Aléatoire

[collapsed title="Asymptotics of Markov processes, Rennes, Brice Franke, (sem2)"] The lectures focuses on large-time asymptotics of Markov processes. Involved topics include : recurrence, transience, invariant measures, etc. [/collapse]

[collapsed title="Dynamical systems and ergodic theory, Rennes, Bachir Bekka, (sem1)"] The lectures focus on dynamical systems given by a measure-preserving transformation T on a probability space (X; B; ). 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 ! 1. Involved topics include : ergodicity, mixing, strong mixing ; ergodic theorems ; unique ergodicity ; entropy. [/collapse]

[collapsed title="Linear and nonlinear filtering & Advanced models in financial engineering, Rennes, Ensai"] [/collapse]

[collapsed title="Machine learning & deep learning, Rennes, Ensai"] [/collapse]

[collapsed title="Mean field games, Rennes, Hu Ying, (sem2)"] The Mean Field Games (MFG in short) describe the evolution in continuous time of one large number of agents interacting among them. Introduced by Lasry and Lions, the models studied in this course are linked with various problems of optimization, with partial differential equations (Hamilton-Jacobi, Fokker-Planck, etc.), with stochastic analysis (Backward Stochastic Differential Equations) or with the theory of games. [/collapse]

[collapsed title="Rough paths, Rennes, Ismaël Bailleul, (sem2)"] Many natural systems are described by Banach space-valued paths whose dynamics is described by differential equations _ yt = 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. [/collapse]

[collapsed title="Statistics of stochastic processes, Rennes, Ronan Le Guével, (sem1)"] 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. [/collapse]

[collapsed title="Statistical parametric estimation, Rennes, Bernard Delyon, sem1)"] Lectures are concerned first with classical estimation methods for parametric models, i.e. when the unknown distribution is parametrized in a finite dimensional space. Asymptotic normality of estimators for dependent observations is studied. Estimator comparisons and optimality are discussed. [/collapse]

[collapsed title="Statistical non-parametric estimation, Rennes, Adrien Saumard, (sem1)"] 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. [/collapse]

[collapsed title="Stochastic calculus, Rennes, Yürgen Angst, (sem1)"] 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. [/collapse]

[collapsed title="Markov process, Nantes, Nicolas Petrelis, (sem1)"] This course will be mainly dedicated to Markov chains and continuous time Markov chains. Along the way we will also study random Poisson measures. These are very important mathematical tools to model stochastic phenomenona. In particular, this course will be a good preparation for those willing to follow "Stochastic models of population dynamics" at the second semester. [/collapse]

[collapsed title="Stochastic models of population dynamics, Nantes, Philippe Carmona (sem2)"] We shall consider models for evolving populations beginning with classical models : birth and death processes, and branching processes. Then we shall consider structured population models, where the population is modeled as a finite measure on a state space that can model both continuous and discrete characteristics of individuals. The evolution of individuals is a mix of deterministic and stochastic behaviour, and the population process is then a continuous time Markov process defined on the space of finite measures. Eventually, scaling limits for large initial populations will give back some classical deterministic models of population dynamics. [/collapse]

[collapsed title="Stochastic processes, Rennes, Jean-Christophe Breton, (sem1)"] 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. [/collapse]

###Algèbre et Géométrie

[collapsed title="Algèbre homologique, Nantes, Friedrich Wagemann, (sem1)"] Nous allons traiter dans ce cours les sujets classiques de l’algebre homologique, comme illustrés dans [2], [3] ou encore [4]. Il s’agit d’abord d’introduire le langage des catégories et des foncteurs, en mettant l’accent sur les foncteurs adjoints et les foncteurs exacts a gauche ou `a droite. Ensuite viendra un chapitre plus spécifiquement sur le produit tensoriel et le bifoncteur Hom. Après cela,nous allons parler du formalisme g´en´eral des foncteurs dérivés, en illustrant la théorie sur les foncteurs Ext et Tor. L’étude du foncteur Ext se prolongera dans des rudiments de la cohomologie des groupes ainsi que la cohomologie des algèbres de Lie. Nous terminerons sur la construction de la catégorie dérivéev(comme construite dans [1]). Si le temps le permet, nous essayerons de caser une ébauche de la cohomologie des faisceaux. References [1] Gelfand, Sergei I.; Manin, Yuri I. Methods of homological algebra. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. [2] Hilton, P. J.; Stammbach, U. A course in homological algebra. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997 [3] Mac Lane, Saunders, Homology. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995 [4] Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 19941 [/collapse]

