{"id":40,"date":"2023-09-20T16:20:41","date_gmt":"2023-09-20T14:20:41","guid":{"rendered":"https:\/\/blog.u-bourgogne.fr\/master-pmg\/?p=40"},"modified":"2025-09-05T13:01:42","modified_gmt":"2025-09-05T11:01:42","slug":"cours-en-m1","status":"publish","type":"post","link":"https:\/\/blog.ube.fr\/master-pmg\/2023\/09\/20\/cours-en-m1\/","title":{"rendered":"Cours en M1"},"content":{"rendered":"

Programme de la 1\u00e8re ann\u00e9e de Master en math\u00e9matiques fondamentales (PMG)
\n<\/strong><\/h1>\n

Master in fundamental Mathematics (PMG) 1st year – Program<\/em><\/p>\n

\u00a0<\/strong><\/p>\n

Cours du semestre 1 (septembre – janvier)<\/strong><\/h3>\n

Courses of Semester 1 (September – January)<\/em><\/p>\n

\u00a0<\/strong><\/p>\n

    \n
  • MG1-1, Alg\u00e8bre 1 (Cours obligatoire – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

    \u00a0\u00a0\u00a0\u00a0 MG1-1, Algebra 1 (Compulsory course – 6 ECTS)<\/em><\/p>\n

    Responsables : Daniele Faenzi (CM), Mattia Cavicchi (TD)<\/strong><\/p>\n

    Dans ce cours, on \u00e9tudie les anneaux commutatifs, notamment les anneaux de polyn\u00f4mes en plusieurs variables, les extensions de corps, les modules sur un anneau et la th\u00e9orie de Galois. Les notions de base de la th\u00e9orie des anneaux, telles que les id\u00e9aux, les quotients, les \u00e9l\u00e9ments irr\u00e9ductibles ; puis concernant les anneaux euclidiens, factoriels et principaux, sont revues rapidement. On \u00e9tudie les polyn\u00f4mes sym\u00e9triques, le r\u00e9sultant de deux polyn\u00f4mes, le discriminant d’un polyn\u00f4me. On \u00e9tudiera en d\u00e9tail les extensions de corps (\u00e9l\u00e9ments alg\u00e9briques et transcendants, polyn\u00f4mes minimaux, corps de rupture, corps de d\u00e9composition). On apprendra la classification des corps finis.<\/p>\n

    On pourra \u00e9tudier ensuite la notion de module sur un anneau, qui g\u00e9n\u00e9ralise le concept d’espace vectoriel. En alternative, on \u00e9tudiera quelques \u00e9l\u00e9ments de la th\u00e9orie de Galois reliant les extensions de corps \u00e0 la th\u00e9orie des groupes.<\/p>\n

    This course is about commutative rings, polynomial rings in several variables, field extensions, modules over a commutative ring and Galois theory. We will start with a review of basic notions of ring theory, such as ideals, ring quotients, irreducible elements; then we will have a quick overview of Euclidean rings, principal ideal domains and unique factorisation domains. We will then study symmetric polynomials, resultants and discriminants. We will study in detail field extensions, starting with the notions of algebraic and transcendental elements, the minimal polynomial, splitting fields. We will learn the classification of finite fields. <\/em><\/p>\n

    We will then choose a further topic. We will either study the notion of modules over a commutative ring (a generalisation of the concept of vector space over a field). Otherwise, we will analyze the basic aspects of the Galois theory of equations, relating field extensions to group theory.
    \n<\/em><\/p>\n

    R\u00e9f\u00e9rences :<\/strong><\/p>\n

    Gilles Bailly-Ma\u00eetre, Philippe du Bois, Lionel Ducos, Henri Lombardi, Alg\u00e8bre. Tome 3, anneaux, polyn\u00f4mes, modules,<\/em> Enseignement des math\u00e9matiques, Cassini,
    \nParis, 2023, Sous la direction d\u2019Aviva Szpirglas<\/p>\n

    Jean-Pierre Escofier, Th\u00e9orie de Galois,<\/em> seconde \u00e9dition, Sciences Sup, Dunod, Paris, 2000.<\/p>\n

    Felix Ulmer, Anneaux, corps, r\u00e9sultants,<\/em> Alg\u00e8bre pour L3\/M1\/agr\u00e9gation, Ellipses, Paris, 2018<\/p>\n

