Mathematics and Applied Mathematics - Master’s Year 1

UE fondamentales

• Discrete processes
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  Discrete processes

  Ects : 8

  Lecturer :

  • JULIEN CLAISSE

  Total hours : 79

  Overview :

  • Conditional expectation: definition and construction, properties
  • Processes: filtrations, stopping times, sigma-field of the past
  • Martingales: definition, stopping theorems, convergence theorems, maximal inequalities
  • Markov chains: definition, random inductions, properties, Markov properties, recurrence and transience, invariant measures, ergodic theory

  Learning outcomes :

  Discrete-time stochastic processes, including conditional expectation, martingales, and Markov chains and their long-term behavior.

• Optimization
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  Optimization

  Ects : 6

  Lecturer :

  • Idriss MAZARI-FOUQUER

  Total hours : 58.5

  Overview :

  The course focuses on finite-dimensional optimization problems and their numerical resolution.

  • Basic concepts: existence of optimisers; optimality conditions; convexity and strict convexity.
  • Unconstrained optimisation: gradient descent (principles, convex case, extensions); Newton’s method; numerical implementations.
  • Constrained optimisation: Lagrange multipliers for equality and inequality constraints; KKT conditions; numerical methods; duality.
  • Introduction to optimal control: discrete-time problems, dynamic programming principle and Bellman equations. Possible brief outlook toward calculus of variations and continuous-time optimal control.

  Recommended prerequisites :

  Optimisation dans R^n sans contraintes.

  Learning outcomes :

  Finite-dimensional optimization problems and their numerical resolution.

  Assessment :

  Examen sur table (mi-semestre et fin de semestre).

• Functional analysis
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  Functional analysis

  Ects : 10

  Lecturer :

  • ERIC SERE

  Total hours : 78

  Overview :

  The course presents functional analysis methods for solving partial differential equations.

  • Compactness in metric spaces; Riesz compactness theorem; Arzelà-Ascoli theorem.
  • Hahn-Banach theorem, Baire category theorem, theorem of Banach-Steinhaus, open mapping theorem, closed graph theorem.
  • Hilbert spaces: projection, Hilbert bases, Riesz isomorphism theorem (reminders), Lax-Milgram theorem, weak convergence
  • Spectral theory: spectrum of bounded operators in Banach spaces, compact and self-adjoint operators in Hilbert spaces, spectral theorem
  • Sobolev spaces in one space dimension.

  Learning outcomes :

  Classical results in functional analysis and some of their applications.

UE optionnelles (choisir 1 option)

• Geometry and dynamics
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  Geometry and dynamics

  Ects : 6

  Lecturer :

  • ANNA FLORIO

  Total hours : 39

  Overview :

  This course provides an introduction to manifolds and to the geometric viewpoint on dynamical systems and differential equations.

  • Reminders of differential calculus, local inversion
  • Manifolds : local coordinates, examples (parametrization, equation : sphere, torus), tangent vectors and space, maps, derivative
  • Differential equations (flow, phase portrait in dimension 2, conjugacy). Examples: linear differential equations in the plane, gradient flows and optimization problems, geodesic flows and mechanical problems
  • Equilibria of autonomous equations and their stability, periodic orbits and Poincaré maps.

  Learning outcomes :

  Introduction to manifolds and the geometric viewpoint of dynamical systems.

• Modèles statistiques
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  Modèles statistiques

  Ects : 6

  Total hours : 58.5

UE fondamentales

• Calculus of variations
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  Calculus of variations

  Ects : 5

  Lecturer :

  • PAUL PEGON

  Total hours : 39

  Overview :

  • Introduction to variational problems in infinite dimension
    • Classical examples: geodesics, brachistochrone, Ramsey growth model, Bolza problem
    • Relation to optimal control
    • Euler-Lagrange equations : formal derivation, rigorous derivation when there is a smooth solution and resolution
    • Existence of solutions : direct method in the Calculus of Variations.
  • Direct method, weak convergence and weak semicontinuity
    • Weierstrass theorem
    • Weak and weak-* topology, relation with convexity, compactness
    • Weak lower semicontinuity of integral functionals of order 0
  • Convex analysis and optimization
    • Optimization of extended-real-valued functions
    • Convex sets: geometric and topological properties, Hahn–Banach
    • Convex functions: definition, regularity, subgradients and subdifferentials, convex conjugate, convex duality
  • Integral functionals of order 1
    • Sobolev spaces (in dimension 1)
    • Continuous and compact embeddings
    • Semicontinuity of integral functionals of order 1
    • Euler-Lagrange equations : weak formulation, regularity a posteriori
    • Convex duality and applications

  Learning outcomes :

  Introduction to the calculus of variations, with a focus on convex and one-dimensional variational problems.

