Course list

The goal of this course is to give a short but rigourous presentation of the notion of stochastic integral with respect to a (continuous) semimartingale. A particular focus will be made on the Brownian motion which will be a recurrent illustration for the main tools introduced.

This course is the natural continuation of the course of the course Stochastic processes. It will first focus on the fundamental tools of stochastic calculus, such as Itô’s change of variable formula, Girsanov change of measure theorem or the representation of martingales theorem. Then, stochastic differential equations and their solutions will be introduced, as well as some of their properties : regularity, Markovianity, semi-group property.

The lectures focus on dynamical systems given by a measure-preserving transformation T on a probability space (X,B,µ). Such transformations are ubiquitous, typical examples including translation by an irrational number on R/Z or elements of Mn(Z) acting on Rn/Zn. The goal is to understand statistical properties of Tn when n →∞. Involved topics include : ergodicity, mixing, strong mixing; ergodic theorems; unique ergodicity; entropy.

Lectures deal with estimation methods for models with continuous time stochastic processes with jumps. First we consider classical properties of counting processes and renewal processes. After that, we study more carefully the simple Poisson process and its generalisations, as the inhomogeneous Poisson process or the compound Poisson process. The pure jump Markov processes are also studied from a probabilistic and a statistical point of view. Finally, some drift estimation problems of Stochastic Differential Equations are considered.

This course focuses on inference in statistical models where the underlying distribution is described by an unknown finite-dimensional parameter. Classical parametric inference methods, such as maximum likelihood, will be discussed first. Tools making it possible to compare parametric estimators, and to decide whether one of them is optimal, will then be introduced. The last part willconcentrateonhowparametricmethodscanbeusedinsemi-parametricmodels,amotivation being the statistical analysis of extreme values.

This course is devoted to the estimation of infinite dimensional objects, such as the density of a probability measure or a regression function. The course will be divided in three parts. First, we will introduce the main methods of non-parametric estimation, such as the kernel methods, the estimation by projection and the methods of regularization. Then we will focus on optimality of the discussed methods and in particular on the best rates of convergence that can be achieved by any estimator : this is the minimax theory. Finally, in relation with the statistical learning theory, we will discuss model selection procedures, allowing adaptation of the proposed estimators, in the sense that the latter will achieve optimal rates of convergence under various hypotheses on the function to be estimated.

The lectures focuses on large-time asymptotics of Markov processes on general spaces. Involved topics include : transition kernels, irreducibility, small sets, periodicity and aperiodicity, recurrence and transience, invariant measures.

This course will contain a large introduction to the stochastic processes having jumps (essentially Lévy processes and if possible the subordinators) and to their stochastic calculus, SDE driven by jump processes, etc. In a second part we will give some examples of models possibly based on this kind of processes.

The lectures cover fundamentals on large deviations (or, roughly speaking, on the "asymptotic theory of rare events"). First we will show a simplified but rigorous version of the classical Large Deviations Techniques and Applications by Dembo and Zeitouni, aiming at a proof of the Sanov theorem (a must-know of probability). Then we shall study two important applications : i) the derivation of thermodynamics from the statistical mechanics of particle systems, and ii) the small-noise limit of stochastic differential equations of Freidlin-Wentzell type (and their use in physics and engineering).

Fouille du web et traitement du langage: ce cours est une introduction à la collecte de données textuelles et au traitement automatique du language.

Analyse des données fonctionnelles: Dans ce cours les étudiants apprennent les idées principales, la théorie associée et les routines numériques de l’analyse des données fonctionnelles.

We will introduce holomorphic functions and plurisubharmonic functions on open subsets of Cn. Then we will focus on problems of extension. In particular we will study holomorphic functions (Hartogs phenomenon, domains of holomorphy, pseudoconvex domains, Levi problem), analytic subsets (Weierstrass preparation theorem, Remmert-Stein theorem) and currents of integration on analytic subsets (Lelong theorem).