    Daniel Perrin, Cours d\u2019alg\u00e8bre,<\/em> Collection de l\u2019\u00c9cole Normale Sup\u00e9rieure de Jeunes Filles, vol. 18, Paris, 1982<\/p>\n

     <\/p>\n

     <\/p>\n

      \n
    • MG1-2, Analyse 1 (Cours obligatoire – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

      \u00a0\u00a0\u00a0\u00a0 MG1-2, Analysis 1 (Compulsory course – 6 ECTS)<\/em><\/p>\n

      Responsables : Abderrahim Jourani (CM), Giuseppe Dito (TD)<\/strong><\/p>\n

      Les th\u00e8mes abord\u00e9s dans cette unit\u00e9 d\u2019enseignement sont les suivants : rappels des th\u00e9or\u00e8mes fondamentaux de Licence (les th\u00e9or\u00e8mes de Baire, de l\u2019application ouverte, du graphe ferm\u00e9, de Banach-Steinhaus, d\u2019Ascoli et de Stone-Weierstrass, espaces L^p, entre autres), transform\u00e9e de Fourier, th\u00e9orie sp\u00e9ctrale et dualit\u00e9 des espaces L^p, dualit\u00e9 dans l\u2019espace des fonctions continues, th\u00e9or\u00e8mes de Hahn-Banach, applications.<\/p>\n

      The topics covered in this cours are as follows : review of the fundamental theorems seen in the third year of the Bachelor\u2019s degree (theorems of Baire, of closed graph, of Banach-<\/em>
      \nSteinhaus, of Ascoli and of Stone-Weierstrass, Lp-spaces, among others) Fourier transform, spectral theory and duality of Lp-spaces, duality in the space of continuous functions, Hahn-<\/em>
      \nBanach theorems, applications.<\/em><\/p>\n

      R\u00e9f\u00e9rences :<\/strong><\/p>\n

      Ha\u00efm Br\u00e9zis, Analyse fonctionnelle<\/em>, Th\u00e9orie et applications,<\/em> Masson, 1983<\/p>\n

      Walter Rudin, Analyse r\u00e9elle et complexe<\/em>, Masson, 1978.<\/p>\n

      Francis Hirsch et Gilles Lacombe, \u00c9l\u00e9ments d\u2019analyse fonctionnelle<\/em>, Dunod, 2009.<\/p>\n

      Hakim Boumaza, Benjamin Collas, St\u00e9phane Collion, Marie Dellinger, Zo\u00e9 Faget, Laurent Lazzarini, Florent Schaffhauser (sous la direction de Jean-Pierre Marco), Math\u00e9matiques L3 \u2014 analyse, <\/em>cours complet avec 600 tests et exercices corrig\u00e9s, Pearson Education, 2009<\/p>\n

      Jean-Michel Bony, Cours d’analyse : Th\u00e9orie des distributions et analyse de Fourier<\/em>, Les \u00e9ditions de l’Ecole polytechnique, 2001<\/p>\n

       <\/p>\n

       <\/p>\n

        \n
      • MG1-3, Analyse Complexe (Cours obligatoire – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

        \u00a0\u00a0\u00a0\u00a0 MG1-3, Complex Analysis (Compulsory course – 6 ECTS) Fonctions holomorphes et analytiques.<\/em><\/p>\n

        Responsables : Gw\u00e9na\u00ebl Massuyeau (CM) et Olivier Couture (TD)<\/strong><\/p>\n

        Ce cours est une introduction \u00e0 l\u2019analyse complexe d\u2019une seule variable, et il traitera des sujets suivants : rappels sur le plan complexe ; sph\u00e8re de Riemann ; fonctions holomorphes ; fonctions analytiques ; propri\u00e9t\u00e9s remarquables des fonctions holomorphes\/analytiques (dont formule int\u00e9grale de Cauchy) ; groupes de biholomorphismes ; \u00e9l\u00e9ments d\u2019homotopie ; \u00e9tude de singularit\u00e9s de fonctions holomorphes (dont th\u00e9or\u00e8me des r\u00e9sidus).<\/p>\n