[collapsed title="Complex analysis in several complex variables, Rennes, Christophe Dupont (sem1)"] 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). [/collapse]

[collapsed title="Complex Geometry, Rennes, Benoît Claudon, (sem1)"] 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. [/collapse]

[collapsed title="Deformations of complex manifolds, Rennes, Christophe Mourougane (sem2)"] The aim is to define and study the parameter space of universal deformations of a given compact complex manifold. Along the way, we will give a large spectrum of examples of complex manifolds, define and use the notions of sheaves and their cohomology that enable to go from local constructions to global ones, define basic notions of Hermitian geometry that help going from formal to analytic geometry, and, if times permits, display on examples, degeneration phenomenons. [/collapse]

[collapsed title="Differential topology, Rennes, Juan Souto, (sem1)"] 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. [/collapse]

[collapsed title="Exemples de symétrie miroir homologique, Nantes, Paolo Ghiggini (sem2)"] Description : La symétrie miroir est un lien surprenant entre la géométrie symplectique et la géométrie algébrique découvert par des physiciens au début des années '90. La symétrie miroir homologique, proposé quelques années plus tard par M. Kontsevich, conjecture que la catégorie de Fukaya dérivée d'une variété symplectique X devrait être isomorphe à la catégorie dérivée des faisceaux cohérents d'une variété algébrique X'. Plusieurs exemples de variétés miroirs sont connus, mais la conjecture est encore largement ouverte. Dans le cours on s'intéressera à la symétrie miroir pour les variétés symplectiques les plus simples : les surfaces orientable, où la catégorie de Fukaya est combinatoire. [/collapse]

[collapsed title="Fundamentals of differential geometry, Rennes, Juan Souto, (sem1)"] [/collapse]

[collapsed title="Introduction to differential geometry, Nantes, Erwan Brugallé (sem1)"] La géométrie énumérative s'intéresse au comptage d'objets géométriques satisfaisant des contraintes géométriques données. Ce type de problème peut être très simple ("Combien de droites passent pas deux points ?"), mais peut aussi déboucher sur des questions ouvertes et difficiles. Le but de ce cours est d'étudier plusieurs problèmes énumératifs concernant les courbes dans le plan, suivant trois perspectives : en géométrie algébrique complexe, en géométrie algébrique réelle, et en géométrie tropicale. Nous verrons comment ces trois perspectives, à priori différentes, sont en fait intimement reliées. Quelques notions élémentaires de géométrie algébrique et géométrie différentielles sont souhaitables pour suivre ce cours. Aucune connaissance préalable en géométrie tropicale n'est requise. [/collapse]

[collapsed title="Géométrie semi-algébrique et o-minimale, Nantes, Nicolas Dutertre, (sem2)"] Comptage des racines d'un polynôme réel, ensembles semi-algébriques et théorème de Tarski-Seidenberg, applications semi-algébriques, inégalité de Lojasiewicz, propriétés topologiques des semi-algébriques et théorème de Hardt, introduction aux structures o-minimales. [/collapse]

[collapsed title="Intersection theory in algebraic geometry I, Rennes, Julien Sebag, (sem1)"] 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. [/collapse]

[collapsed title="Intersection theory in algebraic geometry II, Rennes, Florian Ivorra (sem1)"] This second set of lectures study the geometric point of view on the intersection theory in algebraic geometry. [/collapse]

[collapsed title="Introduction à la géométrie symplectique, Nantes, Baptiste Chantraine (sem1)"] Objectif: l'objectif du cours est de donner une introduction à la géométrie symplectique. On introduira les équations et transformations canoniques d'un espace de phase en montrant que la forme symplectique est invariante sous le flot de telles équations. On en déduira la définition de variété symplectique et parlera de leurs transformations (Hamiltoniennes et symplectiques). On parlera de sous-variétés lagrangiennes et de leur ubiquité (au sein de la géométrie symplectique et à l'extérieur de celle-ci). La cours sera illustré par de nombreux exemples et constructions, on se concentrera entre autre sur la construction de réduction symplectique. On parlera du morphisme du flux sur le groupe des symplectomorphismes. Si le temps le permet on abordera des questions plus avancées sur les intersections lagrangiennes, la topologie du groupe des difféomorphismes Hamiltoniens et sa simplicité. La fin de cours parlera de certaines conjectures motrices du sujet encore ouvertes. [/collapse]

[collapsed title="K-Théorie, Nantes, Hossein Abbaspour, (sem2)"] K-theory is in some sense a meeting ground for several other mathematical subjects, including number theory, geometric topology, algebraic geometry, algebraic topology and operator algebras, relating to constructions like the ideal class group, Whitehead torsion, coherent sheaves, vector bundles and index theory. The main objective of the course is the construction of characteristics classes (as functor) in various settings with values in homology/cohomology theories such as De Rham, Hochschild and cyclic homology,... and eventually comparing them.

