## Program Year

The academic year starts in September with 2 weeks of Preliminary Courses that are not followed by exams and are intended as a quick review of tools that students should mostly already know from previous studies.

During the M2 year students must pass the exams of 6 courses freely chosen in the list below, the only constraint being that at least 2 courses should be in the sublist of Fundamental Courses. It is also possible to validate up to 2 courses of other masters, prior approval from the program directors.

In addition, students must do a research internship under the supervision of a research director: the outcome will be a written manuscript (master thesis) and an oral discussion.

Each student will be followed by a scientific tutor who will orientate for the choice of the courses and help to find a suitable director for the research internship.**For the updated list of courses, see here**

#### Preliminary courses list

- A review of functional analysis tools for PDEs
#### A review of functional analysis tools for PDEs

Lecturer :

DAVID GONTIERTotal 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.

- A review of numerical methods for PDEs
#### A review of numerical methods for PDEs

Lecturer :

GUILLAUME LEGENDRETotal 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.

- A review of probability theory foundations
#### A review of probability theory foundations

Ects : 6

Lecturer :

PAUL GASSIATTotal 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

- A review of differential calculus for ODEs and PDEs
#### A review of differential calculus for ODEs and PDEs

Ects : 6

Lecturer :

EMERIC BOUINTotal 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.

#### Fundamental courses list

- Stochastic Calculus
#### Stochastic Calculus

Ects : 6

Lecturer :

JUSTIN SALEZTotal 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”.

- Introduction to non linear PDEs
#### Introduction to non linear PDEs

Ects : 6

Lecturer :

ERIC SERETotal 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).

- Introduction to evolution PDEs
#### Introduction to evolution PDEs

Ects : 6

Lecturer :

STEPHANE MISCHLEROverview :

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

- Numerical methods for PDE and control
#### Numerical methods for PDE and control

Ects : 6

Lecturer :

JULIEN SALOMONTotal 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).

- Differential dynamical systems in mechanics and physics
#### Differential dynamical systems in mechanics and physics

Ects : 6

Lecturer :

ABED BOUNEMOURATotal hours : 30

Overview :

Lecturers: Abed Bounemoura and Laurent Niederman

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

- Limit theorems and large deviations
#### Limit theorems and large deviations

Ects : 6

Lecturer :

FRANCOIS HUVENEERS

FRANCOIS SIMENHAUSTotal hours : 30

Overview :

The first part of the course (5*3 hours) is devoted to the study of convergence of probability measures on general (that is not necessarily R or Rn) metric spaces or, equivalently, to the convergence in law of random variables taking values in general metric spaces. If this study has its own interest it is also useful to prove convergence of sequences of random objects in various random models that appear in probability theory. The main example we have to keep in mind is Donsker theorem that states that the path of a simple random walk on Z converges after proper renormalization to a brownian motion. We will start this course with some properties of probability mea- sures on metric spaces and in particular on C([0, 1]), the space of real continuous func- tion on [0, 1]. We will then study convergence of probability measures, having for aim Prohorov theorem that provides a useful characterization of relative compatctness via tightness. Finally we will gather everything to study convergence in law on C([0, 1]) and prove Donsker therorem. If there is still time we will consider other examples of appli- catioin. The main reference for this first part of the course is Convergence of probability measures, P. Billingsley (second edition).

The second part of the course will deal with the theory of large deviations. This theory is concerned with the exponential decay of large fluctuations in random sys- tems. We will try to focus evenly on establishing rigorours results and on discussing applications. First, we will introduce the basic notions and theorems: the large devia- tion principle, Kramer theorem for independent variables, as well as GŁrtner-Ellis and Sanovs theorems. Next, we will see some applications of the formalism. The exam- ples are mainly inspired by equilibrium statistical physics and thermodynamics. They include the equivalence of ensembles, the interpretation of thermodynamical poten- tials as large deviation functionals, and phase transitions in the mean-field Curie-Weiss model. In a third part, we will develop large deviation principles for Markovian dy- namical processes. If times allows, we will present some applications of these results in a last part of the course. There is no explicit prerequisite to follow the classes but students should be well acquainted with probability theory.