        This course is an introduction to complex analysis of a single variable, and it will cover the following topics: review of the complex plane; Riemann sphere; holomorphic functions; analytic functions; fundamental properties of holomorphic\/analytic functions (including Cauchy\u2019s integral formula); groups of biholomorphisms; elements of homotopy; study of singularities of holomorphic functions (including the residue theorem).<\/span><\/em><\/p>\n

        R\u00e9f\u00e9rences :<\/strong><\/p>\n

        Mich\u00e8le Audin, Analyse complexe<\/em>. Universit\u00e9 de Strasbourg, 2011.<\/p>\n

        Henri Cartan, Reiji<\/span><\/span>\u00a0Takahashi, <\/span><\/span>Th\u00e9orie \u00e9l\u00e9mentaire des fonctions analytiques d’une ou plusieurs variables, <\/em>Hermann 1961<\/p>\n

        Pierre Vogel, Fonctions analytiques, <\/em>math\u00e9matiques pour la licence : cours et exercices avec solutions, Dunod 1999<\/p>\n

         <\/p>\n

         <\/p>\n

          \n
        • MG1-4, G\u00e9om\u00e9trie (Option – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

          \u00a0\u00a0\u00a0\u00a0 MG1-4, Geometry (Option – 6 ECTS)<\/em><\/p>\n

          Responsables : Johannes Nagel (CM), Maxime Fairon (TD)<\/strong><\/p>\n

          G\u00e9om\u00e9trie affine : G\u00e9om\u00e9trie affine sur un corps, notamment sur R. Barycentres, convexit\u00e9 ; groupe des transformations affines, points fixes ; quelques th\u00e9or\u00e8mes classiques de g\u00e9om\u00e9trie affine. G\u00e9om\u00e9trie euclidienne : Isom\u00e9tries, leurs d\u00e9compositions en r\u00e9flexions. Groupe orthogonal, compacit\u00e9, connexit\u00e9, simplicit\u00e9. Quaternions. D\u00e9composition polaire. Groupe euclidien affine. G\u00e9om\u00e9trie projective : Espaces projectifs, ouverts affines, compl\u00e9t\u00e9 projectif d’un espace affine, dualit\u00e9 projective, homographies, rep\u00e8res projectifs, birapport, groupe des homographies, points fixes. Groupe lin\u00e9aire : Groupes lin\u00e9aires GLn, PGLn, SLn, PSLn, d\u00e9nombrement sur des corps finis, g\u00e9n\u00e9rateurs du groupe lin\u00e9aire, transvections, simplicit\u00e9 de PSLn, isomorphismes remarquables. Quadriques : formes quadratiques, \u00e9l\u00e9ments de classification (corps alg\u00e9briquement clos, nombres r\u00e9els, corps finis), quadriques projectives (quadriques lisses, hyperplan tangent, polarit\u00e9), quadriques affines et euclidienne (classification en petite dimension). Groupe orthogonal g\u00e9n\u00e9ra : bases hyperboliques, espaces isotropes , th\u00e9or\u00e8me de Cartan-Dieudonn\u00e9, th\u00e9or\u00e8me de Witt.<\/p>\n

          Affine geometry : Affine geometry over a field, especially over R<\/strong>, barycenters, convexity, affine transformation group, fixed points, classical theorems of affine geometry. Euclidean geometry : Isometries, decomposition into reflections, orthogonal group, compactness and connectedness, simplicity, quaternions, polar decomposition, affine euclidean group. Projective geometry : Projective spaces, affine charts, projective completion of affine spaces, projective duality, homographies, projective frames, cross-ratio, group of homographies, fixed points. Linear group : GLn, PGLn, SLn, PSLn, counting on finite fields, generators of the linear group, transvections, simplicity of PSLn, remarkable isomorphisms. Quadrics : quadratic forms, classification over algebraically closed fields, real numbers, finite fields, projective quadrics (smooth quadrics, tangent hyperplane, polarity), affine and euclidean quadrics (classification in low dimension). General orthogonal group : hyperbolic bases, isotropic subspaces, Cartan-Dieudonn\u00e9 Theorem, Witt Theorem.<\/em><\/p>\n