Topics that can be included:Topological K-Theory, Algebraic K-Theory, Higher K-theory, Hochschild and cyclic homology, Chern Character (with value in De Rham cohomology, Hochschild /cyclic homology,..) , HKR Theorem and a few classical results from noncommutative geometry, Euler class and Riemann-Roch-Hirzebruch type theorem for DG-Algebras, (bi)simplicial spaces and the realization lemma and application for Mapping spaces, Sheaves, Hochschild homology for O_x-modules, Todd class and Riemann-Roch type theorems. —————— References: Loday, Jean-Louis Cyclic homology.. . Springer-Verlag, Berlin, 1992.
Karoubi, Max K-theory. An introduction. Springer-Verlag, Berlin-New York, 1978. xviii+308 pp
Damien Calaque -Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry


La K-théorie est un lieu de rencontre pour plusieurs autres matières des mathématiques, notamment la théorie des nombres, la topologie géométrique, la géométrie algébrique la topologie algébrique et les algèbres d’opérateurs concernant des constructions telles que le groupe des classes idéaux, la torsion de Whitehead, les faisceaux cohérents, les fibres vectoriels, classes caractéristiques et la théorie de l’indices. L’objectif principal du cours est la construction des classes caractéristiques (comme foncteur), dans divers contextes, à valeurs dans les théories (co)homologique telles que De Rham,Homologie de Hochschild et cyclique, cohomologie de faisceaux etc.

Les sujets qui pourraient être abordés: K-théorie topologique, K-théorie algébrique, groupes supérieurs de K-théorie, homologie d'Hochschild et cyclique , caractère de Chern (à valeur dans le cohomologie de De Rham et homologie de Hochschild..) , Théorème de HKR et quelques résultats classiques en géométrie non-commutative; Périodicité de Bott , Classe d'Euler et Théorème de type Riemann-Roch-Hirzebruch pour DG-Algèbre, Faisceaux et cohomologie de faisceaux, homologie de Hochschild pour O_x-modules, Classe de Todd et Théorème de Riemann-Roch. [/collapse]

[collapsed title="Pseudo-Riemannian geometry & calculus of variations, Rennes, Éric Loubeau (sem2)"] The lectures provide a common framework to Riemannian and Lorentzian geometries by studying classical notions of semi Riemannian geometry : semi-Riemannian metrics, parallel transport, connexion, curvature, Killing fields, geodesics, etc. In particular the course introduces the variational form of the geodesic problem. In the end it discusses harmonic maps on Riemannian manifolds. [/collapse]

[collapsed title="Real algebraic geometry, Rennes, Goulwen Fichou, (sem 2)"] 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. [/collapse]

###Analyse

[collapsed title="Elements of spectral theory, Nantes, Nicolas Raymond, (sem1)"] 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 bornés (fermés, 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, à 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 sûrement pas), on ébauchera une présentation du calcul fonctionnel des opérateurs auto-adjoints et de la fameuse mesure spectrale. [/collapse]

[collapsed title="Equation de Schrödinger semi-classique, Gabriel Riviere (sem2)"] La limite semi-classique de la mécanique quantique est un régime dans lequel la constante de Planck est négligeable devant les autres actions physiques mises en jeu dans le système. Dans cette limite, le système physique considéré est gouverné par les les équations de la mécanique classique. D'un point de vue mathématique, ce type d'asymptotiques est au cœur de l'analyse microlocale (ou semi-classique) dont le champ d'applications est extrêmement varié: équations aux dérivées partielles, théorie spectrale, topologie symplectique, géométrie aléatoire, systèmes dynamiques hyperboliques, etc.

L'objectif de ce cours sera d'introduire ces outils d'analyse semi-classique et de les illustrer à travers l'étude de la dynamique et du spectre de l'équation de Schrödinger sur le tore. Le cours débutera par des rappels d'analyse de Fourier (périodique et non périodique).

