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 among the Fundamental and Specialized courses in the list below, the only constraint being that at least 2 courses should be Fundamental. It is also possible to validate up to 2 courses picked in other masters of the Paris area, upon prior approval from the program directors. See here for special rules applying to students that intend to validate a minor in physics.
In addition, students must write a memoir on a research or reading project under the supervision of a research director either in PSL or in another institution. This project may also take the form of an internship in a company.
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.
We will revise the main notions and theorems from differential calculus (implicit function theorem, inverse function theorem, Brouwer theorem...), as well as main facts about ODE and results about linear and nonlinear stability and smooth dependance by perturbations.
This course is an introduction to methods for the numerical solution of deterministic and stochastic differential equations and numerical aspects of machine learning. It consists of three distincts parts and includes implementations using Python, FreeFEM++ and Keras/Tensorflow.
See also G. Turinici's web site.
In a first part, we will present several results about the well-posedness issue for evolution PDE.
In a second part, we will mainly consider the long term asymptotic issue.
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 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 measures on metric spaces and in particular on C([0, 1]), the space of real continuous function 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 applicatioin. 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 systems. 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 deviation principle, Kramer theorem for independent variables, as well as GŁrtner-Ellis and Sanovs theorems. Next, we will see some applications of the formalism. The examples are mainly inspired by equilibrium statistical physics and thermodynamics. They include the equivalence of ensembles, the interpretation of thermodynamical potentials 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 dynamical 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.
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".
More informations here https://www.ceremade.dauphine.fr/~salez/stoc.html.
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.
The spectral theory of self-adjoint operators has many applications in mathematics, especially in the field of Partial Differential Equations (PDEs). In this course, we will present the details of this theory, which we will illustrate with various practical examples (Dirichlet and Neumann Laplacians on a bounded domain, for example).
In a second part of the course, we will see that the combination of spectral techniques and variational methods allows to obtain interesting results on linear and nonlinear elliptic problems.
We will illustrate this approach on problems from quantum mechanics, widely used in applications. We will study in particular the N-body Schrödinger equation and its mean field approximations giving rise to a nonlinear Schrödinger equation, as well as the periodic Schrödinger operators used for material modeling. The basic theory of quantum mechanics will be presented, but no physical knowledge is required to follow the course.
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.
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.
The aim of this course is to present recent developments concerning the dynamics of nonlinear wave equations. In the first part of the course, I will present some classical properties of linear wave equations (cf. [3, Chapter 5]): representation of solutions, finite speed of propagation, asymptotic behavior, dispersion and Strichartz inequalities ([7], [5]), as well as a concentration-compactness tool, the profile decomposition [1].
The second part of the course concerns semi-linear wave equations. After a presentation of the basic properties of these equations (cf e.g. [5], [6]): local existence and uniqueness of solutions, conservation laws, transformations), I'll give several examples of dynamics: scattering to a linear solution, self-similar behavior and solitary waves. I'll then outline the proof of soliton resolution for the critical wave equation [2], [4].
The prerequisites are the basics of classical real and harmonic analysis (in particular Fourier transform). This course can be seen as a continuation of the fundamental courses "Introduction to Nonlinear Partial Differential Equations" and "Introduction to Evolutionary Partial Differential Equations", but can also be taken independently of these two courses.
This course will be taught at ENS.
Integrable probability is a relatively new subfield of probability that concerns the study of exactly solvable probabilistic models and their underlying algebraic structures. Most of these so-called integrable models come from statistical physics. They serve as toy models to discover the asymptotic behavior common to large classes of models, called universality classes. The methods used in integrable probability often come from other areas of mathematics (such as representation theory or algebraic combinatorics) and from theoretical physics. In the last twenty years, these methods have been particularly fruitful for studying the Kardar-Parisi-Zhang universality class (named after the three physicists who pioneered the domain in the 1980s). This class gathers interface growth models describing a wide variety of physical phenomena, whose asymptotic behavior is surprisingly related to the theory of random matrices.
This course will focus on a central tool in the eld: Schur and Macdonald processes. This will allow us to study in a uni ed way some of the most emblematic integrable models, and ultimately arrive at the the exact calculation of the law of a solution of the Kardar-Parisi-Zhang equation. Along the way, we will take a few detours through various applications or related concepts: random matrices, Robinson-Schensted-Knuth correspondence, interacting particle systems, Yang-Baxter equation and the six-vertex model, random walks in a random environment.
The course will be taught at ENS.
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.
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).
