The idea of this new project is to trainfinishing-Master and PhD students from Africaduring a two weeks school followed by a short conference. The instructors will not only reinforce the knowledge of the participants in some topics important for their PhD project throughlectures, tutorials and working groups, but will also offer mentoring during the following years. There will be two topics: Analysis of PDE and Probability and Statistics. Each group will have few participants (~10) so the event will have a strong human component: through their active and stimulated participation,the students will work in groups, start their network, and have many informal scientific interactions with the professors.

## What will students get out of it?

- an in-depth knowledge of the concepts, notions and perceptions of a research topic
- experience with several of the most commonly used research methods and skills for conducting research in the area of a research topic
- start working in groups on research level questions
- ability to report research findings in writing and orally at an academic level
- start a research network
- mentoring during the PhD thesis

## Format of the event:

- Week 1: basic courses + basic exercises (possibly on computers) to reinforce the knowledge in the topic for the students at master level; students will be split into smaller groups of 2, 3 to work on the exercises,
- Week 2: more advanced lectures for everyone and exercises sessions for the students at master level with the supervision of a PhD student. Working groups on research problems for the PhD students + preparation/rehearsal of their talks for the conference,
- Week 3: conferences on 3 days with talks by the instructors + talks by the PhD students.

## Location:

The courses will all take place at AIMS Senegal.

## Organization committee:

- Sophie Dabo-Niang (Université de Lille)
- Mouhamed Moustapha Fall (AIMS Senegal)
- Christophe Ritzenthaler (Université de Rennes 1 and Université Côte d'Azur, France)

## Instructors for the Analysis of PDE group (* for the coordinators)

- Mehdi Badsi (Université de Nantes): Variational formulations and finite element methods; Mathematical and numerical tools in plasma physics
- Christophe Berthon* (Université de Nantes): Introduction to fortran 90
- Raphaël Loubere (Université de Bordeaux): Numerical methods dedicated to solve hyperbolic models of Partial Differential Equations
- Xavier Ros-Oton (TBC) (Universitat de Barcelona): Introduction to nonlocal problems
- Diaraf Seck* (Université Cheikh Anta Diop, Sénégal): Analysis of PDEs
- Ousmane Seydi (Ecole Polytechnique de Thiès): Semigroup Theory and Biological systems
## Instructors for the Probability and Statistics group (* for the coordinators)

- Jean-Francois Dupuy* (Université de Rennes 1, France): Asymptotic Statistics
- Valérie Garès (Université de Rennes 1, France): Statistical learning
- Gane Samb Lo* (Université Gaston Berger, Sénégal): Rehearsal in Foundations of probability Theory
- Youssef Ouknine (Université de Marrakech, Marroco): Stochastics processes driven by Levy processes and applications

Lectures for the probability sessionGane Samb Lo: Rehearsal in Foundations of probability Theory

- 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.
- Conditional expectation, continuous and stopping time
- Introduction to general weak convergence theory and applications in special spaces
- Feller-Levy-Lindenberg asymptotic theory of sums of random variables
- Opening of asymptotic theory for dependent data (Mac-Leish approach)
- Martingales (continuous and discrete)
- Arbitrary product of probability spaces et canonical stochastic processes
Youssef Ouknine: Stochastics processes driven by Levy processes and applications

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

## 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.

Lectures for the statistics session

Jean-François Dupuy: Asymptotic Statistics

- Recall of probability, different modes of convergence, symbols
- Delta method and applications
- Classical estimation methods (method of moments, maximum likelihood)
- M and Z estimators (definition, consistency, asymptotic normality)
- 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
- Some examples in regression (generalized linear models), application to handling missing data
- Illustrations on R (convergences, asymptotic distributions) will be done throughout the course.
Valérie Gares: Statistical learning

- Decision discriminant analysis
- High dimensional variable selection and penalization
- Decision trees
- Non-parametric learning in regression: piecewise polynomials, splines, kernels Model aggregation
- SVM algorithm
- Neural networks
- Deep learning
- Practical application with R software

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

- introduction to numerical methods
- finite volumes for simple models: linear, scalar and non-linear scalar
- finite volumes for more complex PDE systems: Saint Venant, Euler, etc.
- 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

- Reminders of Sobolev space in one dimension
- Variational formulations for boundary problems
- Variational approximation methods
- Finite element methodFormulations variationelles et méthodes éléments finis
Mehdi Badsi: Mathematical and numerical tools in plasma physics

- Introduction
- Models
- Transport and Vlasov equation
- Numerical discretization

Lectures for the Introduction to (nonlocal) PDEs and semigroup theoryDiaraf Seck: Analysis of PDEs

- Introduction to different kind of PDEs
- Topological methods in PDEs
- Variational methods and critical point theory
- Ky Fan von Neumann Theorem
- Maximum Principle, Perron's method and regularity Theory
Xavier Ros-Oton: Introduction to nonlocal problems

- Introduction to PDEs driven by Lévy processes
- Regularity theory of nonlocal operators
Ousmane Seydi: Semigroup Theory and Biological systems

- Modelling of some epidemic problem
- Introduction to Semigroup Theory

Deadline for application: May 31.