain.

Lecturer :
GUILLAUME LEGENDRE

Total hours : 15

Overview :

Review of the finite difference and finite element methods, with numerical applications using Python Numpy/Jupyter and freeFEM++.

Learning outcomes :

Basic introduction to numerical methods for the approximation of PDE solutions

Bibliography-recommended reading

R. J. LEVEQUE. Numerical methods for conservation laws, Lectures in Mathematics. ETH Zürich. Birkhäuser,

second edition, 1992.

J. C. STRIKWERDA. Finite difference schemes and partial differential equations. SIAM, second edition, 2004. DOI:

10.1137/1.9780898717938.

Ects : 6

Lecturer :
DJALIL CHAFAI

Total hours : 15

Overview :

- Random variables, expectations, laws, independence

- Inequalities and limit theorems, uniform integrability

- Conditioning, Gaussian random vectors

- Bounded variation and Lebegue-Stieltjes integral

- Stochastic processes, stopping times, martingales

- Brownian motion : martingales, trajectories, construction

- Wiener stochastic integral and Cameron-Martin formula

Ects : 6

Lecturer :
EMERIC BOUIN

Total hours : 15

Overview :

Differential calculus :

- C^1 function, optimization, convex function;

- The implicit function theorem, constrained optimization;

- The inverse function theorem;

- Domain and its boundary, submanifold in R^n;

- The divergence theorem;

- Brouwer Theorem;

- Applications to PDE.

Ordinary differential equations :

- Examples: gradient flow, Hamiltonian flow; others;

- Cauchy-Lipschitz theorem;

- Gronwall lemma;

- Smooth dependence by perturbations;

- Linear stability;

- Nonlinear stability and Lyapunov function;

- Volume preserving flow;

- Variations calculus and Euler-Lagrange equation.

Ects : 6

Lecturer :
JUSTIN SALEZ

Total hours : 24

Overview :

• Probability basics
• Stochastic processes, Brownian motion, continuous semi-martingales
• Stochastic integral, Itô’s formula for semi-martingales and Girsanov’s theorem
• Stochastic differential equations, diffusion processes
• Feynman-Kac formula and link with the heat equation
• Probabilistic representation of the Dirichlet problem.

Learning outcomes :

The first part of the course presents stochastic calculus for continuous semi-martingales. The second part of the course is devoted to Brownian stochastic differential equations and their links with partial differential equations. This course is naturally followed by the course “Jump processes”.

Click here for more information

Ects : 6

Lecturer :
ERIC SERE

Total hours : 30

Overview :

Existence of weak solutions of linear and nonlinear elliptic PDEs by variational methods.

Regularity of weak solutions to linear and nonlinear elliptic PDEs.

Maximum principles and applications.

Brouwer degree, Leray-Schauder degree, fixed-point theorems.

Local and global bifurcation theory applied to nonlinear elliptic PDEs.

Learning outcomes :

Existence of weak solutions of linear and nonlinear elliptic PDEs by variational methods.

Regularity of weak solutions to linear and nonlinear elliptic PDEs.

Maximum principles and applications.

Brouwer degree, Leray-Schauder degree, fixed-point theorems.

Local and global bifurcation theory applied to nonlinear elliptic PDEs.

Bibliography-recommended reading

L.C. Evans, Partial Differential equations (Graduate Studies in Mathematics 19, AMS).

L. Nirenberg, Topics in Nonlinear Functional Analysis (Courant Lecture Notes Series 6, AMS).

Ects : 6

Lecturer :
STEPHANE MISCHLER

Overview :

In a first part, we will present several results about the well-posedness issue for evolution PDE.

- Parabolic equation. Existence of solutions for parabolic equations by the mean of the variational approach and the existence theorem of J.-L. Lions.

- Transport equation. Existence of solutions by the mean of the characterics method and renormalization theory of DiPerna-Lions. Uniqueness of solutions thanks to Gronwall argument and duality argument.

- Evolution equation and semigroup. Linear evolution equation, semigroup and generator. Duhamel formula and mild solution. Duality argument and the well-posedness issue. Semigroup Hille-Yosida-Lumer-Phillips' existence theory.

In a second part, we will mainly consider the long term asymptotic issue.

- More about the heat equation. Smoothing effect thanks to Nash argument. Rescaled (self-similar) variables and Fokker-Planck equation. Poincaré inequality and long time asymptotic (with rate) in L2 Fisher information, log Sobolev inequality and long time convergence to the equilibrium (with rate) in L1.

