Parcours Operations research, combinatorics and optimization (ORCO)

This course is an introduction to the main concepts of object-oriented programming, elaborated on C++. It mainly considers: Basics on classes, instances, constructors and destructors, aggregation. Memory management, pointers, references. Operator overloading. Genericity, template classes. STL (Standard Template Library) objects. Inheritance, polymorphism. The objective of this course is to present the computer sciences basics useful for applied mathematics.

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Give an overview of modelling using partial differential equations.

Types of equations, conservation laws

Finite differences methods

Laplace equation

Parabolic equations (diffusion)

Hyperbolic equations (propagation)

Non linear hyperbolic equations

This course include practical sessions.

Voir la page complète

Voir la page complète

Voir la page complète

The aim of this course is to provide the basics mathematical tools and methods of image processing and applications.

Image definition Fourier transform, FFT, applications Image digitalisation, sampling Image processing: convolution, filtering. Applications Image decomposition, multiresolution. Application to compression This course includes practical sessions.

Voir la page complète

This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.

Differential geometry of curves

Approximation of curves with splines, Bézier and spline curves, algorithms,…

Differential geometry of surfaces, metric and curvature properties,…

This course includes practical sessions.

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Voir la page complète

The aim of this course is to provide basic knowledge of applied probability and an introduction to mathematical statistics.

Applied probability

Estimation (parameter)

Sample comparison

Statistical tests

This course includes practical sessions.

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Ce cours présente des manières géométriques de traiter et résoudre des problèmes décrits par des équations différentielles.

        - chapitre I : Introduction : généralités sur les systèmes dynamiques

        - chapitre II : Systèmes unidimensionnels : Les points fixes, linéarisation et stabilité, Exemple : le modèle logistique, Existence et unicité des solutions d'équations différentielles ordinaires

        - chapitre III :  Bifurcations : Bifurcation selle-nœud, Bifurcation transcritique, Bifurcation transcritique imparfaite, Bifurcation fourche, Bifurcation fourche supercritique, Bifurcation fourche sous-critique, Bifurcation fourche supercritique imparfaite

        - chapitre IV :  Champ de vecteur sur un cercle : Oscillateur uniforme, Oscillateur non-uniforme

        - chapitre V : Flots bidimensionnels et applications : Existence et unicité des solutions et conséquences topologiques, Systèmes linéaires, Systèmes non-linéaires : linéarisation proche des points fixes, Cycles limites, Le théorème de Poincaré-Bendixson, Systèmes Liénard, Systèmes gradients, Fonctions de Liapunov

        - chapitre VI : Bifurcations bidimensionnelles : Bifurcations selle-nœud, transcritique et fourche, Bifurcation de Hopf, Bifurcations globales de cycles

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This course presents basic notions on instabilities and turbulence. We try to be as progressive as possible and to base our presentation on analyses of real experiments and real flows. We review few mathematical methods to analyze nonlinear systems in terms of instabilities. The students have to use their new knowledge to run and analyze numerical simulations of very simple systems. We then study some of the most important physical mechanisms for fluid instabilities and the corresponding criteria. We quickly present a zoology of common fluid instabilities and discuss the mechanisms and the possible technical implications. We give a broad introduction on turbulence and describe few fundamental methods and results, in particular the Richardson cascade, the Reynolds decomposition and the Kolmogorov spectra.

Teaching program:

1. General introduction

- Instabilities and turbulence, interest?

- Reynolds experiment and Reynolds number

- Incompressible Navier-Stokes equations: diffusion and advection

- An example: the wake of a cylinder

2. Instabilities and transition to turbulence

- Systems with few degrees of freedom

- Fluid instability mechanisms and conditions

- Other flows examples

3. Effects of variable density

- Boussinesq approximation

- Unstable stratification, Rayleigh-Taylor instability

- Rayleigh-Benard instability (Ra, Nu)

- Stable stratification, Kelvin-Helmoltz instability and Richardson number

4. Turbulence

- Introduction, Richardson cascade

- Average and Reynolds decomposition

- Experimental and numerical methods to study turbulence

- Statistical descriptions

For this course, the students have to write in LaTeX a report on their practical work. Thus, we spent some time for a first gentle introduction of this tool widely used in scientific academia.

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Ce module est une introduction à la turbulence phénoménologique et statistique. On s’intéresse aux définitions et propriétés de la turbulence en terme de processus physiques et leur description dans des familles types d’écoulements cisaillés que l’on peut retrouver dans la nature et en ingénierie.

jet turbulent
jet turbulent

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The aim of this course is to give an introduction to numerical and computing problematics of large dimension problems.

Contents:

• Introduction to database
• Introduction to big data
• Introduction to high performance computing (HPC)
• Numerical solvers for HPC

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Introduction to high performance computinb (HPC)

Numerical solvers for HPC

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Introduction to database

Introduction to big data

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January science and/or industrial project.

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Industrial and/or research internship.

The students have to do an internship (of at least 8 weeks from mid May to end of August, see the planning) in a company or in a laboratory. No report is required (except for Ensimag students that follow the double diploma, who have to give a report to ensimag).

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This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.

Contents:

• Introduction, classification, examples.
• Theoretical results: convexity and compacity, optimality conditions, KT theorem
• Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
• Algorithms for non differentiable problems
• Algorithms for constrained optimisation: penalisatio, SQP methods
• Applications

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The main objective of this course is to provide basics tools in operations research

- Contents

What is OR?
Linear Programming
Duality
Mixed Integer Programming
Dynamic programming
Constraint Programming
Complexity theory and Scheduling

Voir la page complète

Skills

• Recognize a situation where Operations Research is relevant.
• Know the main tools of Operations Research.
• Have the methodological elements to choose the solution methods and the tools the most adapted for a given practical problem.
• Know how to manipulate the software tools to solve a discrete optimization problem.

The course covers various topics:

• Linear Programming (modelling, solving, duality)
• Mixed Integer Linear Programming (modelling techniques, solving with Branch and Bound)
• Dynamic Programming
• Bonus (riddles, elsewhere on the web, OR News)

More details : https://moodle.caseine.org/course/view.php?id=42

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In this part, we will investigate in more details some mathematical notions related to operations research. We will focus on three aspects: Spectral graph theory, Game theory, and Numerical Optimal transport. For each of these themes, we will see how these theoretical results relate to practical operations research problem and finally illustrate them numerically in Python.

Voir la page complète

The aim of this course is to give mathematical grounds of security, integrity, authentication and cryptology.

Course

1. Binary encoding of information
2. Zn* group, field theory
3. Symmetric cryptography
4. Asymmetric cryptography, RSA
5. Hash, DSA
6. Lossless compression
7. Error correcting codes
8. Linear codes
9. Cyclic codes

This course include practical sessions.

Voir la page complète

To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.

Voir la page complète

Voir la page complète

The aim of this course is to give mathematical grounds and algorithms for the modelling, animation, and synthesis of images.

Content

1. Projective rendering methods
2. Animation, cinematic methods
3. Geometrical modelling, 3D, deformation
4. Case study
5. This course include practical sessions. Implementation using OpenGL.

Voir la page complète

Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.