#### Specialized courses analysis list

- Mean field games theory
#### Mean field games theory

Ects : 6

Lecturer :

PIERRE CARDALIAGUETTotal hours : 18

Overview :

Stochastic Control cours (1rst semester) is a necessary prerequisite.

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

- Théorie spectrale et méthodes variationnelles
#### Théorie spectrale et méthodes variationnelles

Ects : 6

Lecturer :

Eric CANCES

MATHIEU LEWINTotal 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

- Continuous optimization
#### Continuous optimization

Ects : 6

Lecturer :

ANTONIN CHAMBOLLETotal 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.

- Introduction to control theory
#### Introduction to control theory

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 & P. Lissy

- Non-convex inverse problem
#### Non-convex inverse problem

Ects : 6

Lecturer :

IRENE WALDSPURGERTotal hours : 24

Overview :

An inverse problem is a problem where the goal is to recover an unknown object (typically a vector with real coordinates, or a matrix), given a few “ measurements ” of this object, and possibly some information on its struc- ture. In this course, we will discuss examples of such problems, motivated by applications as diverse as medical imaging, optics and machine learning. We will especially focus on the questions: which algorithms can we use to numerically solve these problems? When and how can we prove that the solutions returned by the algorithms are correct? These questions are rel- atively well understood for convex inverse problems, but the course will be on non-convex inverse problems, whose study is much more recent, and a very active research topic.

The course will be at the interface between real analysis, statistics and optimization. It will include theoretical and programming exercises.

See www.ceremade.dauphine.fr waldspurger/enseignement.html for more informations / material on this course

- Entropy methods, functional inequalities and applications
#### Entropy methods, functional inequalities and applications

Ects : 6

Lecturer :

AMIC FROUVELLE

EMERIC BOUIN

JEAN DOLBEAULTTotal hours : 24

Overview :

Various functional inequalities are classically seen from a variational point of view in nonlinear analysis. They also have important consequences for evolution problems. For instance, entropy estimates are standard tools for relating rates of convergence towards asymptotic regimes in time-dependent equations with optimal constants of various functional inequalities. This point of view applies to linear diffusionsand will be illustrated by some results on the Fokker-Planck equation based on the "carré du champ" method introduced by D. Bakry and M. Emery. In the recent years,the method has been extended from linear to nonlinear diffusions. This aspect will be illustrated by results on Gagliardo-Nirenberg-Sobolev inequalities on the sphere and on the Euclidean space. Even the evolution equations can be used as a tool for the study of detailed properties of optimal functions in inequalities and their refinements. There are also applications to other equations than pure diffusions: hypocoercivity in kinetic equations is one of them. In any case, the notion of entropy has deep roots in statistical mechanics, with applications in various areas of science ranging from mathematical physics to models in biology. A special emphasis will be put during the course on the corresponding models which offer many directions for new research development.

- Variational problems and optimal transport in economy
#### Variational problems and optimal transport in economy

Ects : 6

Lecturer :

GUILLAUME CARLIERTotal hours : 18

Overview :

1. Duality 2. Optimal Transport 3. Economic Applications of Optimal Transport technics 4. Calculation of variations 5. The principal-agent problem

Learning outcomes :

Mastering of variational and optimal transport methods used in economy.

- Variational and geodesic methods for Image Analysis
#### Variational and geodesic methods for Image Analysis

Ects : 6

Lecturer :

LAURENT COHENTotal 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.

- Chaire Equation aux dérivées partielles : "Sur les équations de transport"
#### Chaire Equation aux dérivées partielles : "Sur les équations de transport"

- Chaire Sciences des données "Information et Complexité"
#### Chaire Sciences des données "Information et Complexité"

Overview :

Lecturer: Stéphane Mallat (Collège de France)