- Entropy and applications. Dynamical system, equilibrium and entropy methods. Self-adjoint operator with compact resolvent. A Krein-Rutman theorem for conservative operator. Relative entropy for linear and positive PDE. Application to a general Fokker-Planck equation. Weighted L2 inequality for the scattering equation.

- Markov semigroups and the Harris-Meyn-Tweedie theory.

In a last part, we will investigate how the different tools we have introduced before can be useful when considering a nonlinear evolution problem.

- The parabolic-elliptic Keller-Segel equation. Existence, mass conservation and blow up. Uniqueness. Self-similarity and long time behavior.

Learning outcomes :

To acquire fundamental notions and tools in order to be able to analayze the most simple linear and nonlinear evolution PDEs

Ects : 6

Lecturer :
JULIEN SALOMON

Total hours : 30

Overview :

Ce cours est composé de 5 chapitres. Chacun des chapitres est associé à une séance de travail sur machine (TP, en Matlab/GNU Octave et Free Fem).

Optimisation numérique et équations aux dérivées partielles

Méthodes numériques pour le contrôle optimal

Traitement numérique des inégalités variationnelles

Introduction à la méthode des éléments finis

Introduction aux méthodes de réduction

English version :

This course is composed of 5 chapters :

Numerical optimization and partial differential equations

Numerical methods for optimal control

Numerical treatment of variational inequalities

Introduction to the finite element method

Introduction to reduction methods

Each chapter is associated with a working session on the machine (TP, in Matlab/GNU Octave and Free Fem).

Ects : 6

Lecturer :
JACQUES FEJOZ

Total hours : 30

Overview :

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

Learning outcomes :

M2

Bibliography-recommended reading

V.I. Arnold, Ordinary differential equations

V.I. Arnold, Geometric methods in the theory or ordinary differential equations

V.I. Arnold, Mathematical methods in classical mechanics

D. McDuff and D. Salamon, Introduction to symplectic topology

A. Cannas da Silva, Lectures on symplectic geometry

Ects : 6

Lecturer :
CRISTINA TONINELLI
BEATRICE TAUPINART DE TILIERE

Total hours : 30

Overview :

The aim of statistical mechanics is to understand the macroscopic behavior of a physical system by using a probabilistic model containing the information for the microscopic interactions. The goal of this course is to give an introduction to this broad subject, which lies at the intersection of many areas of mathematics: probability, graph theory, combinatorics, algebraic geometry…

In the first part of the course we will introduce the key notions of equilibrium statistical mechanics. In particular we will study the phase diagram of the following models: Ising model (ferromagnetism), dimer models (crystal surfaces) and percolation (flow of liquids in porous materials). In the second part we will introduce interacting particle systems, a large class of Markov processes used to model dynamical phenomena arising in physics (e.g. the kinetically constrained models for glasses) as well as in other disciplines such as biology (e.g. the contact model for the spread of infections) and social sciences (e.g. the voter model for the dynamics of opinions).

Click

here and

here for more information.

Ects : 6

Lecturer :
PIERRE CARDALIAGUET

Total hours : 18

Overview :

Mean field games is a new theory developed by Jean-Michel Lasry and Pierre-Louis Lions that is interested in the limit when the number of players tends towards infinity in stochastic differential games. This gives rise to new systems of partial differential equations coupling a Hamilton-Jacobi equation (backward) to a Fokker-Planck equation (forward). We will present in this course some results of existence, uniqueness and the connections with optimal control, mass transport and the notion of partial differential equations on the space of probability measures.

Recommended prerequisites :

Stochastic analysis, Stochastic control.

Learning outcomes :

Mastering of the mean field games technics.

Bibliography-recommended reading

Notes on the course: www.ceremade.dauphine.fr/~cardaliaguet/Enseignement.html

Ects : 6

Lecturer :
Gregoire NADIN

Total hours : 26

Overview :

Une première partie sera consacrée aux propriétés fondamentales des équations aux dérivées partielles elliptiques et paraboliques linéaires et non linéaires. On étudiera ensuite les états stationnaires de ces équations, les propriétés dynamiques et l’existence de solutions de type ondes progressives. On s’attachera en particulier à en déterminer les vitesses et les formes ainsi que les propriétés qualitatives.

La seconde partie du cours décrira quelques modèles de dynamique des populations pour la biologie et différentes applications. Dans le cadre de ces modèles, on analysera les effets des environnements hétérogènes sur la persistance des espèces et la forme des invasions biologiques en fonction de l’environnement. On développera aussi des modèles permettant de décrire les effets de changements climatiques sur la persistance et la distribution de certaines espèces biologiques.

Enseignant : (H. Berestycki & G. Nadin, cours du M2 Math. Model.)