Voir la page complète

Voir la page complète

Voir la page complète

Ce cours a deux objectifs principaux: 

1- Fournir un panorama assez vaste des applications de la dynamique des fluides neutres (hydrodynamique ou HD) en astrophysique. Seront ainsi abordés:

- l’équilibre des systèmes auto-gravitants, l’effondrement gravitationnel et la formation des disques circumstellaires

- les écoulements supersoniques et points critiques associés, la formation des chocs et ondes de détonation (supernovae)    

- la théorie des disques d’accrétion turbulents autour des trous noirs et des étoiles en formation

2- Fournir les hypothèses et les équations maitresses de la magnéto-hydrodynamique (ou MHD), qui est une description monofluide des plasmas. Quelques applications astrophysiques seront ensuite abordées:
 - ondes magnétiques d'Alfven et magnétosoniques    

- bouclier magnétique s’opposant à l’effondrement des nuages

Nous aborderons également, en fin de cours, ce qui est la certainement l’expérience de physique la plus longue de l’humanité, aux conséquences potentiellement majeures sur le devenir de nos sociétés, à savoir la production d’énergie électrique par fusion thermonucléaire contrôlée (ITER et autres machines à confinement magnétique).  

Au-delà de la compréhension nouvelle de certains phénomènes en astrophysique, ce cours illustre la puissance descriptive des plasmas (ici HD ou MHD). A l'issue de ce cours, des concepts avancés tels que les caractéristiques dans des écoulements hyperboliques, les méthodes perturbatives dans des régimes complexes, la turbulence ou encore les ondes dans des milieux inhomogènes auront été abordés.  

Voir la page complète

Voir la page complète

Voir la page complète

Voir la page complète

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The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Content

1. Introduction to modelling with examples.
2. Boundary problem in 1D, variational formulation, Sobolev spaces.
3. Stationary problem, elliptic equations.
4. Finite element method: algorithm, errors…
5. Evolution models, parabolic equations, splitting methods
6. Extensions and applications, FreeFEM++

This course include practical sessions.

Voir la page complète

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Course contents:

• Introduction to modelling with examples.
• Boundary problem in 1D, variational formulation, Sobolev spaces.
• Stationary problem, elliptic equations.
• Finite element method: algorithm, errors…
• Evolution models, parabolic equations, splitting methods
• Extensions and applications, FreeFEM++

This course include practical sessions.

This is a two parts course:

1. Course mutualized with Ensimag 2A 4MMMVAM (head: Emmanuel Maitre)
2. MSIAM specific course (in-depth and practical session) (head: Clément Jourdana)

A description of the course is available here

Voir la page complète

Voir la page complète

This course is an introduction to the main concepts of object-oriented programming, elaborated on C++. It mainly considers: Basics on classes, instances, constructors and destructors, aggregation. Memory management, pointers, references. Operator overloading. Genericity, template classes. STL (Standard Template Library) objects. Inheritance, polymorphism. The objective of this course is to present the computer sciences basics useful for applied mathematics.

Voir la page complète

Give an overview of modelling using partial differential equations.

Types of equations, conservation laws

Finite differences methods

Laplace equation

Parabolic equations (diffusion)

Hyperbolic equations (propagation)

Non linear hyperbolic equations

This course include practical sessions.

Voir la page complète

Voir la page complète

Voir la page complète

The aim of this course is to provide the basics mathematical tools and methods of image processing and applications.

Image definition Fourier transform, FFT, applications Image digitalisation, sampling Image processing: convolution, filtering. Applications Image decomposition, multiresolution. Application to compression This course includes practical sessions.

Voir la page complète

This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.

Differential geometry of curves

Approximation of curves with splines, Bézier and spline curves, algorithms,…

Differential geometry of surfaces, metric and curvature properties,…

This course includes practical sessions.

Voir la page complète

The aim of this course is to provide basic knowledge of applied probability and an introduction to mathematical statistics.

Applied probability

Estimation (parameter)

Sample comparison

Statistical tests

This course includes practical sessions.

Voir la page complète

Voir la page complète

The aim of this course is to give an introduction to numerical and computing problematics of large dimension problems.

Contents:

• Introduction to database
• Introduction to big data
• Introduction to high performance computing (HPC)
• Numerical solvers for HPC

Voir la page complète

Introduction to high performance computinb (HPC)

Numerical solvers for HPC

Voir la page complète

Introduction to database

Introduction to big data

Voir la page complète

January science and/or industrial project.

Voir la page complète

Industrial and/or research internship.

The students have to do an internship (of at least 8 weeks from mid May to end of August, see the planning) in a company or in a laboratory. No report is required (except for Ensimag students that follow the double diploma, who have to give a report to ensimag).

Voir la page complète

This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.

Contents:

• Introduction, classification, examples.
• Theoretical results: convexity and compacity, optimality conditions, KT theorem
• Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
• Algorithms for non differentiable problems
• Algorithms for constrained optimisation: penalisatio, SQP methods
• Applications

Voir la page complète

Voir la page complète

The aim of this course is to give mathematical grounds of security, integrity, authentication and cryptology.

Course

1. Binary encoding of information
2. Zn* group, field theory
3. Symmetric cryptography
4. Asymmetric cryptography, RSA
5. Hash, DSA
6. Lossless compression
7. Error correcting codes
8. Linear codes
9. Cyclic codes

This course include practical sessions.

Voir la page complète

To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.

Voir la page complète

Voir la page complète

The main objective of this course is to provide basics tools in operations research

- Contents

What is OR?
Linear Programming
Duality
Mixed Integer Programming
Dynamic programming
Constraint Programming
Complexity theory and Scheduling

Voir la page complète

Skills

• Recognize a situation where Operations Research is relevant.
• Know the main tools of Operations Research.
• Have the methodological elements to choose the solution methods and the tools the most adapted for a given practical problem.
• Know how to manipulate the software tools to solve a discrete optimization problem.

The course covers various topics:

• Linear Programming (modelling, solving, duality)
• Mixed Integer Linear Programming (modelling techniques, solving with Branch and Bound)
• Dynamic Programming
• Bonus (riddles, elsewhere on the web, OR News)

More details : https://moodle.caseine.org/course/view.php?id=42

Voir la page complète

In this part, we will investigate in more details some mathematical notions related to operations research. We will focus on three aspects: Spectral graph theory, Game theory, and Numerical Optimal transport. For each of these themes, we will see how these theoretical results relate to practical operations research problem and finally illustrate them numerically in Python.

Voir la page complète

The aim of this course is to give mathematical grounds and algorithms for the modelling, animation, and synthesis of images.

Content

1. Projective rendering methods
2. Animation, cinematic methods
3. Geometrical modelling, 3D, deformation
4. Case study
5. This course include practical sessions. Implementation using OpenGL.

Voir la page complète

Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.

