English

As it is described in the title the CIMPA School on “Differential Galois Theory and Applications to Mechanics” will be devoted to the application of the differential equation coming from Mechanical problems using tools from differential algebra and specifically the Differential Galois group.

Very recently has appeared the techniques which enable the use of the differential Galois theory to study the integrability of Hamiltonian systems. These techniques known as Morales-Ramis-Simó theorems were originally used to study integrability of Hamitonian differential equations describing mechanical systems. Nowadays there exist many generalizations of this seminal work and application beyond classical mechanics.

The aim of the CIMPA school is to provide a complete introduction to the subject starting from the background in differential Galois Theory and classical Hamiltonian differential equation and ending with the newest progress in this area of research.

The lecturers for this CIMPA school are: David Blázquez-Sanz, Juan J. Morales-Ruiz, Guy Casale, María Przybylska, Jacques-Arthur Weil and Maria-Ángeles Zurro. The speakers are Jean-Pierre Ramis, Andrzej Maciejewski, Chara Pantazi, Teresa Crespo, Primitivo Acosta-Humánez The School starts with the opening speaker Jean-Pierre Ramis and will close with a panel about women in mathematics.

Official language of the school: English

Contact: https://ecimpa.spc.edu.do

The N'Djamena 2023 CIMPA research school aims to introduce graduate students and young mathematicians to mathematical and statistical tools to model complex systems. The goal is to prepare the participants to model concrete problems rooted in African realities by addressing timely challenges such as infectious diseases, drought or the effects of human migration on ecosystems. This school will present an opportunity for participants to expand their knowledge, find new axes of research and brainstorming with world-class scientific leaders on the exciting topic of mathematical modeling of complex systems.

Official language of the school: English and French

A blockchain is a digital list of data records, comprised by blocks which are organized in chronological order and are linked and secured by cryptographic proofs.

It started to secure digital documents from data tampering, and quickly lead to the creation of Bitcoin, the first decentralized electronic cash system, or cryptocurrency.

The novelty of blockchain technology is providing securily a decentralized, distributed and public digital ledger containing all previously confirmed transactions. Some experts consider that decentralization will avoid third party authorities such as banks, corporations, and insurance companies, changing dramatically the organizational structures in our global society.

In this school we will focus on two branches of blockchain technology. First, the development of the particular algorithms to execute transactions consensually, and second the mathematical tools and cryptographic protocols providing the security issues behind this technology.

The school is aimed to make the participants potential contributors to a blockchain society.

Official language of the school: English

The school aims to introduce graduate students and young researchers to the modern theory of non-associative algebras and their applications, with a focus on Lie algebras, Leibniz algebras, incidence algebras, genetic algebras and their deformations. The courses will be focused on some key topics in description of algebraic systems with certain properties, algebraic systems over positive characteristic, deformations of algebraic systems, classifications of algebraic systems. The participants will be provided with an introduction to the basic material and with the necessary background, before proceeding with the more advanced topics. The courses will survey a range of applications including classical methods of research in certain areas and applications in other modern areas, such as geometry, physics, biology and some other. Besides lectures, we are also planning sessions devoted to solving exercises and discussing open problems.

Official language of the school: English

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;

• hyperbolic geometry;
• probabilistic algebraic geometry;
• differential geometry.
The scientific program is divided into
• 5 courses of 5 lectures each, plus 8 sessions of exercises;
• 4 research seminars;
• 6 sessions in which the students will present their research.
The idea is to keep the level of the courses quite “introductory”, however, on topics that are subject of current research.
While the school is concerned with pure geometry, it will remain highly connected to applications: for example, sub-Riemannian geometry is applied in robotics, in neurophysiology and in medical imaging, or, the approach of probabilistic algebraic geometry promises connections with physics, biology and social science.
Although the courses are intended to give a panorama of different subjects in geometry, nonetheless they are interconnected.

Official language of the school: English

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:

• Asymptotic symmetries and integrable systems in gravity, by M. Cárdenas,
• Groupoids and stacks in generalized geometry, by M. Gualtieri,
• Cobordism and K-theory in the string swampland, by O. Loaiza,
• Perturbative quantum field theory, by K. Rejzner,
• The topology of T-duality, by T. Schick.

Additional introductory courses will be offered that are meant to fill in specific knowledge gaps for students of one discipline mathematics or physics, in the other discipline.

Official language of the school: English

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. Starting from basics, covering a large spectrum of topics in error-correcting codes, finite geometry and combinatorics, we aspire to give the participants an exposure to some of the most current research problems in this area.

The lectures will be supplemented by discussions/tutorial sessions that will provide the participants ample opportunities to work out problems, and clarify their doubts, with the lecturers who are leading experts in this area of research.

Official language of the school: English

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

fields. Applied mathematics plays a key rôle in these areas and the goal of this CIMPA school is to provide a high class training program. In order to have a good balance between diversity and deepness of topics we have selected the following key topics and fundamental solution methods as the focus of this school:

Topics:
1.Inverse problems/parameter identification, 2.Simulation of heterogeneous biomaterials, 3.Mathematical image processing.
Fundamental Solution Methods:
a. Optimization methods,
b. Multiphysics, multiscale, and multigrid methods, c. Regularization methods,
d. Composite Finite Elements..

Official language of the school: English

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:

1. 1)  Exploiting graph structure to efficiently solve combinatorial problems

2. 2)  Extending graph structural analysis to complex network studies

This 12 days program enriched with wonderful social activities will take place in the extremely stimulating environment of the Nesin Mathematics Village (Leelavati prize 2018:https://www.mathunion.org/imu-awards/leelavati-prize/leelavati-prize-20…), and will offer the possibility of starting new and innovative collaborations with young researchers.

Official language of the school: English

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. While the capabilities of such futuristic machine would render all of Elliptic Curves Cryptography and Pairing Based Cryptography suddenly worthless, Isogeny Based Cryptography seems to resist much better to the cryptanalytic powers of the quantum computer'' (Mathematics of Isogeny Based Cryptography, arXiv:1711.04062).

The first week of the school will be devoted to lay down the basis for the more advanced courses of the second week: Algebraic Number Theory, Finite Fields, Elliptic Curves, Graph theory (split between the 2 weeks). All this courses are introductory and there will have exercise/training session. In the second week we have one introductory course Elliptic curves over Finite fields and their Endomorphisms Rings, and two advanced courses Isogenies of Elliptic curves and Isogeny based cryptography. For the courses of the second week we plan to have hands on training sessions (i.e. on computers) as well as exercises sessions.

Official language of the school: English