2025

The research school's program is oriented in such a way as to make listeners benefit from the latest advances in the field of Machine Learning, Deep Learning and other data science techniques. The artificial intelligence and statistics community, and that of scientific computing have come together in recent years to give birth to new algorithms useful to both themes. Indeed, the first community by being interested in large-scale problems, appropriated a certain number of methods usually used by the second.

The goal of the summer school is to introduce students and junior scientists to the basics of the theory of elliptic curves and their applications in modern number theory and cryptography. The origins of the theory of elliptic curves go back to the 19th century, but it has become a central area of number theory only in the 20th century with the work of Mordell, Hasse, Weil and many others. Particularly prominent developments were the formulation of the conjecture of Birch and Swinnerton-Dyer, and the discovery of connections between elliptic curves and modular forms.

An $L$-function is a function defined additively by a Dirichlet series with a multiplicative Euler product.
The initial work of Dirichlet was generalized to number fields by Hecke and given an adelic interpretation by Tate
which paved the way to move from $GL(1)$ to higher degree $L$ functions associated to automorphic forms of $GL(n)$ for a general n .

The school aims to introduce graduate students and young researchers to modern algebra and its applications, with a focus on Tits-Kantor-Koecher construction, Jordan algebras, Rota-Baxter Lie algebras and some non-associative algebras, such as Lie algebras, Jordan algebras, automorphisms, derivations and cohomology of classical and quantum algebras and their applications in other areas of mathematics. The courses will deal with a description of certain algebraic systems, their classifi cation, and their connection with other algebraic systems.

The purpose of the school is to present a self-contained proof of a famous theorem of Serre in 1972. This theorem tells us that the representation associated to the Galois action on the p-torsion points of an elliptic curve is surjective for p great enough. This theorem had a very great impact in the field of arithmetic geometry and opened the field to numerous problems that are, for some, still open today. After introducing the students to the topics needed to understand the proof, illustrating the theory through exercises and sessions on computer, we will present the proof itself.

Les activités scientifiques prévues sont organisées sous forme de cours, travaux dirigés et travaux pratiques. On prévoit éventuellement des tables rondes (une ou deux) autour de la thématique de l’école le soir. La thématique s’articule autour de quelques outils mathématiques qui interviennent en intelligence artificielle. Il s’agit des statistiques, optimisation linéaire, analyse numérique.

Mathematical modelling in biology and related domains is becoming of increasing importance in the mathematical community as well as for biologists, physicians, engineers in environmental science from the point of view of possible applications.
The recent progress in the development and the analysis of these mathematical models have to be shared with Cuba and other countries of the Caribbean area.

The aim of this CIMPA school will be to familiarise graduate students and young researchers with the field of enumerative and analytic combinatorics, and to show its many connections to other areas, especially computer science.

Les mathématiciens sont sollicités par les médecins pour l’optimisation du traitement de maladies gravesoude gestes opératoires, par les généticiens pour le séquençage du génome, etc.
Cette école CIMPA a pour vocation de former des étudiants de masters et des jeunes chercheurs mathématiciens modélisateurs pouvant s’impliquer dans des secteurs d’activités variés.
Elle est caractérisée par une forte interaction entre cours fondamentaux et appliqués, ce qui permet de rassembler les connaissances théoriques et pratiques sur un problème concret.

The main focus of the school will be the mathematics around the L-functions and modular forms database (LMFDB). It will introduce the students to the main ideas and philosophies around modularity theorems, which connect algebraic curves and abelian varieties on one side with modular forms and L-functions on the other. Theoretically, such ideas are part of the Langlands program; the LMFDB can be seen as a way to make this program accessible and concrete by means of a huge treasure trove of examples.