Voir la page complète

Voir la page complète

Voir la page complète

Ce cours a deux objectifs principaux: 

1- Fournir un panorama assez vaste des applications de la dynamique des fluides neutres (hydrodynamique ou HD) en astrophysique. Seront ainsi abordés:

- l’équilibre des systèmes auto-gravitants, l’effondrement gravitationnel et la formation des disques circumstellaires

- les écoulements supersoniques et points critiques associés, la formation des chocs et ondes de détonation (supernovae)    

- la théorie des disques d’accrétion turbulents autour des trous noirs et des étoiles en formation

2- Fournir les hypothèses et les équations maitresses de la magnéto-hydrodynamique (ou MHD), qui est une description monofluide des plasmas. Quelques applications astrophysiques seront ensuite abordées:
 - ondes magnétiques d'Alfven et magnétosoniques    

- bouclier magnétique s’opposant à l’effondrement des nuages

Nous aborderons également, en fin de cours, ce qui est la certainement l’expérience de physique la plus longue de l’humanité, aux conséquences potentiellement majeures sur le devenir de nos sociétés, à savoir la production d’énergie électrique par fusion thermonucléaire contrôlée (ITER et autres machines à confinement magnétique).  

Au-delà de la compréhension nouvelle de certains phénomènes en astrophysique, ce cours illustre la puissance descriptive des plasmas (ici HD ou MHD). A l'issue de ce cours, des concepts avancés tels que les caractéristiques dans des écoulements hyperboliques, les méthodes perturbatives dans des régimes complexes, la turbulence ou encore les ondes dans des milieux inhomogènes auront été abordés.  

Voir la page complète

Voir la page complète

Voir la page complète

Voir la page complète

Voir la page complète

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Content

1. Introduction to modelling with examples.
2. Boundary problem in 1D, variational formulation, Sobolev spaces.
3. Stationary problem, elliptic equations.
4. Finite element method: algorithm, errors…
5. Evolution models, parabolic equations, splitting methods
6. Extensions and applications, FreeFEM++

This course include practical sessions.

Voir la page complète

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Course contents:

• Introduction to modelling with examples.
• Boundary problem in 1D, variational formulation, Sobolev spaces.
• Stationary problem, elliptic equations.
• Finite element method: algorithm, errors…
• Evolution models, parabolic equations, splitting methods
• Extensions and applications, FreeFEM++

This course include practical sessions.

This is a two parts course:

1. Course mutualized with Ensimag 2A 4MMMVAM (head: Emmanuel Maitre)
2. MSIAM specific course (in-depth and practical session) (head: Clément Jourdana)

A description of the course is available here

Voir la page complète

Voir la page complète

I. Compléments sur les anneaux

1. Groupe des éléments inversibles. (ℤ/nℤ)^∗, fonction d’Euler. Éléments irréductibles et éléments premiers. Pgcd et ppcm.
2. Notion d’algèbre. Algèbre des polynômes en n indéterminées. Polynômes symétriques. Liens entre coefficients et racines d’un polynôme. En TD : séries formelles en une variable. Corps des fractions d’un anneau intègre.
3. Anneaux noethériens, théorème de la base de Hilbert.
4. Anneaux factoriels. Lemme de Gauss et lemme d'Euclide. Exemple : les anneaux principaux. Théorème de Gauss sur A[X], pour A factoriel. Polynômes irréductibles, critères d’irréductibilité sur A factoriel (Eisenstein, etc.).

II. Corps (les corps considérés sont commutatifs)

1. Extensions de corps, degrés, multiplicité. Éléments algébriques, éléments transcendants, polynôme minimal, extension algébrique.
2. Corps de rupture, corps de décomposition d’un polynôme.
3. Clôture algébrique (définition), le corps ℂ des nombres complexes est algébriquement clos. Énoncé du théorème de Steinitz.
4. Corps finis, existence et unicité, structure multiplicative. Racines de l’unité, polynômes cyclotomiques, irréductibilité sur ℤ.

III. Représentations des groupes finis sur ℂ

1. Représentations d’un groupe fini. Représentations par permutations, représentations régulières.
2. Représentations irréductibles, Théorème de Maschke.
3. Morphismes de représentations. Lemme de Schur.
4. Caractères. Caractère de Hom(V;W). Orthogonalité et décomposition des représentations. Formule de Burnside. Théorème fondamental de Frobenius et corollaires. Table des caractères. Orthogonalité des colonnes.
5. Exemple : table de 𝔖₄. Noyau d’un caractère. Application : critère de simplicité.
6. Le cas des groupes abéliens. Groupe dual d’un groupe abélien fini. Transformée de Fourier discrète, cas de ℤ/nℤ et (ℤ/nℤ)². Structure des groupes abéliens finis.

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1. Fonctions holomorphes et analytiques, en particulier l’équivalence entre les deux notions, fonction exponentielle et logarithme, principe du prolongement analytique, principe des zéros isolés, formule de Cauchy pour le disque
2. Propriétés élémentaires des fonctions holomorphes (inégalités de Cauchy, suites et séries de fonctions holomorphes, propriété de la moyenne et principe du maximum)
3. Théorie de Cauchy (existence de primitives, théorèmes de Cauchy)
4. Fonctions méromorphes (classification des singularités isolées, fonctions méromorphes, théorème des résidus, séries de Laurent)
5. Théorème de la représentation conforme de Riemann

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1. Rappels élémentaires de théorie des probabilités
2. Éléments de statistique
3. Espérance conditionnelle
4. Processus à temps discret
5. Martingales
6. Chaînes de Markov

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Espaces de Lebesgue L^p(Omega)
 Densité des fonctions C-infini à support compact dans Lp(Ω)
 Théorème de Riesz-Fréchet-Kolmogorov (compacité dans Lp(Ω)).
 Eventuellement : convolution L^p-L^q (dans R^n) inégalités de Young. ;  dual de L^p (montré pour \ell^p)
  

Analyse de Fourier
  Rappels sur la transformation de Fourier sur L^1(R^n)
  Transformation de Fourier sur L^2(R^d)
  Espace de Schwartz sur R^d, transformation de Fourier et convolution.
  Eventuellement : brève présentation des distributions tempérées.
  Séries de Fourier sur L^1(T), L^2(T), voire sur T^n
  Application à la résolution des équations de la chaleur, des ondes, de Schrödinger, avec donnée initiale dans S(R^d), L^2(R^d) L^2(T). (et L^p(R^d), pour l’équation de la chaleur, par convolution avec le noyau de la chaleur)
   Résolution de −∆u + u = f dans S(R^d). Terminologie EDP elliptiques, hyperboliques, paraboliques.

Solutions faibles

- pour les équations de transport (∂tu + c · ∂xu = f, c ∈ Rd), avec condition initiale, après résolution du cas classique (C^1) ;

- pour des EDP elliptiques, avec conditions au bord (Dirichlet homogène) : espace de Sobolev H^1 sur I ou Ω, et Rd : caractérisation par Fourier; espace H_0^1 ; lemme de Lax- Milgram.

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Cette UE propose une découverte de la recherche en mathématiques à travers l'étude d'un sujet décrivant un résultat ou une théorie mathématique, avec lesquels l'étudiant.e devra se familiariser afin de se les approprier et de pouvoir en rendre compte par un rapport écrit et un exposé oral.

