Paris Sciences et Lettres (PSL) Research University
Research Master 2 in Applied and Theoretical Mathematics
Courses for 2025-2026

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.

• 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
  

• \(L^p\) 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
  

The first part of the course (\(5 × 3\) hrs) is devoted to the study of convergence of probability measures on general (that is not necessarily \(\R\) or \(\R^n\)) 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 application. 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.

• Examples of dynamical systems in discrete and continuous time:
  pendulum, circle rotation, shift, hyperbolic dynamical system, horseshoe, flow, section and suspension, attractor, 3-body problem
  
• Topological dynamics, circle homeomorphisms and Poincaré classification, hyperbolic dynamics (geodesic flow, horocyclic flow)
  

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".

• 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
  

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.

Part 1: Numerical methods for deterministic partial differential equations

• finite difference methods
  
• finite element methods
  
• spectral methods
  
• review of numerical methods for ordinary differential equations
  

Part 2: Monte Carlo methods for particle transport

• Monte Carlo integration
  
• convergence and variance reduction
  
• transport equations starting from probability measures: examples and numerical methods including particle methods
  

Part 3: machine learning and numerical statistics

• high-dimensional statistics and machine learning
  
• stochastic optimization : SGD, Adam, RMSProp, etc.
  
• neural networks: architecture, generative paradigms (VAE, GANs, "stable diffusion")
  

• 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 \(L^2\) Fisher information, log Sobolev inequality and long time convergence to the equilibrium (with rate) in \(L^1\).
    
• Markov semigroups and the Harris-Meyn-Tweedie theory.
  
• 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 \(L^2\) inequality for the scattering equation.
  
• 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.
    

Informally, a continuous-time Markov process is a random function of time whose future is conditionally independent of the past, given the present. In a sense, Markov processes are the stochastic analogs of ordinary differential equations. As such, they are widely used to model complex random evolutions in our physical world. They also serve as ef cient stochastic algorithms for the exploration of massive networks or the generation of random high-dimensional data. The aim of this course is to provide an introduction to their general theory, with an emphasis on examples and applications. We will cover the following notions:

• General definition and characterization via generators and martingales
  
• Fundamental examples: Poisson processes, jump processes, Brownian motion, Markov diffusions, Levy processes
  
• Convergence to equilibrium: functional inequalities, mixing times and the cutoff phenomenon
  
• Modern applications: queuing theory, interacting particle systems and MCMC algorithms
  

• 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
  

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.

The contents may be found here. The course will be taught in French, at Collège de France.

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 the Paris campus of École des Mines.

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, 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.

Hyperbolic conservation laws are a class of nonlinear partial differential equations that reflect standard conservation laws of physics (such as conservation of mass, momentum, and energy) and contain many classical models, such as Euler's equations for compressible flows, as well as more modern models for traffic flows, supply chains, etc.

One of the main aspects of these systems is that, regardless of the regularity of the initial data, their solutions generally develop discontinuities in finite time (this mechanism is known as shock formation). Thus, one should consider discontinuous solutions (in the sense of distributions). However, it has been known since Riemann that uniqueness is lost in this context. This motivates the introduction of the concept of entropy solutions: weak solutions fulfilling additional conditions (connected to the second law of thermodynamics in the case of gas dynamics), aimed at recovering uniqueness.

The theory of entropy solutions is now well developed when the space dimension is 1 (but even this case leaves many open questions!) and solutions are of bounded variation. I will mainly focus on this case.

This course aims to cover both classical and modern topics in the field of random walks in random and non-random media. These topics at least include:

• Random walks in random environment are random processes obtained after launching a Markovian walker on \(\Z^d\) equipped with a random field of transition probabilities. We will review classical results (recurrence / transience, LLN, Sinai regime, Kesten Kozlov Spitzer regime) in dimension \(d=1\) where the behaviour of the walk is well understood but also study the difficult multidimensional case \(d \geq 2\) where even simple questions (as LLN) remains open.
  
• Potential theory and electrical networks : the analogy with electrical networks gives a physical insight as well as a robust method for proving recurrence or transience of reversible random walks on the Euclidean lattice or more general graphs.
  
• Random interlacement, introduced by Sznitman in the early 2010, may be seen as a « soup » of random walk paths. It plays an decisive role both as a limit object for many random walk models and also as a tractable long range correlated random field.
  

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.

This 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 using a probabilistic model containing information about its 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 course we will study several important models from the theory of equilibrium statistical mechanics. In particular we will study the phase diagram of the Ising model (ferromagnetism) and of dimer models (crystal surfaces). We will also study uniform spanning trees, including their links to electrical networks, sampling algorithms and connectivity properties.

The purpose of this course is to study random configurations of points (or particles) in continuous space, called point processes. We begin by studying finite systems, starting with a set of N independent and identically distributed points in a compact space, and their natural generalization to an infinite number of points, the Poisson process, which serve as a “reference law”.

The \(N\) eigenvalues of \(N \times N\) real or complex random matrices provide models of highly dependent points, asymptotically related to some determinantal point processes, particularly the “sine process” (translation invariant on \(\R\)) or the “infinite” Ginibre process (translation invariant on \(\C\)). Determinantal processes form a very rich class of models in any dimension, but these two experience a variance cancellation phenomenon that gives them particular macroscopic properties. This phenomenon is called hyperuniformity.

Depending on time, we may study other systems with this particular property, such as the zeros of the planar Gaussian analytic function, related to the study of zeros of random polynomials (this will require some complex analysis, but it is a very nice theory), or certain particle systems other than Ginibre, or study what hyperuniformity implies macroscopically (optimal transport, rigidity, which requires working with tempered distributions).