Learning outcomes :

Des phénomènes observés dans des contextes très variés sont représentés par des équations de type réaction-diffusion : dynamique des populations, écologie, épidémiologie, invasions biologiques, comportements collectifs, diffusion d’opinions ou de normes sociales. Ce cours développera des méthodes mathématiques pour analyser ce type d’équations. Elles seront ensuite mises en œuvre pour établir les principales propriétés de ces équations.

Ects : 6

Lecturer :
MATHIEU LEWIN
Eric CANCES

Total hours : 20

Overview :

La théorie spectrale des opérateurs auto-adjoints a de nombreuses applications en mathématiques, notamment dans le domaine des équations aux dérivées partielles (EDP). Dans ce cours, nous présenterons les détails de cette théorie, que nous illustrerons par divers exemples intervenant dans la théorie et la simulation numérique des EDP (Laplaciens de Dirichlet et de Neumann par exemple).

Dans une deuxième partie du cours, nous verrons que la combinaison de techniques spectrales et de méthodes variationnelles permet d'obtenir des résultats intéressants sur des problèmes elliptiques linéaires et non linéaires.

Nous illustrerons cette approche sur des problèmes issus de la mécanique quantique, extrêmement utilisés dans les applications. Nous étudierons en particulier l’équation de Schrödinger à N corps et ses approximations de champ moyen donnant lieu à une équation de Schrödinger non linéaire, ainsi que les opérateurs de Schrödinger périodiques utilisés pour la modélisation des matériaux. Les éléments de base de la mécanique quantique seront présentés, mais aucune connaissance physique n'est requise pour suivre le cours.

Recommended prerequisites :

Espaces de Sobolev

Learning outcomes :

Base de la théorie spectrale et de l'analyse non linéaire

Assessment :

Examen écrit

Bibliography-recommended reading

hal.archives-ouvertes.fr/cel-01935749

Ects : 6

Lecturer :
ANTONIN CHAMBOLLE

Total hours : 24

Overview :

This course will cover the bases of continuous, mostly convex optimization. Optimization is an important branch of applied industrial mathematics. The course will mostly focus on the recent development of optimization for large scale problems such as in data science and machine learning. A first part will be devoted to setting the theoretical grounds of convex optimization (convex analysis, duality, optimality conditions, non-smooth analysis, iterative algorithms). Then, we will focus on the improvement of basic first order methods (gradient descent), introducing operator splitting, acceleration techniques, non-linear (”mirror”) descent methods and (elementary) stochastic algorithms.

Ects : 6

Total hours : 24

Overview :

A control system is a dynamical system depending on a parameter called the control, which one can choose in order to influence the behaviour of the solution. It often takes the form of an ordinary differential equation or a partial differential equation, in which the control appears as an additive term or in the coefficients. The goal of this class is to present several problems associated with these systems.

A preliminary plan of the course follows :

(1) Finite-dimensional control, some results on control systems governed by ODEs

(2) Linear control systems in infinite dimension: observability, controllability and stabilization via semigroup theory

(3) Examples: heat equation, PDE/ODE couplings, backstepping methods.

Teachers : D. Bresch-Pietri & O. Glass

Ects : 6

Lecturer :
LAURENT COHEN

Total hours : 21

Overview :

This course, after giving a short introduction to digital image processing, will present an overview of variational methods for Image segmentation. This will include deformable models, known as active contours, solved using finite differences, finite elements, level sets method, fast marching method. A large part of the course will be devoted to geodesic methods, where a contour is found as a shortest path between two points according to a relevant metric. This can be solved efficiently by fast marching methods for numerical solution of the Eikonal equation. We will present cases with metrics of different types (isotropic, anisotropic, Finsler) in different spaces. All the methods will be illustrated by various concrete applications, like in biomedical image applications.

Ects : 6

Total hours : 20

Overview :

Le cours aura lieu probablement le mercredi matin au Collège de France au second semestre

Pour plus d'information consulter le site : www.college-de-france.fr/site/stephane-mallat/_course.htm

La validation se fera sur projet, en travaillant sur l'un des challenge proposés sur le site : challengedata.ens.fr/en/home

Ects : 6

Lecturer :
PIERRE-LOUIS LIONS

Total hours : 20

Overview :

Le cours aura lieu probablement le vendredi matin au Collège de France

Pour plus d'information consulter le site : www.college-de-france.fr/site/en-pierre-louis-lions/_course.htm

Ects : 6

Total hours : 20

Overview :

The purpose of this course is to study a class of partial differential equations (PDE) systems used in population dynamics to describe the spatial distribution of different (interacting) animal species. In a classical way, populations are described through two fundamental mechanisms: the dispersion of individuals (modeled by a diffusion operator) and their reproduction or death (modeled by a reaction term). The specificity of the systems on which we will focus lies in the expression "cross-diffusion": for such systems the diffusivity (or mobility) of a species depends - potentially in a non-linear way - on the presence of its competitors.