En pratique, une liste de sujets est proposée au cours du premier semestre. Chaque étudiant.e sélectionne dans cette liste quatre sujets, classés de 1 à 4, puis le responsable de la formation attribue à chaque étudiant.e un sujet figurant dans la mesure du possible parmi ces quatre-là. Dès les attributions connues, chaque étudiant.e contacte l'auteur.e de son sujet, qui va l'encadrer pour ce travail tout au long du second semestre. Une fois que l'encadrant.e a présenté à l'étudiant.e le sujet et les détails du travail attendu, le binôme se rencontre régulièrement afin que l'étudiant.e puisse rendre compte de l'avancement de son travail et progresser dans celui-ci.

Le TER donne lieu à la rédaction d'un rapport écrit, rédigé en utilisant le logiciel LaTeX, comportant obligatoirement un résumé et une bibliographie, et à une soutenance orale d'une durée de 20 à 30 minutes, souvent suivie de questions, devant un jury qui comprend l'encadrant.e. Le rapport et la soutenance contribuent conjointement à l'évaluation du travail réalisé.

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Nous étudierons certains exemples de groupes Fuchsiens, et les pavages du disque qui leur sont associés. Si le temps le permet, nous aborderons la géométrie hyperbolique de dimension trois sous l'angle des déformations de groupes Fuchsiens.

Descriptif

• Partie I. Géométrie des transformations de Möbius et géométrie hyperbolique
  • Transformation de Möbius, inversions par rapports aux sphères, dans Rⁿ, pour n surtout égal à 1, 2, 3.
  • Plan et espace hyperbolique de dimension 3 : demi-plan et demi espace supérieurs
  • Leurs isométries : interprétation matricielle et holomorphe, interprétation dynamique du spectre des matrices

• Partie II. Groupes Fuchsiens
  • Actions discontinues et groupes discrets sur le plan hyperbolique
  • Théorème de combinaison de Klein. Ping-pong, et sous-groupes libres de PSL(2,R)
  • Cas de PSL(2, Z)

• Partie III. Ensembles limites et fractales
  • Ensemble limite d'un sous-groupe discret de PSL(2,R), et de PSL(2,C)
  • Déformations fractales d'ensembles limites
  • Rencontre et familiarisation avec des outils logiciels d'illustration rigoureuse d'ensembles limites fractales

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L'algèbre effective est le domaine des mathématiques où on s'intéresse au calcul exact des objets intervenant en algèbre au sens large (arithmétique des entiers, arithmétique des polynômes et algèbre linéaire sur un corps fini et sur les rationnels), avec l'objectif de les rendre efficaces par rapport à la taille des données, en estimant leur complexité. Les applications sont nombreuses : calcul formel, cryptographie, codes correcteurs (par exemple QR codes)... On montrera plusieurs exemples où des calculs modulo un nombre premier permet d'accélérer les calculs sur les rationnels.
Une partie des exercices nécessite l'utilisation d'un logiciel de calcul formel tel que Xcas sur PC, mobile ou calculatrice CAS. 

Descriptif

1. Arithmétique des polynômes à 1 variable (dont interpolation et FFT), arithmétique des entiers et liens entre eux. Puissance modulaire rapide, application: test de primalité, RSA.
2. PGCD dans Z/pZ[X]. Application à la simplification dans Q[X]. Irréductibilité dans Z/pZ[X], application à la représentation des corps finis, application à la factorisation dans Q[X]. Calcul efficace dans GF(2,n).
3. Théorème fondemental de l'algèbre : localisation de racines de polynômes dans C[X] (Newton, Aberth ; Sturm, Descartes). Résultant, algorithmes de calcul, application au calcul de primitives de fractions rationnelles, à la résolution de certains systèmes polynomiaux. Générateurs effectifs d'extensions de Q.
4. Matrice à coefficients dans un corps fini et sur les rationnels: réduction de Gauss, déterminant, polynôme caractéristique. Applications : codes correcteurs.

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1. Rappels sur le calcul différentiel dans l’espace euclidien ^{n n m}
2. Courbes et repères
3. Calcul différentiel sur les surfaces de ℝ^{3 3}
4. Variétés abstraites
5. Courbure
6. Géométrie des surfaces de ℝ³ theorema egregium
7. Théorème de Gauss-Bonnet

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La première partie de l'UE sera consacrée à la théorie des chaînes de Markov en temps discret et à espace d'états discret ; il s'agira donc de couvrir un chapitre initialement prévu au programme de l'UE Probabilités du premier semestre que le manque de temps et les lacunes des étudiant·es concernant le programme de L3 empêchent régulièrement de traiter.
La deuxième partie constituera une introduction à la mécanique statistique, plus particulièrement à l'étude mathématique des transitions de phase dans des modèles sur réseau. On étudiera en détail le cas de référence du modèle d'Ising, l'un des modèles les plus célèbres et les plus étudiés dans ce domaine.
Si le temps le permet, une troisième partie présentera une introduction à des modèles de chaînes de Markov indexées par les arbres (percolation et modèle d'Ising sur des arbres).

Insertion dans le cursus de master Une partie du contenu de cette UE est au programme de l'agrégation : traitement complet des chaînes de Markov dans ce cadre, espérances conditionnelles, éventuelle\-ment un peu de martingales (convergence de martingales rétrogrades pour l'existence de mesures en volume infini à travers les équations DLR), topologies faibles, etc.
Il peut également servir à renouveler les exemples mobilisables dans ce contexte pour l'option A : conditionnements spatiaux, situations de TCL/non-TCL pour des variables aléatoires non indépendantes (convergence ou non de la magnétisation moyenne à haute température ou à basse température), dynamique de Glauber et algorithme de Metropolis-Hastings, etc.
Enfin, il vise à aborder, spécialement dans la troisième partie, des situations de recherche actuelles.

Simulations du modèle d'Ising sur le réseau carré à basse température (à gauche, phase ferromagnétique) et à haute température (à droite, phase paramagnétique)

Programme prévisionnel

• Partie I. Chaînes de Markov
  • Classification des états
  • Propriété de Markov forte
  • Mesure invariante (conditions d'existence, unicité)
  • Théorème ergodique

• Partie II. Modèle d'Ising
  • Le modèle de Curie-Weiss -- Un modèle sans géométrie
  • Mesures de Gibbs en volume fini
  • Limite thermodynamique -- Pression et magnétisation
  • Le modèle d'Ising sur Z -- Matrices de transfert
  • Mesures en volume infini
  • Fonctions locales et inégalités de corrélation
  • Diagramme de phase du modèle d'Ising sur Z^d, d>1
  • Critères d'unicité
  • Brisure de symétrie à basse température -- Argument de Peierls
  • Unicité à haute température
  • Unicité en champ magnétique non nul -- Théorème de Lee-Yang

• Partie III. Chaînes de Markov indexées par les arbres
  • Mesures de Gibbs sur les arbres
  • Lois de bord
  • Percolation, Ising
  • Extrémalité et critères de reconstruction

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Le but de ce cours est d'introduire les étudiants aux bases de la théorie spectrale des opérateurs elliptiques sur des domaines, et de voir quelques applications, notamment dans le cadre de la mécanique quantique.
Un des résultats clés du cours sera le fait que le Laplacien avec condition de Dirichlet dans un domaine Ω borné et lisse de R^d admet une base hilbertienne de fonctions propres régulières e_n vérifiant -Δ e_n = λ _ne_n, où λ _n → +∞ avec n∈ N.
Ce résultat sera un prétexte à l'étude de différents concepts : opérateurs compacts, ou à résolvante compacte, dans les espaces de Hilbert et leur "diagonalisation", espaces de Sobolev H^k( Ω) et leurs propriétés (injections de Sobolev par exemple), régularité des solutions d'EDP elliptiques.