Le cours introduit une approche mathématique de l'apprentissage statistique à travers l ’ estimation par maximum de vraisemblance, la théorie de l'information et la construction de modèles d'approximation. Les apprentissages non supervisé et supervisé passent par l'estimation de distributions de probabilité en grande dimension, à partir des données d'apprentissage. Cela nécessite de construire des modèles paramétrés, définis par une information a priori. Cela peut être des réseaux de neurones profonds dont l'architecture est spécifiée. Le cours soulève les questions fondamentales de modélisation en grande dimension, et leur formalisation mathématique à travers des mesures d ’ information. Il introduira les notions d ’ information de Fisher pour l ’ inférence de modèle par maximum de vraisemblance, et d ’ information de Shannon pour la prédiction et le codage. L ’ information de Shannon est basée sur une notion de concentration et de mesure d ’ incertitude par l ’ entropie. La construction de classes de modèles se base sur des hypothèses concernant la structure des distributions et leurs invariants. Les liens avec la physique statistique seront explorés. On s'intéressera particulièrement aux données « complexes » qui mettent en jeu de nombreuses échelles de variabilité, que ce soit des images, des sons, des séries temporelles ou des données qui proviennent de la physique. On étudiera des applications à la compression de signaux et d ’ images et à l'apprentissage non supervisé.

- Numerical methods for PDE and control
#### Numerical methods for PDE and control

Ects : 6

Lecturer :

JULIEN SALOMONTotal 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).

#### Spécialized courses probability list

- Jump processes
#### Jump processes

Ects : 6

Lecturer :

JULIEN POISATTotal 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.

- Mixing times of Markov chains
#### Mixing times of Markov chains

Ects : 6

Lecturer :

JUSTIN SALEZTotal 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)

- Stochastic Control
#### Stochastic Control

Ects : 6

Lecturer :

PIERRE CARDALIAGUETTotal 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.

- Monte-Carlo and Finite Differences Methods with Applications to Finance
#### Monte-Carlo and Finite Differences Methods with Applications to Finance

Ects : 6

Lecturer :

JULIEN CLAISSETotal 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.

- Random geometric models
#### Random geometric models

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.

- Convergence en temps long pour les processus de Markov
#### Convergence en temps long pour les processus de Markov

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

- Products of random matrices and disordered systems in statistical mechanics
#### Products of random matrices and disordered systems in statistical mechanics

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/

- Opérateurs aléatoires
#### Opérateurs aléatoires

Ects : 6

Total hours : 20

Overview :

Lecturer : Laure Dumaz (ENS)

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).

- Introduction to statistical mechanics
#### Introduction to statistical mechanics

Ects : 6

Lecturer :

CRISTINA TONINELLI

BEATRICE TAUPINART DE TILIERETotal 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).

- Continuous optimization
#### Continuous optimization

Ects : 6

Lecturer :

ANTONIN CHAMBOLLETotal 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.

- Cross-diffusion systems
#### Cross-diffusion systems

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-compa ctness, 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)

- Dynamique des systèmes gravitationnels à grand nombre de particules
#### Dynamique des systèmes gravitationnels à grand nombre de particules

Ects : 6

Total hours : 21

Overview :

Teacher : Jean-Pierre Marco Avenue : Observatoire de Paris

- Mean field games theory
#### Mean field games theory

Ects : 6

Lecturer :

PIERRE CARDALIAGUETTotal hours : 18

Overview :

Stochastic Control cours (1rst semester) is a necessary prerequisite.

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

- Non-linear models for quantum physics
#### Non-linear models for quantum physics

Ects : 6

Lecturer :

DAVID GONTIERTotal hours : 24

Overview :

In this course, we are interested in the derivation and study of nonlinear models that appear in quantum mechanics. In the first part of the course, we will show how, starting from the linear Schrödinger equation, we find nonlinear models in certain regimes or after certain approximations. We will introduce in particular the Hartree-Fock and Thomas-Fermi models. The study of these nonlinear models will allow us in a second time to obtain information on the Schrödinger equation. We will find certain more or less known phenomena of quantum mechanics. In particular, we will demonstrate the "stability of matter", and describe what a "heavy atom" looks like. Bibliography: * “Analysis” (Elliott H. Lieb and Michael Loss) * “Stability of Matter” (Elliott H. Lieb and Robert Seiringer)

- Brownian continuum objects
#### Brownian continuum objects

Overview :

Lecturer: Jan Swart

#### Specialized courses geometry and dynamical systems list

- Gravitation classique et mécanique céleste
#### Gravitation classique et mécanique céleste

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.