          R\u00e9f\u00e9rences :<\/strong><\/p>\n

          Mich\u00e8le Audin, G\u00e9om\u00e9trie, <\/em>EDP Sciences, 2006<\/p>\n

          \u00a0<\/strong><\/em><\/p>\n

           <\/p>\n

            \n
          • MIGS1-2, Probabilit\u00e9s (Option – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

            \u00a0\u00a0\u00a0\u00a0 MIGS1-2, Probability (Option – 6 ECTS)<\/em><\/p>\n

            Responsables\u00a0: Yoann Offret (CM), Samuel Herrmann (TD)<\/strong><\/p>\n

            La premi\u00e8re moiti\u00e9 du cours concerne la caract\u00e9risation des lois de probabilit\u00e9 et la caract\u00e9risation de la convergence en loi. On traitera notamment les caract\u00e9risations par la fonction de r\u00e9partition, la fonction caract\u00e9ristique ou plus g\u00e9n\u00e9ralement diverses caract\u00e9risations fonctionnelles. On illustrera ces r\u00e9sultats en d\u00e9montrant le th\u00e9or\u00e8me de L\u00e9vy que l’on appliquera pour d\u00e9montrer le th\u00e9or\u00e8me central limite. Ceci sera \u00e9galement l’occasion d’aborder la notion de vecteurs Gaussiens. La seconde moiti\u00e9 du cours porte sur la notion fondamentale d’esp\u00e9rance et de lois conditionnelles, notions \u00e9l\u00e9mentaires dans le cadre discret qui s’av\u00e8rent bien plus th\u00e9oriques dans le cadre g\u00e9n\u00e9ral. Ces deux notions sont des pr\u00e9requis essentiels pour introduire les notions de martingales et de cha\u00eenes de Markov du cours d’Algorithmes Stochastiques au second semestre.<\/p>\n

            To begin with, the course deals with the characterization of probability distributions and their convergence. For instance with the distribution function, the characteristic function or more generally with various other functional characterizations. We will illustrate these results by demonstrating L\u00e9vy’s theorem and the central limit theorem. This will be the opportunity to study Gaussian vectors. The second half of the course deals with the fundamental notions of conditional expectation and transition kernels. Those are elementary in the discrete framework which turn out to be much more theoretical in the general one. These two notions are essential prerequisites to introduce the notions of martingales and Markov chains of the Stochastic Algorithms course in the second semester.<\/em><\/p>\n

            R\u00e9f\u00e9rences<\/strong><\/p>\n

            Girardin, Val\u00e9rie, Limnios, Nikolaos, Probabilit\u00e9s : en vue des applications : cours & exercices,<\/em> licence & ma\u00eetrise de math\u00e9matiques, \u00e9coles d’ing\u00e9nieurs, Vuibert, 2001<\/p>\n

            Foata, Dominique & Fuchs, Aim\u00e9, Processus stochastiques : processus de Poisson, cha\u00eenes de Markov et martingales<\/em> : cours et exercices corrig\u00e9s, Dunod, 2004<\/p>\n

            Norris, James, Markov chains<\/em>, Cambridge University Press, 1997<\/p>\n

            <\/h3>\n

            <\/h3>\n

            Cours du semestre 2 (janvier – mai)<\/strong><\/h3>\n

            Courses of Semester 2 (January – May)<\/em><\/p>\n

            \u00a0<\/strong><\/p>\n

              \n
            • MG2-1, Alg\u00e8bre 2 (Cours obligatoire – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

              \u00a0\u00a0\u00a0\u00a0 MG2-1, Algebra 2 (Compulsory course – 6 ECTS)<\/em><\/p>\n