Dans un second temps, nous souhaitons aussi présenter des applications à la théorie spectrale des opérateurs emblématiques de la mécanique quantique.
Le formalisme mathématique de la mécanique quantique sere présenté, puis nous discuterons le spectre de certains opérateurs de Schrödinger -Δ +V tels l'oscillateur harmonique ou le Laplacien sur le tore. Nous aborderons également l'existence et les propriétés de fonctions propres associées à la plus basse valeur propre d'un opérateur de Schrödinger en fonction du potentiel V via l'approche variationnelle.

Plus généralement, nous présenterons la caractérisation du spectre d'opérateurs par le min-max des quotients de Rayleigh, ainsi que le théorème de Courant sur le nombre de domaines nodaux des fonctions propres du Laplacien, et l'asymptotique de Weyl pour les valeurs propres sur un domaine. Des exemples simples de problèmes d'optimisation spectrale pourront aussi être présentés en TD, en fonction du temps disponible.

programme préliminaire

• Préliminaires sur les espaces de Hilbert : bases hilbertiennes, Lax-Milgram (en fonction de ce qui sera fait dans le cours du S1).
• Opérateurs compacts. Définition et propriétés du spectre. Théorème spectral pour les opérateurs compacts.
• Espaces de Sobolev H^k( Ω) sur des ouverts bornés réguliers de R^d, et leurs propriétés.
• Existence d'une base hilbertienne de fonctions propres pour le Laplacien sur un domaine borné régulier.
• Régularité elliptique et régularité des fonctions propres.
• Principes variationnels, min-max pour les valeurs propres.
• Le formalisme de la mécanique quantique : vecteur d'état/fonction d'ondes, opérateur Hamiltonien, évolution/équation de Schrödinger, observable, processus de mesure en mécanique quantique.
• Le spectre de certains opérateurs de Schrödinger -Δ +V dans R^d et les propriétés de son état fondamental. Exemple de l'oscillateur harmonique, du Laplacien sur le tore, de la particule dans un champ magnétique.
• Un peu de géométrie spectrale : théorème de Courant sur les domaines nodaux, asymptotique de Weyl. Optimisation spectrale dans des cas simples.

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The main objective of this course is to provide basics tools in operations research

- Contents

What is OR?
Linear Programming
Duality
Mixed Integer Programming
Dynamic programming
Constraint Programming
Complexity theory and Scheduling

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Skills

• Recognize a situation where Operations Research is relevant.
• Know the main tools of Operations Research.
• Have the methodological elements to choose the solution methods and the tools the most adapted for a given practical problem.
• Know how to manipulate the software tools to solve a discrete optimization problem.

The course covers various topics:

• Linear Programming (modelling, solving, duality)
• Mixed Integer Linear Programming (modelling techniques, solving with Branch and Bound)
• Dynamic Programming
• Bonus (riddles, elsewhere on the web, OR News)

More details : https://moodle.caseine.org/course/view.php?id=42

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In this part, we will investigate in more details some mathematical notions related to operations research. We will focus on three aspects: Spectral graph theory, Game theory, and Numerical Optimal transport. For each of these themes, we will see how these theoretical results relate to practical operations research problem and finally illustrate them numerically in Python.

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Il s'agit de viser le niveau de qualification B2 du Conseil de l'Europe, défini par ALTE, dans trois champs de compétences :

1. Être capable de faire un exposé clair sur un sujet connu et répondre à des questions factuelles prévisibles
2. Être capable de parcourir un texte pour retrouver l'information pertinente et en saisir l'essentiel
3. Être capable de prendre des notes simples et en faire un usage raisonnable pour écrire une dissertation ou faire une révision

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This course presents advanced methods and technics for Operations Research.

Reminder :

Linear Programming, Dynamic Programming, MIP modelling and BB
Complexity (P, NP, Co-NP)

Advanced MIP :

formulation, cuts, bounds
applications
lagragian relaxation
column generation
Benders decomposition
Solvers

Constraint Programming

Heuristics

local search
approximation algorithms

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The aim of this course is to provide a broad knowledge of fundamental problems in Combinatorial Optimization to show their algorithmic solutions and to derive min-max results on them. In order to achieve this goal a new object called a polyhedron is introduced. This polyhedral approach helps to shed new light on some classic results of Combinatorial Optimization.

Syllabus: Study of polyhedra associated to problems of Combinatorial Optimization ; Theory of blocking polyhedra ; Connectivity: shortest paths, spanning trees and spanning arborescences of minimum weight ; Flows: Edmonds-Karp Algorithm, Goldberg-Tarjan Algorithm, minimum cost flows ; Matchings: Hungarian method, Edmonds' Algorithm, Chinese postman problem; Matroids: greedy algorithm, intersection of two matroids ; Graph coloring ; Applications coming from various areas of Operations Research.

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The objective of this course is to present different techniques to handle uncertainty in decision problems. These techniques will be illustrated on several applications e.g. inventory control, scheduling, energy, machine learning.

https://moodle.caseine.org/course/view.php?id=1084

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The aim of this course is to learn how to use the structure of graphs and other discrete objects to obtain general results on them, and in particular efficient algorithms solving important problems.

We will cover the following topics:

• Structural Graph Theory: we will study the structure of important graph classes with nice algorithmic properties (planar graphs, interval graphs, comparability graphs, ...) and show several concrete problems that can be solved thanks to their structural properties.
• Graph Drawing: with the rise of Big Data, representing huge data sets is a fundamental challenge. Efficient ways to represent large graphs will be presented.
• Codes: we will see various codes (dominating, locating, identifying, ...) in graphs and their applications.
• Extremal combinatorics: the typical question in this field is "what global condition do we need to impose in some graph in order to make sure that some nice structure appears locally?" We will introduce a powerful tool called "the probabilistic method", an show how it can be applied to solve problems in this important area of research.

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This course offers the possibility for the students to gain some experience by facing open/difficult combinatorial problems.

The goal is to model and solve a combinatorial problem with direct industrial applications. We expect the students to take a variety of approaches (local search, compact/extended linear programming formulations, constraint programming, ...) and establish useful results (lower bounds, cuts, complexity,...).

The experimental results will be compared to the litterature (a known academic open benchmark will be available in this case) or will be validated by the industrial partner.

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Presenting models and tools for solving problems of company location and transport management in a supply chain.

Description:

This course presents two aspects: strategic design of a distribution network in a supply chain and tactical-operational aspects of network management.

For each aspect the course develops two key points:

• in network design, the choice of location and the establishment of services (production site, warehouse, logistics platform) and the synthesis of the transport network (connections / roads between production sites, warehouses, logistics centres)
• in network management, planning and modes of distribution transport (planning delivery rounds, number and types of vehicles).

