Lectures for the probability session

Gane Samb Lo: Rehearsal in Foundations of probability Theory

1. A tour on probability laws on Rd (A measure theory and integration point of view, independence, dependence with Sklar’s Theorem). Focus on (i) Lebesgue-Stieltjes measures (ii) Kolmogorov Existence Theorem on Rd (iii) Diffeomorphic transforms. Study of Gaussian family and Generalized Hyperbolic Family.
2. Conditional expectation, continuous and stopping time
3. Introduction to general weak convergence theory and applications in special spaces
4. Feller-Levy-Lindenberg asymptotic theory of sums of random variables
5. Opening of asymptotic theory for dependent data (Mac-Leish approach)
6. Martingales (continuous and discrete)
7. Arbitrary product of probability spaces et canonical stochastic processes

Youssef Ouknine: Stochastics processes driven by Levy processes and applications

1. Arbitrary Product Probability measure (Kolmogorov, Skorohod and Canonical construction of Markov processes theorems)
2. Advanced Properties of Brownian and Poisson processes
3. Levy processes
4. Stochastic calculus with respect to Levy process
5. Applications

Lectures for the statistics session

At the end of these two courses, the student will master the classical tools of learning for decision support and be able to put into practice modern high dimensional modeling techniques from machine learning.

Jean-François Dupuy: Asymptotic Statistics

1. Recall of probability, different modes of convergence, symbols
2. Delta method and applications
3. Classical estimation methods (method of moments, maximum likelihood)
4. M and Z estimators (definition, consistency, asymptotic normality)
5. Useful notions of empirical processes (Glivenko-Cantelli and Donsker theorems, GC and Donsker classes, entropy and bracketed entropy) will be introduced as we go along
6. Some examples in regression (generalized linear models), application to handling missing data
7. Illustrations on R (convergences, asymptotic distributions) will be done throughout the course.

Valérie Gares: Statistical learning

1. Decision discriminant analysis
2. High dimensional variable selection and penalization
3. Decision trees
4. Non-parametric learning in regression: piecewise polynomials, splines, kernels Model aggregation
5. SVM algorithm
6. Neural networks
7. Deep learning
8. Practical application with R software

Lectures for the Modeling and numerical analysis of some Complex systems session

Christophe Berthon: Introduction to fortran 90
This is a series of computer tutorials to introduce participants to fortran 90.
Raphael Loubere: Numerical methods dedicated to solve hyperbolic models of Partial Differential Equations

1. introduction to numerical methods
2. finite volumes for simple models: linear, scalar and non-linear scalar
3. finite volumes for more complex PDE systems: Saint Venant, Euler, etc.
4. extension of the finite volume method: the discontinuous Galerkin method.Numerical methods dedicated to solve hyperbolic models of Partial Differential Equations

Mehdi Badsi: Variational formulations and finite element methods

1. Reminders of Sobolev space in one dimension
2. Variational formulations for boundary problems
3. Variational approximation methods
4. Finite element methodFormulations variationelles et méthodes éléments finis

Mehdi Badsi: Mathematical and numerical tools in plasma physics

1. Introduction
2. Models
3. Transport and Vlasov equation
4. Numerical discretization

Lectures for the Introduction to (nonlocal) PDEs and semigroup theory

Diaraf Seck: Analysis of PDEs

1. Introduction to different kind of PDEs
2. Topological methods in PDEs
3. Variational methods and critical point theory
4. Ky Fan von Neumann Theorem
5. Maximum Principle, Perron's method and regularity Theory

Xavier Ros-Oton: Introduction to nonlocal problems

1. Introduction to PDEs driven by Lévy processes
2. Regularity theory of nonlocal operators

Ousmane Seydi: Semigroup Theory and Biological systems

1. Modelling of some epidemic problem
2. Introduction to Semigroup Theory