The first article proposing such a system was published in 1979, in the Journal of Theoretical Biology. The authors, Shigesada, Kawasaki and Teramoto, proposed this type of system (henceforth called "SKT") to capture the phenomenon of species segregation, i.e. an almost disjoint occupation of the available space between the different species. It is often the case in applied mathematics that an efficient modeling tool leads to interesting and surprisingly difficult mathematical questions; we will see in this course that cross-diffusion systems are a nice illustration of this fact.

After a quick introduction that will (formally) unveil the link between cross-diffusion and segregation, the course will first focus on the so-called Kolmogorov equation, a parabolic PDE whose diffusion operator is adapted to the behavior of sensitive individuals (as opposed to Fick's law, for the diffusion of inert matter). This equation being the constituting block of several cross-diffusion systems, we will try to understand it in a framework of very low regularity. Then, we will switch to the study of cross-diffusion systems. As it is often the case for nonlinear PDEs, we will see that the very question of the existence of solutions is not trivial for those non-linear systems. We will provide a scheme for the construction of global weak solutions by approximation-compactness, based on the dissipation over time of a functional, called the entropy of the system. Depending on the remaining time, we will then show the existence of more regular (but local) solutions, some properties of strong-weak uniqueness and eventually more difficult results: the realization of the SKT system as an asymptotic limit of more elementary equations or the rigorous analysis of some equilibrium states offering a segregation of species.

In addition to exploring cross-diffusion systems, this course will illustrate some standard methods in the study of parabolic equations (maximum principle, Aubin-Lions lemma, approximation-compactness, infinite-dimensional fixed point, asymptotic analysis) that we will present in the specific framework that interests us while underlining the broader scope of these tools.

Teacher : Ayman Moussa (ENS)

Ects : 6

Overview :

This course presents an introduction to master students to a recent line of research within kinetic theory that has been recently blossoming : the theory of the geometric constraints i.e. the confining properties of the kinetic « gas » (of molecules, stars, electrons...). This is a fundamental question for the physical relevance of the mathematical theory. We shall first revisit the works from the 70s and 80s regarding the entropy in- equality at the boundary, and the works from the 90s and early 2000s regarding the subtle functional analysis issues with the trace at the boundary. Then we shall focus on more recent works devoted to the relaxation to equilibrium in low regularity, the propagation of regularity by the transport flow, and finally we shall discuss the new theory of geometric control for collisional kinetic equations.

Teacher : Clément Mouhot (Cambridge U)

Ects : 6

Total hours : 20

Overview :

Conformal invariance plays an important role in physics and geometry: conformal field theory, general relativity, superconductivity, Riemann surface, Yang-Mills fields. In this course we will study the analytical aspect of some of these problems. More precisely, we will be interested in the analysis of nonlinear PDEs resulting from conformal invariant problem: harmonic maps, prescribed curvature problem, Ginzburg-Landau and Yang-Mills. We will start with the constant mean curvature equation which will allow me to introduce the phenomena of compactness by compensation, then we will develop the theory via the general approach of Riviere [2]. Then we will focus on the Ginzburg-Laundau problem [3], which can be considered as an abelian version of Yang-Mills. Finally, we will study Uhlenbeck’s work on the Yang-Mills equation and if time permits we will give geometric applications [1]. Background: elliptic PDE, differential geometry.

REFERENCES [1] Daniel S. Freed and Karen K. Uhlenbeck. Instantons and four-manifolds, volume 1 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, second edition, 1991. [2] Tristan Riviere. Conformally invariant variational problems. 2012. [3] Etienne Sandier and Sylvia Serfaty. Vortices in the magnetic Ginzburg-Landau model, volume 70 of Progress in Nonlinear Differential Equations and their Applications. Birkhauser Boston, Inc., Boston, MA, 2007.

Teacher : Paul Laurain (ENS)

Ects : 6

Lecturer :
JULIEN POISAT

Total hours : 18

Overview :

Poisson process, compound Poisson process,

Infinitely divisible distributions,

Random measures of Poisson,

Lévy process,

Decomposition of Lévy-Khintchine,

Itô's formula for Lévy processes,

Stochastic differential equations driven by a Lévy process,

Equivalence of measures, Doleans-Dade exponential, Girsanov's theorem

Merton’s Model

Hawkes' Process

Learning outcomes :

This course aims to master the techniques of analysis and stochastic calculation specific to jump processes. It complements the "Stochastic Calculation" course which is limited to processes with continuous paths.