For each of the three problem areas: location, delivery rounds and network design, students will study both modelling and techniques for accurate resolution (Branch-and-Bound) plus techniques for approximate resolutions (Lagrangian relaxation , metaheuristics) and how they are set up in commercial software (OPL-CPLEX Studio, Excel solver), or free software (COIN, lp-solve) or ad-hoc.

Evaluation :

Individual written exam (1/2 of the final mark) and group project (1/2 of the final mark).

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Today, parallel computing is omnipresent across a large spectrum of computing platforms, from processor cores or GPUs up to Supercomputers and Cloud platforms. The largest Supercomputers gather millions of processing units and are heading towards Exascale (a quintillion or 10^18 flops - http://top500.org). If parallel processing was initially targeting scientific computing, it is now part of many domains like Big Data analytics and Deep Learning. But making an efficient use of these parallel resources requires a deep understanding of architectures, systems, parallel programming models, and parallel algorithms.

Overview:

The class is organized around weekly lectures, discussions and help time. The presented materials that will be available each week on the class web page. To get practical experience and good understanding of the main concepts the students are expected to develop short programs and experiments. Students will also have to prepare a presentation or a written report on research articles. Students will have access to Grid'5000 parallel machines and the SimGrid simulator for experiments.

This class is organized around 2 main blocks:

Overview of parallel systems:

1. • Introduction to parallelism from GPU to supercomputers.
   • Hardware and system considerations for parallel processing (multi-core architectures, process and thread handling, cache efficiency, remote data access, atomic instructions)
   • Parallel programming: message passing, one-sided communications, task-based programming, work stealing based runtimes (MPI, Cilk, TBB, OpenMP).
   • Modeling of parallel programs and platforms. Locality, granularity, memory space, communications.
   • Parallel algorithms, collective communications, topology aware algorithms.
   • Scheduling: list algorithms; partitioning techniques. Application to resource management (PBS, LSF, SGE, OAR).
   • Large scale scientific computing: parallelization of Lagrangian and Eulerian solvers, parallel data analytics and scientific visualization.
   • AI and HPC: how parallelism is used at different levels to accelerate machine learning and deep learning using supercomputers.

Functional parallel programming:

We propose to study a clean and modern approach to the design of parallel algorithms: functional programming. Functional languages are known to provide a different and cleaner programming experience by allowing a shift of focus from "how to do" on "what to do".
If you take for example a simple dot product between two vectors. In c language you might end up with:

unsigned int n = length(v1); double s = 0.0; for (unsigned int i = 0 ; i < n ; i++) { s += v1[i] * v2[i]; }

In python however you could write:

return sum(e1*e2 for e1, e2 in zip(v1, v2))

You can easily notice that the c language code displayed here is highly sequential with a data-flow dependence on the i variable. It intrinsically contains an ordering of operations because it tells you how to do things to obtain the final sum. On the other end the python code exhibits no dependencies at all. It does not tell you how to compute the sum but just what to compute: the sum of all products.

In this course we will study how to express parallel operations in a safe and performant way. The main point is to study parallel iterators and their uses but we will also consider classical parallel programming schemes like divide and conquer. We will both study the theoretical complexity of different parallel algorithms and practice programming and performance analysis on real machines.
All applications will be developed in the RUST programming language around the Rayon parallel programming library.
No previous knowledge of the rust language is required as we will introduce it gradually during the course. You need however to be proficient in at least one low level language (typically C or C++)

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This course introduces the field of MAS, various theoretical aspects (agent architectures, reasoning, interactions, game theory, social choice, etc), as well as practical applications from recent research. The focus is mostly on agent-based social simulation, and how to integrate psychological aspects in agents (so-called “human factors”: emotions, biases…) to make them more human-like and realistic. Applications discussed include epidemics modelling, computational economy, crisis management, urban planning, serious games, etc. The practical part of the course comprises several tutorials with various agent-based modelling platforms (in particular GAMA and Netlogo), scientific papers discussions, and analysis and/or extension of existing models. 

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Modern computing increasingly takes advantage of large amounts of distributed data and knowledge. This is grounded on theoretical principles borrowing to several fields of computer science such as programming languages, data bases, structured documentation, logic and artificial intelligence. The goal of this course is to present some of them, the problems that they solve and those that they uncover. The course considers two perspectives on data and knowledge: interpretation (what they mean), analysis (what they reveal) and processing (how can they be traversed efficiently and transformed safely).

The course offers a semantic perspective on distributed knowledge. Distributed knowledge may come from data sources using different ontologies on the semantic web, autonomous software agents learning knowledge or social robots interacting with different interlocutors. The course adopts a synthetic view on these. It first presents principles of the semantics of knowledge representation (RDF, OWL). Ontology alignments are then introduced to reduce the heterogeneity between distributed knowledge and their exploitation for answering federated queries is presented. A practical way for cooperating agents to evolve their knowledge is cultural knowledge evolution that is then illustrated. Finally, the course defines dynamic epistemic logics as a way to model the communication of knowledge and beliefs.

The course also introduces a perspective on programming language foundations, algorithms and tools for processing structured information, and in particular tree-shaped data. It consists in an introduction to relevant theoretical tools with an application to NoSQL (not only SQL) and XML technologies in particular. Theories and algorithmic toolboxes such as fundamentals of tree automata and tree logics are introduced, with applications to practical problems found for extracting information. Applications include efficient query evaluation, memory-efficient validation of document streams, robust type-safe processing of documents, static analysis of expressive queries, and static type-checking of programs manipulating structured information. The course also aims at presenting challenges, important results, and open issues in the area.

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The course aims to provide the fundamental basis for a sound scientific methodology of experimental evaluation in computer science. This lecture emphasizes on methodological aspects of measurement and on the statistics needed to analyze computer systems, human-computer interaction systems, and machine learning systems. We first sensibilize the audience to reproducibility issues related to empirical research in computer science as well as to ethical and scientific integrity aspects. Then we present tools that help address the aforementioned issues and we give the audience the basis of probabilities and statistics required to develop sound experiment designs. The content of the lecture is therefore both theoretical and practical, illustrated by a lot of case studies and practical sessions. The goal is not to provide analysis recipes or techniques that researchers can blindly apply but to make students develop critical thinking and understand some simple (and possibly not-so-simple) tools so that they can both readily use and explore later on.

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The course is divided in two complementary parts: distributed systems and data management.

Part 1: Distributed systems

Summary

Distributed systems are omnipresent. They are formed by a set of computing devices, interconnected by a network, that collaborate to perform a task. Distributed systems execute on a wide range of infrastructures: from Cloud datacenters to wireless sensor networks.

The goal of this course is to study the main algorithms used at the core of Distributed systems. These algorithms must be efficient and robust. An algorithm is efficient when it sustains a high level of performance. Performance can be measured using various metrics such as throughput, latency, response time. An algorithm is robust when it is able to operate despite the occurrence of various types of (network and/or machine) and attacks.

Content

During the course, we will cover several topics that are listed below:

• Event-driven formalisms for distributed algorithms
• Basic abstractions: processes, links
• Failure detector algorithms
• Leader election algorithms
• Broadcasting algorithms
• Distributed shared memory algorithms
• Consensus algorithms
• Epidemic algorithms
• Performance models for distributed systems

Part 2: Data management

Summary

The ability to process large amounts of data is key to both industry and research today. As computing systems are getting larger, they generate more data that need to be analyzed to extract knowledge.

