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 will concentrate on how parametric methods can be used in semi-parametric models,a motivation 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.

A stochastic process is a phenomenon which evolves through time with a random trajectory, and as such, is represented as a family of random variables indexed by time. In this course, we propose to study the various implications brought by such random object, and hence introduce the concepts of the stochastic process and the stochastic integration. We will first construct and review some basic facts about the Brownian motion, including its Markov property, as it represents a major stochastic process example. Next, we will introduce the concept of the continuous semi-martingale in order to finally construct the stochastic integral with respect to a continuous semi-martingale..

Many natural systems are described by Banach space-valued paths whose dynamics is described by differential equations y˙t = f(yt)ht. Such an equation provides a description of the infinitesimal increment of the path y in terms of the infinitesimal increment of a driving signal h. Usual ordinary differential equations correspond to ht = t, but numerous situations require that we consider non-smooth signal that may not be differentiable, like in stochastic differential equations, where h is a Brownian trajectory. What is then the meaning of the above equation ? Rough path theory provides an optimal setting for such question, and has applications to controlled deterministic and stochastic systems, noisy PDEs or machine learning ! Two thirds of the lectures will deal with the purely analytic side of the story, so the students from the Analysis pathway are most welcome to attend the course.

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.

We shall introduce the main concepts of random Gaussian fields and review classical related inequalities such that Slepian, Sudakov, Fernique, isoperimetric... Then, we will study criteria of smoothness of theses processes as well as the distribution of the maximum on a compact set, the number of local extrema, the number of roots by using the Kac-Rice formulas or else Wiener chaos expansion. Several applications will be discussed such that random trigonometric polynomials, algebraic polynomials ("Kac", "Elliptic", "Flat"...)

The main objective of this course consists in introducing stochastic mean field interacting particles systems as well as their asymptotic counterpart known as McKean-Vlasov stochastic systems. These systems describe, respectively, the random evolution of a large number of particles/agents which interact with each other through the empirical measure of the system ; and the dynamic of one of these particles in the asymptotic (with respect to the number of agents) regime. This relies on the theory of Stochastic Dierential Equations which will play a central role along the lectures. If time allows, connections with (non-linear) PDEs will be discussed.

Webmining: This course is an introduction to webmining and natural language processing.

Functional Data Analysis: In this course students learn the main ideas, the related theory and the numerical routines to perform function data analysis.

This course will be given in French, but questions and discussions in English or other languages are welcome, and additional material and references in English will be given, office hours in English could be proposed. The course (I, II) will focus on technics from arithmetic, harmonic analysis, and p-adic analysis, applied to group theory and dynamical systems. The main applications will concern — the dynamics of polynomial transformations, and phenomena of unlikely intersections — linear groups and the Tits alternative ; — the distribution and the arithmetic growth of orbits of algebraic transformations (Tate height, extensions of the Skolem-Mahler-Lech theorem, etc).

After recalling basic facts on manifolds and coverings, we will introduce the notion of complex manifold and holomorphic functions on it. Riemann surfaces are connected complex manifolds of dimension 1 (real surfaces with a complex structure). We will start with examples : - the Riemann sphere and its automorphisms, - complex torus, automorphisms and classification, - algebraic curves defined over complex numbers. We will detail the case of elliptic curves, and their group law. We will state (without proving it) Uniformization Theorem of Poincare-Koebe : the universal cover of a Riemann surface is biholomorphic to either the complex disc, the complex plane or the Riemann sphère. We will discuss consequences. We will introduce the notions of divisors, line bundles and Picard group. We will introduce the notion of sheaves, Cech cohomology, and arrive to the Riemann-Roch Theorem. Finally, we will end, if we have time, by Abel-Jacobi map.