              Responsables\u00a0: Luis Paris (CM), Renaud Detcherry (TD)<\/strong><\/p>\n

              Rappels sur les groupes et exemples : groupes di\u00e9draux, groupes sym\u00e9triques, groupes lin\u00e9aires (sur des corps finis), produit semi-direct, groupes de permutations sign\u00e9es. Produit tensoriel d\u2019espaces vectoriels : propri\u00e9t\u00e9 universelle, bases, produit tensoriel d\u2019applications lin\u00e9aires. Repr\u00e9sentations lin\u00e9aires des groupes finis : d\u00e9finition, exemples, repr\u00e9sentation r\u00e9guli\u00e8re, sous-rep\u00e9sentation, d\u00e9composition en somme directe de repr\u00e9sentations irr\u00e9ductibles, produit tensoriel, carr\u00e9 sym\u00e9trique, carr\u00e9 altern\u00e9. Th\u00e9orie des caract\u00e8res : D\u00e9finition, caract\u00e8re d\u2019une somme, caract\u00e8re d\u2019un produit tensoriel, Lemme de Schur, orthonormalit\u00e9 des caract\u00e8res simples, d\u00e9composition de la repr\u00e9sentation r\u00e9guli\u00e8re. Groupes et g\u00e9om\u00e9trie : Espace affine euclidien, groupe des isom\u00e9tries, coordonn\u00e9es barycentriques, groupes des isom\u00e9trie d\u2019un solide, cas des polygones r\u00e9guliers, des simplexes et des ncubes. Sous-groupes de SO(3) : classification des poly\u00e8dres r\u00e9guliers, classification des sous-groupes finis de SO(3).<\/p>\n

              Reminders on groups and examples: dihedral groups, symmetric groups, linear groups (on finite fields), semi-direct product, groups of signed permutations. Tensor product of vector spaces: universal property, bases, tensor product of linear applications. Linear representations of finite groups: definition, examples, regular representation, subrepresentation, decomposition as a direct sum of irreducible representations, tensor product, symmetric square, alternating square. Character theory: Definition, character of a sum, character of a tensor product, Schur lemma, orthonormality of simple characters, decomposition of a regular representation. Groups and geometry: Euclidean affine space, group of isometries, barycentric coordinates, groups of isometries of a solid, cases of a regular polygons, of an nsimplex and of an n-cube. Subgroups of SO(3): classification of regular polyhedra in dimension 3, classification of finite subgroups of SO(3).<\/em><\/p>\n

              \u00a0<\/strong><\/p>\n

               <\/p>\n

                \n
              • MG2-2, Analyse 2 (Cours obligatoire – 6 ECTS)<\/strong><\/li>\n<\/ul>\n

                \u00a0\u00a0\u00a0\u00a0 MG2-2, Analysis 2 (Compulsory course – 6 ECTS)<\/em><\/p>\n

                Responsables : Shizan Fang (CM), Giuseppe Dito (TD)<\/strong><\/p>\n

                \u00a0<\/strong>L’objectif de ce cours est de pr\u00e9senter la th\u00e9orie des distributions (L. Schwartz), qui en g\u00e9n\u00e9ralisant la notion de fonction, permet de r\u00e9soudre de nombreuses \u00e9quations de la physique (\u00e9quations diff\u00e9rentielles ou aux d\u00e9riv\u00e9es partielles). Programme : Rappels de calcul diff\u00e9rentiel, construction de fonctions test, les distributions (d\u00e9finition, d\u00e9rivation, support, produit de convolution), la famille des distributions temp\u00e9r\u00e9es (espace de Schwartz, transform\u00e9e de Fourier d’une distribution), la r\u00e9solution d’\u00e9quations en utilisant les solutions fondamentales.<\/p>\n

                The aim of this course is to present the theory of distributions (introduced by L. Schwartz). It is not always convenient to solve equations associated with physical problems (differential or partial differential equations) using classical functions, it could be easier to deal with more general mathematical objects called “distributions”. We need therefore to define carefully all their properties. Contents: Smooth functions, test functions, distributions (definition, differentiation, support, convolution) and temperate distributions (space of rapidly decreasing functions, Fourier transform), fundamental solutions of differential operators.<\/em><\/p>\n

                R\u00e9f\u00e9rences :<\/strong><\/p>\n

                Claude Zuily, \u00c9l\u00e9ments de distributions et d’\u00e9quations aux d\u00e9riv\u00e9es partielles<\/em>, Dunod 2021<\/p>\n

                Ha\u00efm Br\u00e9zis, Analyse fonctionnelle<\/em>, Dunod 2020<\/p>\n

                Jean-Michel Bony, Cours d’analyse : Th\u00e9orie des distributions et analyse de Fourier<\/em>, Les \u00e9ditions de l’Ecole polytechnique, 2001<\/p>\n

                 <\/p>\n

                 <\/p>\n

                \u00a0<\/strong><\/em><\/p>\n