- Generic properties of Hamiltonian Systems
#### Generic properties of Hamiltonian Systems

Ects : 6

Lecturer :

PATRICK BERNARDOverview :

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 means, that is to define some notions of small sets in the space of all systems, or in some 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.

- Dynamique des systèmes gravitationnels à grand nombre de particules
#### Dynamique des systèmes gravitationnels à grand nombre de particules

Ects : 6

Total hours : 21

Overview :

Teacher : Jean-Pierre Marco Avenue : Observatoire de Paris

- Géométrie de relations aux dérivées partielles
#### Géométrie de relations aux dérivées partielles

Overview :

Lecturer: Emmanuel Giroux

La première partie du cours sera une introduction à l'étude des variétés et de leur topologie qui commencera avec les bases de la théorie de l'homotopie et de l'homologie. On se concentrera ensuite sur les variétés et les fibrés lisses en s'intéressant à la géométrie des formes différentielles et des connexions. La seconde partie du cours sera consacrée à l'étude des relations aux dérivées partielles et du h-prinicpe, théorie qui permet de mettre à jour de nombreux phénomènes géométriques frappants : théorème de Smale sur le retournement de la sphère, théorèmes de Nash sur les plongements isométriques...

- Applications harmoniques et connections de Yang-Mills
#### Applications harmoniques et connections de Yang-Mills

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)

#### Specialized courses macroscopic physics list

- Instabilities and nonlinear phenomena
#### Instabilities and nonlinear phenomena

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).

- Numerical methods for fluid dynamics
#### Numerical methods for fluid dynamics

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.

- Non-linear models for quantum physics
#### Non-linear models for quantum physics

Ects : 6

Lecturer :

DAVID GONTIERTotal hours : 24

Overview :

In this course, we are interested in the derivation and study of nonlinear models that appear in quantum mechanics. In the first part of the course, we will show how, starting from the linear Schrödinger equation, we find nonlinear models in certain regimes or after certain approximations. We will introduce in particular the Hartree-Fock and Thomas-Fermi models. The study of these nonlinear models will allow us in a second time to obtain information on the Schrödinger equation. We will find certain more or less known phenomena of quantum mechanics. In particular, we will demonstrate the "stability of matter", and describe what a "heavy atom" looks like. Bibliography: * “Analysis” (Elliott H. Lieb and Michael Loss) * “Stability of Matter” (Elliott H. Lieb and Robert Seiringer)

- Plasma Physics and advanced fluid dynamics
#### Plasma Physics and advanced fluid dynamics

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)

- Systems out of equilibrium and nonlinear dynamics
#### Systems out of equilibrium and nonlinear dynamics

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).

- Théorie des systèmes complexes : des verres de spins aux réseaux de neurones
#### Théorie des systèmes complexes : des verres de spins aux réseaux de neurones

Overview :

Lecturer: B. Derrida (Collège de France)

- Turbulence
#### Turbulence

Ects : 3

Overview :

Enseignants : Alexandros Alexakis et Berengere Dubrulle

- Soft solids
#### Soft solids

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

*Academic Training Year 2023 - 2024 - subject to modification*

## Internships and Supervised Projects

During the second semester students will do a supervised research project (internship) of variable duration (minimum 3 months, average 4/5 months).

The subject of the internship will be freely chosen by the students according to their interest. If needed, tutors will orient in identifying a suitable research subject and/or project supervisor.

We warmly encourage our students to visit the research pages of the laboratories associated to the master (Ceremade - PSLand DMA - PSL) as well as of other research laboratories to identify potential directors and contact them directly. Students who want to conduct their research internship outside the Paris region can apply for financial assistance from the Paris Mathematical Sciences Foundation (more details here).

With prior written approval from the program director, the research internship may be replaced by an internship in a private company, as long as it has a sufficient research component.

**Research-driven Programs **

Training courses are developed in close collaboration with Dauphine's world-class research programs, which ensure high standards and innovation.

Research is organized around 6 disciplines all centered on the sciences of organizations and decision making.

**Learn more about research at Dauphine **