Various families of harmonic functions reflect the asymptotic geometry of graphs. There are several concepts for the boundary of infinite metric spaces and graphs, in particular Cayley graphs of groups. The Martin boundary is a topological space which provides us with the description of non-negative harmonic functions. Due to the convgerence theorem for martingales, such functions converge along one-sided infinite trajectories of the random walk on a graph. If we suppose that functions are bounded, they can be recovered from these limit values, which lead to one of the definitions of the Poisson boundary. It consists in a probability space, which is key to understanding the asymptotic behavior of random walks on graphs and groups. Other classes of functions which were studied recently include Lipschitz harmonic functions, which appear in Kleiner's proof of the polynomial growth theorem.
In this course, we will study harmonic functions and various notions of boundary for random walks on graphs, by focusing on the case where graphs are transitive and operators are homogeneous. This latter class comprise random walks on groups. We will study fundamental results due to Doob, Furstenberg, Azencott, Cartier, Margulis, Guivarch, Kaimanovich, Vershik and Derriennic. At the end we will discuss some very recent results in this domain.
No prerequisites are required.
The course will be taught at ENS.
Diverses familles de fonctions harmoniques reflètent la géométrie asymptotique des graphes. Il existe différents concepts du bord d'espaces métriques infinis et des graphes, en particulier pour des graphes de Cayley de groupes. Le bord de Martin est un espace topologique qui fournit la description de fonctions harmoniques positives. Par le théorème de convergence des martingales, de telles fonctions convergent le long de trajectoires infinies unilatérales de la marche aléatoire sur un graphe. Si nous supposons que les fonctions sont bornées, elles peuvent être récupéré à partir de ces valeurs limites, ce qui conduit à l'une des défnitions du bord de Poisson. Il s'agit d'un espace de probabilité, qui est essentiel pour comprendre le comportement asymptotique des marches aléatoires sur les graphes et les groupes. D'autres classes des fonctions étudiées récemment incluent les fonctions harmoniques lipshitziennes, qui apparaîssent dans la preuve de Kleiner du théorème de croissance polynomiale.
Dans ce cours, nous étudions les fonctions harmoniques et différentes notions de bords pour les marches aléatoires sur les graphes, en nous concentrant sur le cas où les graphes sont transitifs et les opérateurs sont homogènes. Cette dernière classe comprend les marches aléatoires sur des groupes. Nous étudierons des résultats fondamentaux du à Doob, Furstenberg, Azencott, Cartier, Margulis, Guivarch, Kaimanovich, Vershik et Derriennic, et à la fin du cours nous prévoyons de discuter quelques résultats très récents dans ce domaine.
Aucun prérequis spéci que en probabilité n'est requis.
Le cours sera enseigné à l'ENS.
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.
This course is part of the partner master Probabilités et Modéles Aléatoires of Sorbonne Université.
This course aims to master the techniques of analysis and stochastic calculation specific to jump processes. It complements the course "Stochastic Calculation", which is limited to processes with continuous paths.
How many times must one shuffle a deck of 52 cards? This course is a self-contained introduction to the modern theory of mixing times of Markov chains. It consists of a guided tour through the various methods for estimating mixing times, including couplings, spectral analysis, discrete geometry, and functional inequalities. Each of those tools is illustrated on a variety of examples from different contexts: interacting particle systems, card shufflings, random walks on groups, graphs and networks, etc. Finally, a particular attention is devoted to the celebrated cutoff phenomenon, a remarkable but still mysterious phase transition in the convergence to equilibrium of certain Markov chains.
Pour en savoir plus : www.ceremade.dauphine.fr/~salez/mix.html
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).
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.
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.
The course on Stochastic Control (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.
Lecture notes on www.ceremade.dauphine.fr/~cardaliaguet/Enseignement.html
This course provides an in-depth presentation of the main techniques for the evaluating of options using Monte Carlo techniques.
This course is provided by the M2 Mathématiques de la Modélisation of Sorbonne Université.
The interaction between partial differential equations and probability is very active research field with fundamental breakthroughs in the last fifteen years.
Let us give three examples.
In these three examples of very different natures, the difficulty of the analysis comes from the nonlinearity of the interaction between the differential operator and the randomness (due to the nonlinearity of the equation in the first two cases eg).
The aim of this course is to make a short tour of the interaction between PDEs and randomness by treating an example (as simple as possible) of each type.
This course is an introduction of some recent (and less recent) trends in stochastic analysis going beyond the standard techniques of stochastic calculus and going under the umbrella name of “rough analysis”. Starting from rough path theory, these techniques allow to resolve the singularities of the non-linear interactions of random fields, and identify new “building blocks” which can then be used to synthetize solutions of stochastic differential equations which are ill-posed wrt standard “linear” approaches, or just not very robust. The plan of the course is to explain the basic ideas of rough path theory, pathwise regularization by noise and some paracontrolled calculus applied to the construction of singular unbounded operators. The different settings will allow to present the basic ideas from various perspectives.
This course is taught in French at Observatoire de Paris.
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.
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.
Plan:
This course is taught at Observatoire de Paris.