Ects : 6

Lecturer :
STEFANO OLLA

Total hours : 21

Overview :

Large deviations are at the center of modern probability and statistics. The theory originated form the risk analysis for insurance companies. Today there are applications in almost all domains of applied mathematics.

This course will revise various generalizations of Cramer and Sanov Theorems and will illustrate applications to Statistical Physics and Analysis, in particular :

- Statistical Physics: Gibbs probability distributions for mean field models (Curie-Weiss) and for models with local interaction (Ising Model).

Local Large deviations and equivalence on ensembles.

- Non-linear PDE: Viscous solutions in Hamilton-Jacobi equations.

- Random perturbations of dynamical systems: Freidlin–Wentzell theory.

Ects : 6

Lecturer :
JUSTIN SALEZ

Total hours : 24

Overview :

Combien de fois faut-il battre un paquet de 52 cartes pour que la permutation aléatoire obtenue soit à peu près uniformément distribuée ? Ce cours est une introduction sans pré-requis à la théorie moderne des temps de mélange des chaînes de Markov. Un interêt particulier sera porté au célèbre phénomène de "cutoff", qui est une transition de phase remarquable dans la convergence de certaines chaînes vers leur distribution stationnaire. Parmi les outils abordés figureront les techniques de couplage, l'analyse spectrale, le profil isopérimétrique, ou les inégalités fonctionnelles de type Poincaré. En guise d'illustration, ces méthodes seront appliquées à divers exemples classiques issus de contextes variés: mélange de cartes, marches aléatoires sur les groupes, systèmes de particules en intéraction, algorithmes de Metropolis-Hastings, etc. Une place importante sera accordée aux marches sur graphes et réseaux, qui sont aujourd'hui au coeur des algorithmes d'exploration d'Internet et sont massivement utilisées pour la collecte de données et la hiérarchisation des pages par les moteurs de recherche.

Learning outcomes :

Pour en savoir plus : www.ceremade.dauphine.fr/~salez/mix.html

Bibliography-recommended reading

Notes de cours, examen 2019 et correction (J. Salez)

Markov Chains and Mixing Times (D. Levin, Y. Peres & E. Wilmer)

Mathematical Aspects of Mixing Times in Markov Chains (R. Montenegro & P. Tetali)

Mixing Times of Markov Chains: Techniques and Examples (N. Berestycki)

Reversible Markov Chains and Random Walks on Graphs (D. Aldous & J. Fill)

Ects : 6

Total hours : 18

Overview :

Relationship between conditional expectations and parabolic linear PDEs.

Formulation of standard stochastic control problems: dynamic programming principle.

Hamilton-Jacobi-Bellman equation

Verification approach

Viscosity solutions (definitions, existence, comparison)

Application to portfolio management, optimal shutdown and switching problems

Teacher : Bruno BOUCHARD

Learning outcomes :

PDEs and stochastic control problems naturally arise in risk control, option pricing, calibration, portfolio management, optimal book liquidation, etc. The aim of this course is to study the associated techniques, in particular to present the notion of viscosity solutions for PDEs.

Ects : 6

Lecturer :
JULIEN CLAISSE

Total hours : 30

Overview :

Generalities on Monte-Carlo methods

1. Generalities on the convergence of moment estimators

2. Generators of uniform law

3. Simulation of other laws (rejection method, transformation, …)

4. Low discrepancy sequences

Variance reduction

1. Antithetical control

2. Payoff regularization

3. Control Variable

4. Importance sampling

Process simulation and payoff discretization

1. Black-Scholes model

2. Discretisation of SDEs

3. Diffusion’s bridges and applications to Asian, barrier and lookback options.

Calculation of sensitivities (greeks)

1. Finite differences

2. Greeks in the Black-Scholes model

3. Tangent process and Greeks

4. Malliavin calculus, Greeks, conditional expectations and pricing of American options

Calculation of conditional expectations and valuation of American options.

1. Nested Monte Carlo approach

2. Regression Methods (Tsitsiklis Van Roy, Longstaff Schwartz)

3. Rogers’ Duality

Finite difference methods: the linear case

1. Construction of classical schemes (explicit, implicit, theta-scheme)

2. Conditions for stability and convergence

Finite difference methods: the non-linear case

1. Monotonous schemes: General conditions of stability and convergence

2. Examples of numerical schemes: variational problems, Hamilton-Jacobi-Bellman equations.

Learning outcomes :

This course provides an in-depth presentation of the main techniques for the evaluating of options using Monte Carlo techniques.

