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.