Data management infrastructures are growing fast, leading to the creation of large data centers and federations of data centers. Suitable software infrastructures should be used to store and process data in this context. Big Data software systems are built to take advantage of a large set of distributed resources to efficiently process massive amounts of data while being able to cope with failures that are frequent at such a scale.

In addition to the amount of data to be processed, the other main challenge that such Big Data systems need to deal with is time. For some use cases, the earlier the results of a data analysis is obtained, the more valuable the result is. Some Big Data systems especially target stream processing where data are processed in real time.

Through lectures and practical sessions, this course provides an overview of the software systems that are used to store and process data at large scale. The following topics will be covered:

• Map-Reduce programming model
• In-memory data processing
• Stream processing (data movement and processing)
• Large scale distributed data storage (distributed file systems, NoSQL databases)

Throughout the lectures, the challenges associated with performance and fault tolerance will also be discussed.

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The course present the main cryptographic primitives and security protocols, focusing on security parameters and properties.

Pedagogical goals:

- generic cryptographic primitives: one-way, trap-door and hash functions; random generators; symmetric and assymertic cipher; interactive protocols;

- security properties : complexity and reduction proofs; undistinguidhability; non-malleability; soundness, completeness and zero-knowledge; confidentiality; authentication; privacy; non-repudiation

- use, deployment and integration of protocols in standard crypro lib (eg open-ssl)

- security proofs : fundations and verification based on tools (eg avispa)

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Statistical learning is about the construction and study of systems that can automatically learn from data. With the emergence of massive datasets commonly encountered today, the need for powerful machine learning is of acute importance. Examples of successful applications include effective web search, anti-spam software, computer vision, robotics, practical speech recognition, and a deeper understanding of the human genome. This course gives an introduction to this exciting field. In the first part, we will introduce basic techniques such as logistic regression, multilayer perceptrons, nearest neighbor approaches, both from a theoretical and methodological point of views. In the second part, we will focus on more advanced techniques such as kernel methods, which is a versatile tool to represent data, in combination with (un)supervised learning techniques that are agnostic to the type of data that is learned from. The learning techniques that will be covered include regression, classification, clustering and dimension reduction. We will cover both the theoretical underpinnings of kernels, as well as a series of kernels that are important in practical applications. Finally we will touch upon topics of active research, such as large-scale kernel methods and the use of kernel methods to develop theoretical foundations of deep learning models.

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Machine Learning is one of the key areas of Artificial Intelligence and it concerns the study and the development of quantitative models that enables a computer to perform tasks without being explicitly programmed to do them. Learning in this context is hence to recognize complex forms and to make intelligent decisions. Given all existing entries, the difficulty of this task lies in the fact that all possible decisions is usually very complex to enumerate. To get around that, machine learning algorithms are designed in order to gain knowledge on the problem to be addressed based on a limited set of observed data extracted from this problem. To illustrate this principle, consider the supervised learning task, where the prediction function, which infers a predicted output for a given input, is learned over a finite set of labeled training examples, where each instance of this set is a pair constituted of a vector characterizing an observation in a given vector space, and an associated desired response for that instance (also called desired output). After the training step, the function returned by the algorithm is sought to give predictions on new examples, which have not been used in the learning process, with the lowest probability of error. The underlying assumption in this case is that the examples are, in general, representative of the prediction problem on which the function will be applied. We expect that the learning algorithm produces a function that will have a good generalization performance and not the one that is able to perfectly reproduce the outputs associated to the training examples. Guarantees of learnability of this process were studied in the theory of machine learning largely initiated by Vladimir Vapnik. These guarantees are dependent on the size of the training set and the complexity of the class of functions where the algorithm searches for the prediction function. Emerging technologies, particularly those related to the development of Internet, reshaped the domain of machine learning with new learning frameworks that have been studied to better tackle the related problems. One of these frameworks concerns the problem of learning with partially labeled data, or semi-supervised learning, which development is motivated by the effort that has to be made to construct labeled training sets for some problems, while large amount of unlabeled data can be gathered easily for these problems. The inherent assumption, in this case, is that unlabeled data contain relevant information about the task that has to be solved, and that it is a natural idea to try to extract this information so as to provide the learning algorithm more evidence. From these facts were born a number of works that intended to use a small amount of labeled data simultaneously with a large amount of unlabeled data to learn a prediction function.

The intent of this course is to propose a broad introduction to the field of Machine Learning, including discussions of each of the major frameworks, supervised, unsupervised, semi-supervised and reinforcement learning.

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Causality is at the core of our vision of the world and of the way we reason. It has long been recognized as an important concept and was already mentioned in the ancient Hindu scriptures: “Cause is the effect concealed, effect is the cause revealed”. Even Democritus famously proclaimed that he would rather discover a causal relation than be the king of presumably the wealthiest empire of his time. Nowadays, causality is seen as an ideal way to explain observed phenomena and to provide tools to reason on possible outcomes of interventions and what-if experiments, which are central to counterfactual reasoning, as ‘‘what if this patient had been given this particular treatment?’’

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This course is related to mathematical and statistical methods which are very used in supervised learning.

It contains two parts.

In the first part, we will focus on parametric modeling. Starting with the classical linear regression, we will describe several families of estimators that work when considering high-dimensional data, where the classical least square estimator does not work. Model selection and model assessment will particularly be described.

In the second part, we shall focus on nonparametric methods. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for non parametric regression from independent to correlated training dataset. We shall also study some methods to avoid the overfitting in supervised learning.

This course will be followed by practical sessions with the R software.

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This course deals with

• Topic 1: convex analysis

• Topic 2: convex programming

• Basic notions: vector space, affine space, metric, topology, symmetry groups, linear and affine hulls, interior and closure, boundary, relative interior

• Convex sets: definition, invariance properties, polyhedral sets and polytopes, simplices, convex hull, inner and outer description, algebraic properties, separation, supporting hyperplanes, extreme and exposed points, recession cone, Carathéodory number, convex cones, conic hull

• Convex functions: level sets, support functions, sub-gradients, quasi-convex functions, self-concordant functions

• Duality: dual vector space, conic duality, polar set, Legendre transform

• Optimization problems: classification, convex programs, constraints, objective, feasibility, optimality, boundedness, duality

• Linear programming: Farkas lemma, alternative, duality, simplex method

• Algorithms: 1-dimensional minimization, Ellipsoid method, gradient descent methods, 2nd order methods

• Conic programming: barriers, Hessian metric, duality, interior-point methods, universal barriers, homogeneous cones, symmetric cones, semi-definite programming

• Relaxations: rank 1 relaxations for quadratically constrained quadratic programs, Nesterovs π/2 theorem, S-lemma, Dines theorem Polynomial optimization: matrix-valued polynomials in one variable, Toeplitz and Hankel matrices, moments, SOS relaxations

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The automatic processing of languages, whether written or spoken, has always been an essential part of artificial intelligence. This domain has encouraged the emergence of new uses thanks to the arrival in the industrial field of many technologies from research (spell-checkers, speech synthesis, speech recognition, machine translation, …). In this course, we present the most recent advances and challenges for research. We will discuss discourse analysis whether written or spoken, text clarification, automatic speech transcription and automatic translation, in particular recent advances with neural models.

