2023

Purpose of the school is to bring together some research top- ics in geometry together with their interconnections, in particular:

• sub-Riemannian geometry;

• Poisson geometry;

This school, whose topics lie at the threshold of geometry, topology, algebra and quantum field theory, is the eleventh of a series of summer schools organized in Colombia every other year since July 1999. It is addressed to both physicists and mathematicians with a master’s level in either of the fields and offers courses on the following topics:

The study of error-correcting codes has posed a large number of intriguing and important questions in several areas of mathematics, such as algebra, number theory, combinatorics, algebraic geometry etc. In addition it is vital tool in making transmission of data robust against noise. In this school, we are going to present several lectures on the usage of techniques in finite geometry in the study of error-correcting codes.

In modern medicine, reliable computer simulation and medical imaging which are based on mathematical algorithms and numerical methods gain significantly importance for diagnosis and individualized therapies. In order to keep up with these developments it is of utmost importance to provide state-of-the art tertiary education and training in these important

Since the works of Watts and Strogatz (1998) on one hand, and Barabási and Albert (1999) on the other, graph theory has become a major mathematical field that provides a framework to handle network properties theoretically and enables us with very powerful tools to model and solve problems on networks. Understanding their graph structure is a key point in deriving efficient algorithms in large networks. In this school, we will cover theoretical aspects of graph structure analysis as well as applications on complex network studies with 9 lectures in two main axes:

Isogenies of elliptic curves, or more generally of abelian varieties, are surjective homomorphism having finite kernel and they play an important role in the study of arithmetic and geometric properties of elliptic curves. Moreover in recent years there has been an increasing interest in isogeny of elliptic curves from cryptographers. The main reason lies in quantum computer as Luca de Feo eloquently puts it: ''The main reason for this is the sudden realization by the cryptographic community of the very possibly near arrival of a general purpose quantum computer.

The school will address a selection of topics of importance in modern research in combinatorics, representation theory, higher structures via the prism of geometry, and their interrelation. Specific themes to be covered are geometric representation theory, quantum groups, the use of higher structures to study the geometry of various spaces, categorification, combinatorial aspects of geometry and cluster algebras.

Langue officielle de l'école : anglais

Two centuries after its origin, Complex Analysis remains a vi- brant and fruitful field of research. Its applications go well beyond the realm of Mathematics and range from Physics to Engineering. Current research in Complex Analysis blends techniques from a variety of Mathematical areas and thus requires a broad expertise. Following this philosophy, the school will expose students to a wide spectrum of fields interacting with Complex Analysis. Special emphasis will be put on Differential Geometry, Partial Differential Equations, Dynamical Systems and Algebraic Geometry.

Cette École, résolument pluridisciplinaire, a pour but l’apprentissage, par les étudiants, des outils mathématiques propres à aborder des questions de développement durable comme l’urbanisme, l’épidémiologie, le climat. Ces outils inclueront la statistique, l’Analyse multi- échelle, le traitement d’images, l’Analyse des données et des équations différentielles.