• Continuous processes
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  Continuous processes

  Ects : 10

  Total hours : 78

  Overview :

  • Part I. Continuous-Time Markov Chains (15-18 hours)
    • Càdlàg and increasing processes: definition. Poisson process (on RR) and counting processes.
    • Continuous-time Markov chains (countable state space): embedded Markov chain, jump times, Kolmogorov equations, generator, recurrence vs. transience, invariant measures.
    • Possible applications (in lectures and/or tutorials): M/M/s queues, birth and death chains, branching processes.
  • Part II. Brownian Motion and Diffusions (21-24 hours)
    • Continuous processes: definition. Random continuous functions. Brownian motion: construction of Brownian motion (as a Gaussian process), existence of a continuous version (via Kolmogorov’s criterion or admitted). Fundamental properties.
    • Introduction to stochastic calculus (continuous martingales, Itô integral and Itô’s formula) and stochastic differential equations. One may restrict to the one-dimensional case for simplicity.
    • Application in physics: Langevin equation (in lectures or tutorials).
    • Application in finance and asset pricing: Black-Merton-Scholes model.

  Learning outcomes :

  Continuous-time Markov chains and their applications, Brownian motion, stochastic calculus, stochastic differential equations, and applications to physics and finance.

• Preparation to pure and applied research
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  Preparation to pure and applied research

  Ects : 5

  Overview :

  This course consists of a year-long project carried out in groups of one to three students on an assigned topic, under the supervision of a faculty member or a professional. It is intended as an introduction to research or to the application of research methods to the resolution of a concrete problem. The work concludes with the submission of a written dissertation and an oral defense during which the results are presented and discussed.

  Learning outcomes :

  Students will develop the ability to conduct a supervised research project or apply research methods to a concrete problem, and to present their work effectively in both written and oral form.

UE optionnelles (choisir 2 options)

• Numerical methods
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  Numerical methods

  Ects : 5

  Lecturer :

  • GABRIEL TURINICI

  Total hours : 40.5

  Overview :

  FRENCH VERSION ((ENGLISH VERSION below): Volume horaire détaillé : CM : 16h30, TD : 12h00, TP : 12h00

  • Introduction
  • Équations Différentielles Ordinaires : Euler Implicite, Runge Kutta, consistance, stabilité, A-stabilité
  • Applications des EDO : épidémiologie
  • Calcul automatique de dérivée (back-propagation) et contrôle: graphe computationnel, différentiation automatique
  • Application du calcul de dérivée: réseaux neuronaux et deep learning, contrôle
  • Équations Différentielles Stochastiques : Euler Maruyama, Milstein
  • Applications de EDS: calcul d'options en finance sur modèle log-normal

  ENGLISH VERSION: Detailed hourly volume: CM: 16:30, TD: 12:00, TP: 12:00

  • Introduction
  • Ordinary Differential Equations: Implicit Euler, Runge Kutta, Consistency, Stability, A-Stability
  • Applications of ODE: Epidemiology
  • Automatic derivative calculation (back-propagation) and control: computational graph, automatic differentiation
  • Application of derivative calculus: neural networks and deep learning, control
  • Stochastic differential equations: Euler, Maruyama, Milstein
  • Applications of EDS: calculation of options in finance on log-normal model

  Require prerequisites :

  python, algèbre matricielle,

  Learning outcomes :

  (FR) : Présentation de méthodes de résolution numérique des problèmes d’évolution et d’éléments d’analyse numérique. Cours théorique mais aussi une forte partie implementation (en python).

  (EN) : Presentation of numerical methods for solving evolution problems and elements of numerical analysis. A theoretical course with a strong implementation component (in Python).