These lectures aim at introducing some basic notions of complex geometry, such as complex manifolds, holomorphic vector bundles and hermitian metrics on them. In this framework, the notion of differential forms of type (p,q) is used to define Dolbeault cohomology groups (the analogue of De Rham cohomology). We will study the consequences of the positivity of a metric in this setting : for metrics on line bundles, Kodaira vanishing (of cohomology groups) and embedding theorems; for Kähler metrics, Hodge decomposition.

This first set of lectures study algebraic tools involved in the intersection theory for algebraic geometry. The roadmap is : Introduction; Basic notions in commutative algebra; Artinians and Noetherian rings and modules; Algebraic theory of dimension; Regular rings; Flitrations, graduations, polynomials of Hilbert-Samuel.

This second set of lectures study the geometric point of view on the intersection theory in algebraic geometry.

This is a sequel to Fundamentals of differential geometry The two foregoing courses form a pair of complementary courses. The first one will begin by discussing basic concepts of differential topology such as what are manifolds, bundles, maps between manifolds, etc. It will give examples showing that such objects appear naturally. The second one will then focus on differential forms and de Rham cohomology, proving that it is a topological invariant, computing it in a few cases, and obtaining some of the standard applications such as the Browder fixed point theorem. To conclude the class we will discuss general position and some of its applications such as Whitney’s (weak) embedding theorem or the Hopf index formula. If time allows, we will also discuss the notion of degree, both from the point of view of de Rham cohomology as from the point of view of general position.

The purpose of this course is to introduce, through many examples, differentiable manifolds and classical differentiable objects that lives on these : vector fields, diffeomorphisms, differential forms, de Rahm Cohomology.

We will introduce affine algebraic varieties over the complex numbers, which are basically zero sets of polynomials over the complex numbers with extra structure. The course will consist of getting to know basic concepts such as morphisms, isomorphisms, irreducibility, tangent space and dimension, all accompanied by examples. Concepts from

In real algebraic geometry, the objects one considers can be seen in some Rn, defined by the vanishing of a — single! — polynomial. The study of these objects combines both algebraic properties (like the Nullstellensatz) and topological ones. Several rings of functions are associated with these sets, and they are related to their different properties : of course polynomial functions, but more generally Nash functions (analytic and satisfying a polynomial equation with polynomial coefficients), or even arc-analytic functions. Some of these rings are noetherian, others not, but all of them enable to understand better real algebraic sets.

In this class, we will study varieties obtained by compactification of a complex torus. We will give the combinatorial description of these varieties in terms of fans. This will provide a link between algebraic geometry and convex geometry. We will then focus on divisors, sheaves and their cohomology on these varieties. If time allows, we will also study toric singularities and their resolutions.

, The aim of the course is to describe symmetric spaces, underlying real semisimple groups and their representations. Involved topics include complex Lie algebras, root systems, real Lie groups, the Cartan decomposition, the Iwasawa decomposition, representation theory, the Peter-Weyl theorem.

This course is an introduction to the applications of simplicial technics in algebra, topology and algebraic geometry. We will begin with the basic notions2 In the theory of simplicial objects and homological algebra. This will be followed by an introduction to model categories and in particular the model category of simplicial (commutative) rings. As an interlude, we will discuss the differential graded algebras and the normalized chain functor to the category of DGA. If time permits we will give an introduction to the ideas of derived algebraic geometry.

We will study branched covers of smooth manifolds of dimensions 2, 3, and 4. We will start with the Riemann–Hurwitz formula for surfaces; we will then look at covers of 3-manifolds branched over links and the 4-manifolds they bound. We will then prove the G-signature theorem in dimension 4, and discuss some applications, in particular to embeddings of (possibly singular) surfaces in 4manifolds. Depending on time and on the audience’s interests, we might discuss some complex-geometric, contact, or symplectic aspects of branched covers. Prior knowledge of differential topology and algebraic topology, at the level of the first-semester courses and student seminars, is required; the more low-dimensional notions will be reviewed along the way.