Ects : 6

Overview :

This course provides a quick access to some popular models in the theory of random graphs, point processes and random sets. These models are widely used for the mathematical analysis of networks that arise in different applications: communication and social networks, transportation, biology...

We will discuss among the others: the Erdos-Reny graph, the configuration model, unimodular random graphs, Poisson point processes, hard core point processes, continuum percolation, Boolean model and coverage process, and stationary Voronoi percolation.

Our main goal will be to discuss the similarities and the fundamental relationships between the different models.

Teacher : Bartolomej Blaszczyszyn.

Ects : 6

Total hours : 24

Overview :

Ce cours fait partie de l’offre du Master partenaire Probabilités et Modéles Aléatoires et sera enseigné à Sorbonne Université.

Pour l’abstract voir le site web www.lpsm.paris/formation/masters/m2-probabilites-et-modeles-aleatoires/

Enseignant : Pierre Monmarché

Ects : 6

Total hours : 24

Overview :

Several problems in statistical mechanics of disordered systems boil down to questions about products of random matrices. This is true to the point that certain products of random matrices are the prototype models for wide classes of disordered systems and, in some cases, they are even the standard models. Moreover, in most of the physically relevant cases the matrices that appear are either two by two or the reduction to the the two by two case still captures the essence of the problem.

Several questions are still open even in this (apparently) elementary context.

The course is organized in two parts:

Part I. Introduction to the theory of products of independent and identically distributed random matrices, with focus on the two by two case.

Part II. Disordered models and products of random matrices.

1a. Transfer matrices: the classical Ising chain (i.e., d=1) with random external field, the quantum Ising chain with random transverse field and the classical Ising model in d=2 with columnar disorder.

2b. Anderson localization: the Schrödinger equation with random potentials in the strong coupling limit.

The prerequisites are limited to the mathematics (notably, probability) known by second semester M2 students. A vast literature is available on the subject: there will be notes for the course.

Teacher : Giambattista Giacomin.

Learning outcomes :

This course is part of the partner master Probabilités et Modéles Aléatoires of Sorbonne Université.

www.lpsm.paris/formation/masters/m2-probabilites-et-modeles-aleatoires/

Ects : 6

Lecturer :
DJALIL CHAFAI

Total hours : 18

Overview :

- Analysis and geometry of Markov processes.

- Functional inequalities and long time behavior

- Concentration of measure and transportation of measure

- Entropy and large deviations

- Random matrices and universality phenomena

- Lindeberg and Stein methods for central limit phenomena

Ects : 6

Total hours : 20

Overview :

Since Anderson’s works in the 1950s, localization in disordered systems has been the subject of many studies in physics and mathematics literature. From a mathematical point of view, the question is to know if the self-adjoint operator representing the Hamiltonian of the system has a pure point spectrum.

At the same time, the theory of random matrices has been developed following the work of Wigner, who observed that the energy levels of heavy atoms is well modeled by the eigenvalues of large random matrices. The studies focus in this case on the statistical distribution of the eigenvalues of these large matrices and in particular the repulsion between them.

The object of this course is the study of random operators coming from these two theories. These operators belong to the class of generalized first or second order Sturm Liouville operators. We will explain which operators appear in these models and then we will study some of their spectral properties, using in particular tools from stochastic calculus.

The important notions of the theory of the self-adjoint operators will be recalled in the first courses (they are therefore not a necessary prerequisite for this course).

Teacher : Laure Dumaz (ENS)

Ects : 6

Ects : 6

Total hours : 30

Overview :

La mécanique céleste est plus vivante que jamais. Après un renouveau résultant de la conquête spatiale et de la nécessité des calculs des trajectoires des engins spatiaux, un deuxième souffle est apparu avec l’étude des phénomènes chaotiques. Cette dynamique complexe permet des variations importantes des orbites des corps célestes, avec des conséquences physiques importantes qu’il faut prendre en compte dans la formation et l’évolution du système solaire. Avec la découverte des planètes extra solaires, la mécanique céleste prend un nouvel essor, car des configurations qui pouvaient paraître académiques auparavant s’observent maintenant, tellement la diversité des systèmes observés est grande. La mécanique céleste apparaît aussi comme un élément essentiel permettant la découverte et la caractérisation des systèmes planétaires qui ne sont le plus souvent observés que de manière indirecte.