  Learn more about the course :

  turinici.com

  Bibliography-recommended reading

  site de Gabriel Turinici (aller au cours en question)

• Statistiques non paramétriques
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  Statistiques non paramétriques

  Ects : 5

  Lecturer :

  • LAETITIA COMMINGES

  Total hours : 39

  Overview :

  1. Introduction et rappels
  2. Estimation de la fonction de répartition
  3. Tests robustes
  4. Estimation de densités par estimateurs à noyau
  5. Régression non paramétrique

  Learning outcomes :

  Décrire les méthodes d’analyse statistique qui permettent de s’affranchir de la connaissance d’un modèle de forme trop contraint; prise de conscience des hypothèses de modélisation.

• Machine learning
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  Machine learning

  Ects : 5

  Lecturer :

  • GABRIEL TURINICI

  Total hours : 39

  Overview :

  1. Examples and machine learning framework: applications, supervised and non-supervised learning
  2. Useful theoretical objects: predictors, loss functions, bias, variance
  3. K-nearest neighbors (k-NN); Higher dimensions and Curse of dimensionality
  4. Regularization in high dimensions: ridge and lasso (for linear and logistic models)
  5. Stochastic Optimization Algorithms used in machine learning: Stochastic Gradient Descent, Momentum, Adam, RMSProp
  6. Naive Bayesian classification
  7. Deep learning through neural networks : introduction, theoretical properties, practical implementations (Tensorflow, PyTorch depending on acumen)
  8. Generative and non-supervised learning: k-means

  Coefficient : cf. CC

  Require prerequisites :

  Probability (including

  conditional expectation

  ), statistics (undergraduate / L3 level), numerical analysis.

  Learning outcomes :

  Introduction to statistical learning, particularly in a high-dimensional context, including baseline algorithms (k-NN,...) and modern approaches in deep learning (neural networks).

  Bibliography-recommended reading

  See site of the course (site of the teacher); also see textbook by G. Turinici (cf. Amazon)

• Introduction to partial differential equations
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  Introduction to partial differential equations

  Ects : 5

  Total hours : 39

  Overview :

  This course introduces the functional framework of partial differential equations, including distribution theory and Sobolev spaces. It then studies the Laplace, heat, and wave equations, together with analytical and numerical methods for their resolution.

  • Distribution theory and Sobolev spaces (~5 weeks)
    • The space of test functions and of distributions: examples (locally integrable functions, Dirac mass, principal value distributions, etc.) ; jump formula (e.g. Rankine–Hugoniot condition in problem sessions) ; support of a distribution ; convergence of sequences ; operations on distributions ; applications (fundamental solution of the Laplacian, representation of solutions to -?u=f in the whole space when f has compact support).
    • Tempered distributions: Schwartz space ; Fourier transform on tempered distributions (Plancherel theorem, inversion formula, etc.) ; important examples.
    • Sobolev spaces (Hilbert framework): spaces H^s(?) (definition, Fourier characterization when ?=R^d, simple case of compact embedding, Rellich theorem) ; the space H^1_0(?) and Poincaré inequality.
  • Laplace, heat, and wave equations (~7 weeks)
    • Laplace equation: solutions in a bounded domain, Lax–Milgram theorem ; solutions (definitions, existence and uniqueness) of -?u=f in H^1_0 ; H^2 regularity (optional) ; eigenfunction basis of the Laplacian.
    • Heat equation: Fourier method in the whole space (possible reminder) ; Galerkin method ; explicit resolution using the eigenfunction basis of the Laplacian.
    • Numerical simulation of the Laplace and heat equations: time and space discretization ; implicit and explicit Euler schemes ; notions of stability ; consistency ; and convergence.
    • Wave equation: modeling ; particular solutions via Fourier methods and separation of variables ; light cone ; classical solutions ; weak solutions ; superposition principle.

  Learning outcomes :

  Introduction to the fundamental partial differential equations — Laplace, heat, and wave — within the functional framework of distribution theory and Sobolev spaces.

Certificat

• SAS, Excel, Matlab
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  SAS, Excel, Matlab

  Lecturer :

  • JEROME LEPAGNOL

  Total hours : 15

  Overview :

  Apprentissage de SAS, Excel, Matlab.

  Learning outcomes :

  Mise à niveau sur les logiciels SAS, Excel, Matlab, susceptibles d’être utilisés en projet et souvent exigés pour les stages.

  Assessment :

  QCM en fin de cours