The goal of the course is to introduce symplectic geometry. In the first half of the class we will review basics definitions from symplectic geometry and we will stydu Lagrangian submanifolds of symplectic manifold. We will justify the importance of these submanifolds within and outside symplectic geometry. In the second half of the class we will study two tools used to study Lagrangian submanifolds in cotangent bundles : generating family and constructible sheaves.

This course is an introduction to unbounded operators, which generalise matrices to infinite dimensional spaces. We will discuss their spectrum, and we will apply the theoretical results to differential (or pseudo-differential) operators, often arising from physics.

This lecture will be given after the course on Spectral Theory. It will introduce pseudodifferental operators, a generalization of differential operators which provides a particularly pleasant way to solve some linear partial differential equations. The course will focus on the so-called semiclassical version, which nicely highlights geometric aspects, and applies to the spectral theory of Schrödinger type operators.

The first part of the course is focused on Sobolev spaces. It studies embedding theorems on rather generalclassesofopensets,fractionalsspacesandtraces.Thesecondpartconcernsellipticpartial differential equations, beginning with the linear cas, with different kinds of boundary conditions. Intheend,sometoolsfornonlinearellipticequationswillbeintroduced(Galerkinapproximation, fixe-point framework, etc.).

This course takes Sobolev spaces & elliptic equations as prerequisite. It is thought as an introduction to the nonlinear analysis of evolution partial differential equations, carried out on the example of quasilinear hyberbolic systems. Most of the course is devoted to nonlinear scalar conservation laws, studying both strong and entropic solutions, but it also deals with linear hyperbolic systems. Along the way, to consider vanishing viscosity limits, we introduce basics of semilinear parabolic systems.

This lecture is a numerical counterpart to Sobolev spaces and elliptic equations. In the first part, after some reminders on linear elliptic partial differential equations, the approximation of the associated solutions by the finite element methods is investigated. Their construction and their analysis is described in any dimension. The second part of the lectures consists in defining a generic strategy for the implementation of the method based on the variational formulation. The course includes a practical project to be implemented using one of the classical programming languages (Matlab, Octave, Scilab, Python,...).

This lecture is a numerical counterpart to Hyperbolic equations. The first part is devoted to the analysis of finite difference schemes. Both the problematics of stability and consistency for such schemes are presented on infinite or periodic domains and then on bounded domains. In a second step, the approximation of weak entropy solutions to nonlinear hyperbolic conservation laws are presented through the finite volume method. Some recent development of such schemes are also presented.

Le but de ce cours est d’abord de renforcer les connaissances des étudiants concernant intégration, le calcul différentiel et l’analyse fonctionnelle, pour qu’ils maîtrisent les outils comme les troncatures régulières, les partitions de l’ unité, la régularisation par convolution, l’usage des parties denses, les critères

Ce cours de théorie spectrale est basé sur des notes, en anglais, librement accessibles; elles donneront aux étudiants francophones le loisir d’exercer leur anglais : https ://nraymond.perso.math.cnrs.fr/spectral-theory.pdf 1) Le cours débutera avec des considérations très élémentaires dont le but est double : se rafraîchir la mémoire et introduire des idées qu’on retrouvera plus tard dans des situations plus générales. 2) Il se poursuivra par l’ étude des opérateurs non bornes (fermes, fermables, auto-adjoints, etc.) et la définition du spectre (et même de certains types de spectre : discret et essentiel). On en profitera pour (re)visiter un théorème de Lax-Milgram qui permet de définir des opérateurs fermés bijectifs à l’aide de formes sesquilinéaires coercives. 3) Le reste du cours sera consacré à l’ étude des relations entre les propriétés des opérateurs et les propriétés du spectre. Ainsi, on exposera la théorie de Fredholm et, `a cette occasion, on revisitera la description du spectre des opérateurs compacts ou à résolvante compacte (par réduction à la dimension finie et l’utilisation de l’analyse complexe). 4) La fin du cours explorera le cas particulier des opérateurs auto-adjoints (borne sur la résolvante, principe du min-max); on montrera notamment que spectres discret et essentiel sont complémentaires dans le spectre. Des exemples seront fournis tout au long du cours : Laplacien de Dirichlet, oscillateur harmonique, opérateur de Schrödinger. On s’amusera à calculer le spectre essentiel dans certains cas particuliers. Si le temps le permet (et il ne le permettra surement pas), on ébauchera une présentation du calcul fonctionnel des opérateurs auto-adjoints et de la fameuse mesure spectrale.