Références.

  • M. Ruzhansky, V. Turunen. Pseudodifferential operators and symmetries, Birkhäuser Verlag, Basel Boston Berlin (2010)
  • M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. [/collapse]

[collapsed title="Finite element method, Rennes, Éric Darrigrand & Nicolas Seguin, (sem1)"] This lecture is a numerical counterpart to Sobolev spaces & 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 one and two dimensions. The second part of the lectures consists in defining a generic strategy for the implementation of the method based on the variational formulation. A program is written in Matlab (implementable with Matlab or Octave). [/collapse]

[collapsed title="Fourier Analysis and PDEs, Nantes, Nicolas Depauw (sem1)"] [/collapse]

[collapsed title="Hyperbolic equations, Rennes, Miguel Rodrigues, (sem1)"] This course takes Sobolev spaces & elliptic equations as prerequisite. It will first concern entropic solutions of scalar conservation laws and then will focus on strong solutions of quasi-linear hyberbolic systems. [/collapse]

[collapsed title="Introduction to Harmonic Analysis, Nantes, Cristina Benea, (sem2)"]

  • Dyadic analysis/geometry of dyadic sets ; distribution function, (weak) Lp spaces and interpolation
  • Hardy-Littlewood maximal function and Calderon-Zygmund decomposition ; Calderon-Zygmund operators
  • Marcinkiewicz multipliers ; square functions ; some useful inequalities
  • BMO spaces ; good lambda inequalities in harmonic analysis
  • Martingales in (harmonic) analysis ; another look at Doob and Burkholder-Gundy inequality [/collapse]

[collapsed title="Microlocal analysis, Rennes, Zied Ammari, (sem1)"] 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. [/collapse]

[collapsed title="Modelling and mathematical analysis of shallow free-surface fluid flows, Rennes, Vincent Duchêne, (sem2)"] During the lectures, we will discuss modeling techniques inspired by coastal oceanography problems. Specifically, we will explain how one can rigorously justify asympotic models such as Saint-Venant, Green-Naghdi and friends starting from the free-surface Euler equations in the shallow water regime. To this aim, we will employ and adapt tools for elliptic and hyperbolic equations that will have been covered during the first semester. In the way, we will introduce a few keywords such as Hamiltonian equations, dispersive equations... [/collapse]

[collapsed title="Numerics of transport, Rennes, Benjamin Boutin & Mohammed Lemou, (sem1)"] This lecture is a numerical counterpart to Hyperbolic equations. The first part is devoted to the construction and the analysis of finite volume methods for scalar conservation laws. Extensions to systems of conservation laws and to multidimensional cases is also presented. In the second part, semi-Lagrangian methods and particle methods for linear transport equations are investigated, focusing on the Vlasov equation for the kinetic representation of plasma dynamics. [/collapse]

[collapsed title="Optimal control with uncertainty & Hamilton-Jacobi equations, Rennes, Marc Quincampoix, (sem2)"] In these lectures, we will discuss optimal control problems together with the associated Hamilton-Jacobi Equations. We will also study how to cope with uncertainty on initial data of the system, and see why optimal transport and the notion of viscosity solutions provide well adpated tools to investigate these problems. [/collapse]

[collapsed title="Sobolev spaces & elliptic equations, Rennes, Roger Lewandowski, (sem1)"] The first part of the course is focused on Sobolev spaces. It studies embedding theorems on rather general classes of open sets, fractionals spaces and traces. The second part concerns elliptic partial differential equations, beginning with the linear cas, with different kinds of boundary conditions. In the end, some tools for nonlinear elliptic equations will be introduced (Galerkin approximation, fixe-point framework, etc.). [/collapse]

[collapsed title="Spectral theory, Rennes, Christophe Cheverry, (sem1)"] We intend to present in this course the basic tools in spectral analysis and to illustrate the theory with examples from various branches of physics. We also provide some general introduction to (unbounded) operator theory. [/collapse]

[collapsed title="Theoretical and numerical averaging techniques for highly oscillatory equations, Rennes, Florian Méhats, (sem2)"] 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, which enable to write the solution of the problem under a suitable form : after a periodic change of variable, it is the solution of an averaged equation, non stiff in 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 methods well-adapted for such problems. These methods are called uniformly accurate, since their numerical convergence is uniform with respect to the small parameter in the equation. [/collapse]