2025

The relation between modular forms and their corresponding L-functions with various disciplines of mathematics has undergone significant evolution in the past century due to the critical role these complex analytical functions play in resolving essential problems and conjectures. The connection of modular forms and their L-functions with number theory, elliptic curves, representation theory, and algebraic geometry, among others, have resulted in diverse generalizations in different directions.

This summer school focuses on number theory and its practical applications in cryptography and coding theory. Broadly speaking, number theory investigates properties of integers, including primes and solutions of Diophantine equations. Despite being a very ancient branch of mathematics, number theory very much remains a dynamic branch of mathematics, with ongoing research uncovering impressive results every year, and with new puzzling questions regularly surfacing.

The goals of the research school we propose are: first, to gather young researchers in order to provide them the basics on Leavitt path algebras, Hochschild (co)homology, K-theory and related topics and also a glimpse of the state of art in the ongoing research carried out within the fields which comprise the subject of Leavitt path algebras, Hochschild (co)homology and K-theory; second, to provide the audience a general view of the results which have been achieved; and, finally, to give a broad picture of some of the research lines which are currently being pursued.

Mathematical and statistical modeling in oncology is a multidisciplinary field that applies mathematical and statistical techniques to understand, describe, and predict various aspects of cancer biology, epidemiology, and treatment. It plays a crucial role in advancing our understanding of cancer, optimizing treatment strategies, and informing healthcare decision-making.

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.