Solutions of small amplitudes of non-linear dispersive partial differential equations on a compact without boundary (e.g. a torus or a sphere) are subject to two concurrent effects : — wave dispersion, a consequence of the fact that the plane waves solving the linear part of the equation travel with different velocities (the waves move away from each other). — the compactness of the domain which encourages interaction via nonlinearity (the waves have to see each other often!) Which wins? Does the dynamics in long time go towards stability or turbulence? We will try to answer (partially) these questions on a few examples and through normal form methods in the context of Hamiltonian PDEs.

In these lectures, we will study the notion of controllability, which corresponds to the possibility ofchoosing someparametersof anevolution equation(for example, aforce appliedto the system) in such a way that the state is driven to a desired target. The first part will be an introduction in the setting of ODEs, with a strong focus on nonlinear effects, using tools coming from geometry (Lie brackets). The second part will tackle example of problems coming from fluid mechanics and governed by nonlinear PDEs. We will consider the controllability of Burgers and Euler equations.

We will study evolution equations (ODEs or PDEs) highly oscillatory in time, when the non autonomous vector field depends periodically on time. Such problems occur in many interesting physical situations, which will be presented. The so-called averaging techniques will be studied, whichenabletowritethesolutionoftheproblemunderasuitableform:afteraperiodicchangeof variable,it is the solution of an average dequation,non stiffin time. We will study in particular the stroboscopic averaging method, which preserves the geometric structures of the initial equation (Hamiltonian character, preservation of volume). In a last part, we will construct some numerical methodswell-adapted for such problems. These methods are called uniform lyaccurate,sincetheir numerical convergence is uniform with respect to the small parameter in the equation.

In the past 15 years, a new theory has been developped concerning the long time behavior of kinetic equations, under the name ”hypocoercivity”. This approach has now proved to be very efficient in the study of a large range of equations. This course is devoted to the main features of this approach in the hilbertian case : basic notions of coercivity and hypocoercivity, relations between the spectral theory and the long time behavior of linear or linearized models. We shall also - depending on the time remaining - study the nonlinear perturbative case, the case of collision kernels of Boltzmann type, the connection with the theory of hypoellipticity and the recent theory of enlarged spaces.

The Schr¨odinger equation is one of the fundamental equation of quantum mechanics. This course will be devoted to the study of its properties from a mathematical perspective : we will describe its localization and regularizing properties in the periodic framework. This will give us the opportunity to introduce standard tools from Fourier analysis (such as pseudodifferential operators) and control theory. Keywords : Fourier analysis, mathematical physics, spectral theory, semiclassical analysis, pseudodifferential operators, harmonic analysis, control of PDE.

Thanks to the spectral theorem, a self-adjoint (or normal) operator on Hilbert space can be nearly completely understood by studying its eigenvalues and eigenvectors (or their continuous analogues). Non-selfadjoint operators, which appear in many applications including the theory of resonances, kinetic theory, and perturbation theory, are not so easily reduced to their spectral decomposition. We will study ways to measure and understand the distance between the selfadjoint ideal and the true behavior of non-selfadjoint operators.