This series of lectures is meant as the continuation of the course "Riemann surfaces". The main purpose is to provide a complete proof of Kodaira’s embedding theorem that characterizes smooth projective varieties among the complex compact manifolds in terms of the existence of certain line bundles. To achieve this goal, the following points shall be explained : -complex differential calculus and Dolbeault cohomology of holomorphic vector bundles -hermitian and Kähler metrics on complex manifolds -notion of connection on a vector bundle, Chern connection of a hermitian line bundle -Laplace operators and Hodge theory -ample/positive line bundles and vanishing theorems

Keywords : Definition of Lie groups and Lie algebras, classical examples. Representations of sl(2,C) and sl(3,C). Solvable, Nilpotent, semi-simple Lie algebras. Cartan subalgebras. Keywords : Root systems, Weyl group, Dynkin diagramm. Relation between real and complex Lie algebras. Cartan decomposition. Relation between Lie groups and Lie algebras. More on representations. Nilmanifolds.

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

An affne algebraic variety is the zero-set of a polynomial in several variables with complex coeffcients. For instance, any parabola is an affne variety, any line, and any graph of a polynomial. On the other side, we can look at the ring of regular functions defined on an affine variety. It is in fact the quotient ring of a polynomial ring modulo the polynomials defining the affine variety. The geometry of an affne variety and the properties of its ring of regular functions are closely related. This is where geometry and algebra meet. This course in an introduction to affine algebraic geometry, perquisites are some basics in commutative algebra (such as rings and quotients of rings).

The goal of the class is to introduce the first tools of algrebraic topology. We will first study coverings of topological spaces and their transformations. That will allows us to define the fundamental group. We will devellop tools to compute it for instance van Kampen theorem. We will then introduce singular homology of spaces and study its functorial aspects. That will allow us to study some basic aspects of homological algebra. In the end we will define cellular homology and see how to compute it. All classes will be illustrated with lots of explicit examples.

This course is an introduction to Galois theory of differential equation from geometrical perspectives using the notion of foliation. 1- Cauchy theorem at a regular point. Notion of foliation and Frobenius theorem 2- Differential Galois theory - from differential field extension point of view - from principal connection point of view as reduction of the structural group 3- Linear differential equation of a Riemann surface - at regular singular point (and at irregular ones) - monodromy representation (and Stokes phenomenom) - Schelssinger density theorem ( and Ramis’s stronger version) 4- Examples of second order differential equations - hypergeometric equations - Legendre family - Projective structure on a complex algebraic curve

This is a sequel to Fundamentals of differential geometry Hyperbolic dynamical systems are the simplest instance (and the one which is best understood) of chaotic dynamical systems : they are maps of a compact space that, at each point, expand uniformly a direction and contract uniformly a completementary direction. The study of these systems combines arguments originating in different areas of mathematics (mainly geometry, but also analysis, probability theory, algebra, combinatorics). We will cover the fundamental properties of these systems (existence of stable and unstable manifolds, structural stability, gluing orbits, Markov partitions) before considering more advanced theorems : classification up to homeomorphism of Anosov diffeomorphisms, on tori and more generally on nilmanifolds ; absolute continuity of foliations, ergodicity ; growth of the number of periodic orbits (Margulis theorem).

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 study branched covers of smooth manifolds of dimensions 3 and 4. This naturally leads to the study of knots and links in 3-manifolds, of 4-manifolds with boundary, and of surfaces in 4-manifolds. We will then prove the G-signature theorem in dimension 4. We will discuss applications to smooth 4-manifolds topology, in particular to embeddings of (possibly singular) surfaces in 4-manifolds. 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 (smooth manifolds, smooth maps, applications of transversality) and algebraic topology (fundamental group, some homology/cohomology), at the level of the first-semester courses and student seminars, is required ; the more low-dimensional notions will be introduced along the way.