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.

• 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
  

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.

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 aim of this course is to give a synthetic but complete presentation of the rough approach to stochastic analysis, which has been developed since 1998 by T. Lyons, M. Gubinelli, A.M. Davie and others. This approach is an alternative to the method based on classical stochastic calculus and provides a (surprising) answer to the question of the continuity of the Ito application associating the Brownian motion trajectory to the solution of a stochastic differential equation. At the end of the course, we will explain how these very innovative ideas were applied by M. Hairer to the much more complex framework of stochastic differential equations with the theory of regularity structures.

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.

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.

Link to the course

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.

Link to the course

• Reminder on differential equations
  
• Reminder on Hamiltonian systems
  
• Short reminder on measure and integration theories
  
• Elements on distributions
  
• Application to the Vlasov equation
  
• The Vlasov-Poisson system
  
• The BBGKY hierarchy, the hypothesis of molecular chaos
  
• The particular case of a cluster with spherical symmetry, an explicit solution
  

This course is taught at Observatoire de Paris.

The objective of this course is to deal with statistical problems where the studied data are high dimensional, meaning that the number of parameters to infer is very large, and in some situations much larger than the number of observations. In this course, we shall present different statistical frameworks adapted to the high-dimensional paradigm, the statistical problems that arise and the specific methodologies for solving them. More precisely, for simple regression models or for more elaborate settings modelling functional data, we shall consider methods based on penalized criteria (AIC, BIC, Ridge…), with a special focus on Lasso-type approaches and their variations. Wavelet thresholding techniques and FDR approaches for multiple testing will be also topics at the core of this course. Finally, the course includes an introduction to statistics for functional data, which is a branch of statistics that studies data that can be modeled as random curves.

This course will be taught at Prairie Institut, 16 rue de l’Estrapade.

The course will cover different aspects of Bayesian statistics with an emphasis on the theoretical properties of Bayesian methods. The course starts with an introduction Bayesian decision theory from point estimation, to credible regions, testing and model selection and some notion on Bayesian predictive inference. The second part will cover the most important results on Bayesian asymptotics.

1. Bayesian decision theory : an Introduction
   
   • Prior / Posterior, risks and Bayesian estimators
     
   • Credible regions
     
   • Model selection and tests
     
2. Bayesian asymptotics; in this part, both well and mis-specified models will be considered.
   
   • Asymptotic posterior distribution: in this part we will study asymptotic normality of the posterior, the penalization induced by the prior and the Bernstein von - Mises theorem. Regular and nonregular models will be treated.
     
   • marginal likelihood and consistency of Bayes factors/model selection approaches.
     
   • Empirical Bayes methods. This part will review some results on the asymptotic posterior distribution for parametric empirical Bayes methods.
     
   • Bayesian bootstrap.
     
   • Posterior consistency and posterior convergence rates. This part will first cover the case of statistical loss functions using the theory introduced by L. Schwartz and developed by Ghosal and Van der Vaart.
     

This course will be taught at Prairie Institut, 16 rue de l’Estrapade.

Modern machine learning typically deals with high-dimensional data. The fields concerned are very varied and include genomics, image, text, time series, or even socioeconomic data where more and more unstructured features are routinely collected. As a counterpart of this tendency towards exhaustiveness, understanding these data raises challenges in terms of computational resources and human understandability. Manifold Learning refers to a family of methods aiming at reducing the dimension of data while preserving certain of its geometric and structural characteristics. It is widely used in machine learning and experimental science to compress, visualize and interpret high-dimensional data. This course will provide a global overview of the methodology of the field, while focusing on the mathematical aspects underlying the techniques used in practice.

This course will be taught at Prairie Institut, 16 rue de l’Estrapade.

Generative models based on dynamical transport have recently led to significant advances in unsupervised learning. At mathematical level, these models are primarily designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second. While these methods were first introduced in the context of image generation, they have found a wide range of applications, including in scientific computing where they offer interesting ways to reconsider complex problems once thought intractable because of the curse of dimensionality. In this class, we will discuss the mathematical underpinning of generative models based on flows and diffusions, with special focus on understanding how to structure the transport to best reach complex target distributions while maintaining computational efficiency, both at learning and sampling stages. We will also discuss applications of generative AI in scientific computing, in particular in the context of Monte Carlo sampling, with applications to the statistical mechanics and Bayesian inference, as well as probabilistic forecasting, with application to fluid dynamics and atmosphere/ocean science.

1. Mathematical Foundations
   
   • Fundamentals of measure transport theory
     
   • Flow matching with stochastic interpolants
     
   • Diffusive Transport with stochastic interpolants
     
   • Link with normalizing flows and score-based diffusion models
     
   • Link with probabilistic denoising methods
     
   • Connections to optimal transport theory and Schrödinger bridges
     
2. Algorithmic aspects
   
   • Training strategies for flow-based models
     
   • Efficient sampling techniques for diffusion models
     
   • Balancing expressivity and computational tractability
     
   • Recent algorithmic innovations and efficiency improvements
     
   • Evaluation metrics for generative models (likelihood measures, FID, Inception Score)
     
   • Challenges and limitations of current models
     
3. Applications in Scientific Computing
   
   • Monte Carlo sampling applications: Statistical mechanics simulations, Bayesian inference and uncertainty quantification
     
   • Probabilistic forecasting: Fluid dynamics predictions, Climate and atmospheric/oceanic modeling, Domain-specific adaptations and constraints, Key breakthrough papers and state-of-the-art applications, Future research directions and open problems