Thèmes abordés :

• Le problème des deux corps. Aperçu de quelques intégrales premières, réduction du nombre de degrés de liberté, trajectoire, évolution temporelle. Développements classiques du problème des deux corps.
• Introduction à la mécanique analytique. Principe de moindre action, Lagrangien, Hamiltonien.
• Équations canoniques. Crochets de Poisson, intégrales premières, transformations canoniques.
• Propriétés des systèmes Hamiltoniens. Systèmes intégrables. Flot d’un système Hamiltonien.
• Intégrateurs numériques symplectiques.
• Systèmes proches d’intégrable. Perturbations. Série de Lie.
• Développement du potentiel en polynômes de Legendre.
• Évolution à long terme d’un système planétaire hiérarchique, mécanisme de Lidov- Kozai. Application aux exoplanètes.
• Mouvements chaotiques.
• Exposants de Lyapounov. Analyse en fréquence.

Teacher : G. BOUE

Learning outcomes :

Le cours a pour but de fournir les outils de base qui permettront de mieux comprendre les interactions dynamiques dans les systèmes gravitationnels, avec un accent sur les systèmes planétaires, et en particulier les systèmes planétaires extra solaires. Le cours vise aussi à présenter les outils les plus efficaces pour la mise en forme analytique et numérique des problèmes généraux des systèmes dynamiques conservatifs.

Ects : 6

Overview :

Teacher : Nicolas Tholozan (ENS)

Ects : 6

Overview :

The study of general dynamical systems is often difficult and it is sometimes useful to consider only "generic" systems, that is to disregard some "pathological" situations which happen only for few systems. Of course it is necessary to precisely settle what "few" here mens, that is to define some notions of small sets in the space of all systems, or in sime classes of systems. We will study several such definitions in the course. We will then focus the study on properties of periodic orbits, and we'll

see that their generic properties strongly depend on the class of systems considered. We will consider the following more en more restricted classes of systems : general vector fields, Hamiltonian vector fileds and geodesic flows.

Teacher : Patrick Bernard

Ects : 6

Total hours : 21

Overview :

Teacher : Jean-Pierre Marco

Avenue : Observatoire de Paris

Ects : 6

Overview :

The first part of the course concerns bifurcation theory for maps and ordinary differential equations and an introduction to pattern-forming instabilities and reaction-diffusion equations. Nonlinear waves and solitons as well as instabilities in spatially extended systems are considered in the second part of the lectures, mostly using the concept of amplitude equations which is also applied to problems in condensed-matter physics such as commensurate-incommensurate transitions, magnetic domains and superconductivity. Through these lectures, our aim is to show that symmetry arguments together with a qualitative analysis of differential equations and the use of perturbation techniques provide tools that can be used to understand many phenomena in various fields of physics and elsewhere.

Detail of the course here

Enseignants : Laurette Tuckerman (ESPCI), Stephan FAUVE (LPENS)

Learning outcomes :

Most problems in dynamics encountered in physics or in other fields are governed by nonlinear differential equations. In contrast to linear equations, they usually display multiple solutions with different qualitative characteristics, often different symmetries. We study the bifurcations, i.e. the transitions, between these different solutions when a parameter of the system is varied. We show that the dynamics in the vicinity of these bifurcations is governed by universal equations called normal forms that mostly depend on the broken symmetries at the transition. We emphasize the analogy with phase transitions, but also point out differences such as limit cycles or chaotic behaviors which do not occur at equilibrium.

Bibliography-recommended reading

P. G. Drazin, Nonlinear systems, (Cambridge University Press, 1992).

P. Manneville, Dissipative structures and weak turbulence, (Academic Press, 1990).

S. Strogatz, Nonlinear Dynamics and Chaos, (Westview Press, 1994).

S. Fauve, Pattern-forming instabilities, in Hydrodynamics and nonlinear instabilities, edited by C. Godrèche and P. Manneville (Cambridge University Press, 1998) darchive.mblwhoilibrary.org/handle/1912/802.

R. Hoyle, Pattern Formation. An Introduction to Methods, (Cambridge University Press, 2006).

Ects : 6

Overview :

I- Plasma Physics : principles, structures and dynamics, waves and instabilities :

Besides ordinary temperature usual (i) solid state, (ii) liquid state and (iii) gaseous state, at very low and very high temperatures new exotic states appear : (iv) quantum fluids and (v) ionized gases. These states display a variety of specific and new physical phenomena : (i) at low temperature quantum coherence, correlation and indiscernibility lead to superfluidity, superconductivity and Bose-Einstein condensation ; (ii) at high temperature ionization provides a significant fraction of free charges responsible for instabilities, nonlinear, chaotic and turbulent behaviors characteristic of the “plasma state”. This set of lectures provides an introduction to the basic tools, main results and advanced methods of Plasma Physics.

