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 7 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.
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".
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.
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 focuses on an introduction to systems and control theory. It concerns the study of a dynamical system affected by an input signal which we aim at designing to modify the system behavior. It will focus on nonlinear Ordinary Differential Equations (ODEs), but will also include an introduction to the control of Partial Differential Equations.
We will start by reviewing stability notions of nonlinear ODEs (Lyapunov theorems, sufficient and necessary stability conditions, spectral criteria for linear systems, Input-to-State Stability,…). Then, we will study the concepts of controllability/observability of dynamical systems and move to stabilization of equilibrium points, with the presentation of a few control design methodologies (backstepping, forwarding, optimal control, Lie Bracket methods...).
The class will be concluded by a few session on the extension of these concepts to infinite-dimensional linear control systems, namely, Partial Differential Equations. Examples will include in-domain and/or boundary control of the heat equation and the wave equation.
The course will be taught at École des Mines.
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 non-linear 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].
The second part of the course concerns semi-linear wave equations. After a presentation of the basic properties of these equations (local existence and uniqueness of solutions, conservation laws, transformations cf. e.g. [5, 6]), I'll give several examples of dynamics: scattering to a linear solution, self-similar behavior and solitary waves. I will also give results on the classification of the dynamics for the energy critical wave equation following [2, 4], and some elements of proofs, including the profile decomposition introduced by Bahouri and Gérard [1].
The prerequisites are the basics of classical real and harmonic analysis. 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.
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.
Further information: www.ceremade.dauphine.fr/~salez/mix.html
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.
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
The pathwise (or rough) approach to stochastic analysis consists in a series of analytic methods, developed in the last two decades, which are able to deal with the functions of low regularity which arise naturally when considering stochastic objects. The crucial step is typically to identify a natural metric space on which the "noise-to-solution" map is continuous.
These methods allow on the one hand for a new (and often more natural and robust) point of view on classical objects (such as the SDE defined by Itô integration), while also allowing to deal with objects which were previously out of reach (such as ODE driven by fractional Brownian motion, or singular stochastic PDE).
The course will be taught at ENS.
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:
The theory of groups and their representations is a central topic which studies symmetries in various contexts occurring in pure or applied mathematics as well as in other sciences, most notably in physics.
Lie theory (i.e. the study of Lie groups and Lie algebras) has played an important role in mathematic ever since its introduction by the Norwegian mathematician Sophus Lie in the 19th century. It has had a profound impact in physics as well.
The aim of this course is to provide an introduction, from the mathematical perspective, of the classical concepts and techniques of Lie theory. The course will in particular deal with Lie groups, Lie algebras (of finite dimension) and their representations, and include the study of numerous examples.
This course will be taught at ENS.
The goal of these lectures is to define mathematical gauge theory and relate it to gauge theory (from physics).
To achieve this, we will build a mathematical background: a lot of differential geometry, some algebraic topology, a bit of Riemannian geometry, and an ounce of category theory. We will also see how to use these effectively in physics.
This course will be taught at ENS.
This course is taught at Observatoire de Paris.