Information access and retrieval is now ubiquitous in everyday life through search engines, recommendation systems, or technological and commercial surveillance, in many application domains either general or specific like health for instance. In this course, we will cover Information retrieval basics, information retrieval evaluation, models for information retrieval, medical information retrieval, and deep learning for multimedia indexing and retrieval

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Course content

Part 1: Access Control, or how to prevent unauthorized people from entering or accessing a system?

This part deals with:

• Cryptology for authentication and trust;
  • security needs : confidentiality, availability, integrity, non-repudiation (CAI/DICP);
  • cryptology primitives : private and public key ; trusted infrastructure (PKI and ledgers).
  • zero-knowledge protocols; secret sharing and multiparty computation / or Practical work with Open SSL
• Access control mechanisms (MAC, DAC, RBAC, ABAC) and their implementations
• The detection and remediation of security breaches such as intrusions and insider attacks
• The deployment of control filters in applications and proxies.

The presented approach is built on the model-driven security paradigm (MDS). It refers to the process of modeling security requirements at a high level of abstraction, and generating technical security implementations. Security models are transformed into enforceable security rules including the run-time security management (e.g. entitlements/authorisations). Three labs are planned:

• B4MSecure: we will apply a formal language with animation and model-checking facilities to identify security breaches in Access Control policies.
• Snort: in this practical session you will set a local environment to simulate two machines, the target machine and the attacker. You will learn how to create firewall rules, monitor your network and how to react when an attack is detected.
• Metasploit: you will discover technology intelligence for vulnerabilities through a practical session where you will reproduce an exploit to hack and take control over a web-based server.

Part 2: Overview of modern attacks on systems, protocols, and networks and countermeasures

This part is devoted to modern attacks carried out on the Internet scale, in particular attacks on the DNS system (Domain Name System), such as cache or zone poisoning attacks, reflection and amplification of DDoS attacks (Distributed Denial of Service), IP spoofing - the root cause of DDoS attacks, botnets (e.g., Mirai), domain generation algorithms used for command-and-control communications, modern malware (e.g., Emotet trojan, Avalanche), spam, phishing, and business email compromise (BEC) scams.

The module will discuss preventative measures and security protocols to fight modern attacks, such as DDoS protection services, IP source address validation (SAV) known as BCP 38, Sender Policy Framework, and DMARC protocols as the first line of defense against email spoofing and BEC fraud, and DNSSEC to prevent DNS manipulation attacks. It will also discuss large-scale vulnerability measurements (a case study of the zone poisoning attack) and the challenges of deploying current security technologies by the system and network operators.

This part will be concluded with a practical team assignment in which students will be divided into groups and will have to configure a secure system in a real-world environment. The goal is to secure their system against the various types of discussed attacks and exploit other groups' systems.

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Information processing capabilities from the computer side seem to be almost limitless now. However, these capabilities are only useful if users can easily and efficiently express their needs and obtain results. This course addresses Human-Computer Interaction as the process of optimizing humans' sensorimotor capabilities to maximize the communication bandwidth. We review current research on:

• User's sensorimotor capabilities
• Modeling human-computer interaction bandwidth
• Novels forms of efficient HCI:
  • AR, Handheld AR and head-mounted display AR
  • Interaction with non-anthropomorphic AI
  • Discoverability
  • Tangible & shape-changing interaction

Class schedule and notes

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Cette UE est obligatoire pour les étudiant.e.s inscrit.e.s dans le PT MSTIC.
Elle est constituée d'une projet tutoré et d'un MOOC consacré à l'éthique de la Science. 

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This course presents advanced methods and technics for Operations Research.

Reminder :

Linear Programming, Dynamic Programming, MIP modelling and BB
Complexity (P, NP, Co-NP)

Advanced MIP :

formulation, cuts, bounds
applications
lagragian relaxation
column generation
Benders decomposition
Solvers

Constraint Programming

Heuristics

local search
approximation algorithms

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The aim of this course is to provide a broad knowledge of fundamental problems in Combinatorial Optimization to show their algorithmic solutions and to derive min-max results on them. In order to achieve this goal a new object called a polyhedron is introduced. This polyhedral approach helps to shed new light on some classic results of Combinatorial Optimization.

Syllabus: Study of polyhedra associated to problems of Combinatorial Optimization ; Theory of blocking polyhedra ; Connectivity: shortest paths, spanning trees and spanning arborescences of minimum weight ; Flows: Edmonds-Karp Algorithm, Goldberg-Tarjan Algorithm, minimum cost flows ; Matchings: Hungarian method, Edmonds' Algorithm, Chinese postman problem; Matroids: greedy algorithm, intersection of two matroids ; Graph coloring ; Applications coming from various areas of Operations Research.

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The objective of this course is to present different techniques to handle uncertainty in decision problems. These techniques will be illustrated on several applications e.g. inventory control, scheduling, energy, machine learning.

https://moodle.caseine.org/course/view.php?id=1084

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The aim of this course is to learn how to use the structure of graphs and other discrete objects to obtain general results on them, and in particular efficient algorithms solving important problems.

We will cover the following topics:

• Structural Graph Theory: we will study the structure of important graph classes with nice algorithmic properties (planar graphs, interval graphs, comparability graphs, ...) and show several concrete problems that can be solved thanks to their structural properties.
• Graph Drawing: with the rise of Big Data, representing huge data sets is a fundamental challenge. Efficient ways to represent large graphs will be presented.
• Codes: we will see various codes (dominating, locating, identifying, ...) in graphs and their applications.
• Extremal combinatorics: the typical question in this field is "what global condition do we need to impose in some graph in order to make sure that some nice structure appears locally?" We will introduce a powerful tool called "the probabilistic method", an show how it can be applied to solve problems in this important area of research.

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This course offers the possibility for the students to gain some experience by facing open/difficult combinatorial problems.

The goal is to model and solve a combinatorial problem with direct industrial applications. We expect the students to take a variety of approaches (local search, compact/extended linear programming formulations, constraint programming, ...) and establish useful results (lower bounds, cuts, complexity,...).

The experimental results will be compared to the litterature (a known academic open benchmark will be available in this case) or will be validated by the industrial partner.

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Presenting models and tools for solving problems of company location and transport management in a supply chain.

Description:

This course presents two aspects: strategic design of a distribution network in a supply chain and tactical-operational aspects of network management.

For each aspect the course develops two key points:

• in network design, the choice of location and the establishment of services (production site, warehouse, logistics platform) and the synthesis of the transport network (connections / roads between production sites, warehouses, logistics centres)
• in network management, planning and modes of distribution transport (planning delivery rounds, number and types of vehicles).

For each of the three problem areas: location, delivery rounds and network design, students will study both modelling and techniques for accurate resolution (Branch-and-Bound) plus techniques for approximate resolutions (Lagrangian relaxation , metaheuristics) and how they are set up in commercial software (OPL-CPLEX Studio, Excel solver), or free software (COIN, lp-solve) or ad-hoc.

Evaluation :

Individual written exam (1/2 of the final mark) and group project (1/2 of the final mark).

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