II- Advanced fluid dynamics :

These lectures aim to bridge the gap between classical introductory lectures on fluid mechanics and the more advanced problems addressed in academic research. The first part of the course will be devoted to fundamental aspects, including the interplay between statistical physics and fluid dynamics. In particular, we will briefly discuss the theoretical process leading to the Navier-Stokes equations from the Boltzmann equation of the classical kinetic theory of gases. This approach highlights the relation between transport coefficients (such as heat conduction) and microscopic data and will lead us to describe some features of the theory of thermal conduction and diffusion in fluids. Similarly, a part of the course will focus on the fundamentals of compressiblefluid motions, which usually remain on the fringes of introductory courses. We will see that there exists a strong analogy between gas dynamics and shallow-depth interfacial waves, leading to interesting results on shock waves and solitary waves. An introduction to complex fluids, such as magnetohydrodynamics or quantum hydrodynamics will also be given.

Teachers : Jean-Marcel RAX (Univ. Paris-Saclay, Ecole Polytechnique), Christophe Gissinger (LPENS).

Learning outcomes :

I- Plasma Physics : principles, structures and dynamics, waves and instabilities

II- Advanced fluid dynamics

Bibliography-recommended reading

J. M. Rax, Physique des Plasmas, (Dunod Sciences Sup, 5e tirage, 2018).

J. M. Rax, Physique des Tokamaks, (Editions de l’Ecole Polytechnique, 2011).

L Chen, Waves and Instabilities in Plasmas, (World Scientific, 1987).

L. Landau, E. Lifchitz, Course of theoretical physics - Fluid mechanics(1971)

P. A. Thompson, Compressible Fluid Dynamics (1972)

H.K. Moffatt, magnetic field generation in electrically conducting fluids, Cambridge Press (1978)

J. Lienhard IV & J. Lienhard V A Heat Transfer Textbook, Cambridge Massachussets (1981)

Ects : 6

Overview :

Many natural phenomena are far from thermodynamic equilibrium and keep on exchanging matter, energy or information with their surroundings, producing currents that break time-reversal invariance. Such systems lie beyond the realm of traditional thermodynamics: the principles of equilibrium statistical mechanics do not apply to them. At present, there exists no general conceptual framework `a la Gibbs-Boltzmann to describe their physics from first principles. The last two decades, however, have witnessed remarkable progress.

Details of the course here

Enseignants : Kirone Mallick (CEA, IPht), François Petrelis (LPENS)

Learning outcomes :

The aim of these lectures is to present a unified view on non-equilibrium physics using the framework of statistical and non-linear physics.

Bibliography-recommended reading

N. Van Kampen, Stochastic Processes in Physics and Chemistry, (NorthHolland, Amsterdam, 2007).

P. L. Krapivsky, S. Redner and E. Ben Naim, A Kinetic View of Statistical Physics, (Cambridge University Press, 2010).

R. Elber, D. E. Makarov and H. Orland, Molecular Kinetics in Condensed Phases: Theory, Simulation, and Analysis, (Wiley-Blackwell, 2020).

P. Brémaud, Initiation aux Probabilités: et aux chaînes de Markov (Springer, 2009).

Ects : 6

Overview :

Overview of discretisation in time and space for pdes,

Stokes equation and splitting algorithms,

Transport schemes, numerical diffusion and dispersion,

Compressible flows,

Spectral methods and turbulent flows,

Waves and numerical anisotropy,

Open domains and boundary conditions,

Complex domains,

Prospects.

Validation: Validation will take the form of a mid-term problem and a final project in pairs.

Teacher : Emmanuel Dormy (DMA ENS)

Learning outcomes :

Detail of the course : www.phys.ens.fr/nonlinear_master/program.html

Numerical simulation is playing an expanding role in the study of fluid dynamics in scientific research. In this course, we will develop and analyse the various methods available to solve the partial differential equations relevant to computational fluid dynamics (elliptic, parabolic and hyperbolic). The emphasis will be placed on the algorithms and the convergence properties, as well as their application to a wide variety of problems in fluid dynamics.

Ects : 6

Overview :

1. How to describe deformations : stresses and strains

2. Deformable slender objects.

3. What is a solid?

4. Engineering point of view on rods and truss.

5. Elastic energy

6. Elastic Instabilities : avoiding and using them.

7. Fracture. how things break.

8. Fracture and adhesion of thin objects.

9. Soft solids.

10. interacting fluid and solids.

11. Plasticity.

Details of the course here : www.phys.ens.fr/nonlinear_master/program.html