The study of Riemann surfaces is a subject at the crossroads of geometry, algebra, group theory, topology, complex analysis and analysis on manifolds. The objective of this course is therefore to introduce several geometric, algebraic and analytical tools to understand several aspects of Riemann surfaces. Riemann surfaces can be defined as complex varieties of dimension 1, so they are topological spaces which are locally homeomorphic to an open of C and we have on them notions of holomorphic and meromorphic functions. Historically, the definition of Riemann surfaces emerged to give a geometrical and topological meaning to multivalued holomorphic functions like √ z, log z. A second equivalent denition is that of a real manifold of dimension 2 (hence the name of surface) where each tangent space is endowed with a rotation of angle +π/2 or a way to measure oriented angles. We will study for example the meromorphic functions and the links of the study of the meromorphic functions with the classification of complex lines bundle, with the topology of the surface and this study will lead to the Riemann-Roch theorem, it is a formula for the dimension of the space of the meromorphic functions whose poles are prescribed. This theorem allows a classication of certain surfaces (of genus 0and 1). The uniformisation theorem for Riemann surfaces is a major result that have proven at the beginning of the XX's century by Koebe and Poincaré. This theorem allows to geometrize Riemann surfaces and to classify compact Riemann surfaces. We will try to apprehend some facets of this fascinating theorem with, according to the development of the course, approaches via complex analysis, or potential theory (study of harmonic functions) or global analysis by studying the Gauss curvature. Bibliographic references : - Ahlfors, L., Complex Analysis, McGraw Hill, 1966. - Farkas, H., and Kra, I., Riemann surfaces, Springer, 1980. - Forster, O. : Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol.81, Springer-Verlag, Berlin,1981. - Griffiths, P., and Harris, J., Algebraic geometry, Wiley-Interscience, 1978. - Jost J., Compact Riemann Surfaces, An Introduction to Contemporary Mathematics, Universitext, Springer, 2002. - de Saint-Gervais H.-P., Uniformisation des surfaces de Riemann, Retour sur un théorème centenaire, ENS Édition, 2011. (An english translation is also available)

This is an introductory course to Riemannian Geometry with a specialization towards its complex version, the Kähler geometry. We are going to cover the following classic material. Riemannian manifolds, the Levi-Civita connection, geodesics. Examples of riemannian manifolds. The exponential map, Hopf-Rinow theorem and Gauss Lemma. Parallel transport, riemannian holonomy, the deRham decomposition theorem. The Riemannian curvature tensor, sectional curvature, Ricci curvature, scalar curvature. Jacobi fields, the Bonnet-Myers, Synge and Cartan-Hardamard theorems. Uniformization of space forms. Killing vector fields and Bochner's Theorem. Hodge-DeRham theory. Introduction to Kähler geometry : The complex projective space and the Fubini-Study metric. Kähler metrics, Ricci form and the complex Monge-Ampère PDE's. Calabi-Yau manifolds and Kähler-Einstein metrics. Scalar curvature and extremal Kähler metrics. Examples : the Hirzebruch complex surfaces.

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.

Several classical partial differential equations (Laplace, Poisson, heat, wave and Schrodinger equations) will be studied from the perspective of Fourier analysis. For this, we will need to introduce the formalism of spaces of distributions (or generalized functions), the associated operations and transformations (among which, the Fourier transform), and Sobolev spaces. With this set of tools, we can address questions regarding the existence and the uniqueness of solutions for the equations mentioned above.

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.

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.

In this course, we will study kinetic equations and their numerical approximation. These microscopic equations enable to describe large particles systems and are used in a number of applications like plasma physics, biology, astrophysics,... First, we will focus on modelling aspects. Several properties of kinetic equations will be highlighted and using asymptotic expansions (with respect to a small parameters in the equation), macroscopic models will be derived. Second, some numerical methods dedicated to the numerical approximation of kinetic equations will be designed and analyzed. In particular, the so-called multi-scale methods, which make the link at the discrete level between microscopic and macroscopic descriptions will be presented.

First, we give that we will be insisting on the modeling aspects leading to structured equations in the biological context. We give some classical results in the theory of Volterra integral equations. We then present the renewal equation and the generalized entropy method. In a last chapter we introduce adhesion models in the framework of cell motility and show some related mathematical results.

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.