English

The school will present modern tools of number theory and algebra, with a focus on their use in cryptography. Although cryptography has a long history, it has developed during the 20th century into a modern science with the help of computer science and mathematical tools coming from algebra, number theory, combinatorics, geometry. The minicourses will present various aspects of the subject: elementary and algebraic number theory, elliptic curves, primality tests, representation theory, lattices, computational aspects, applications to cryptography. Training sessions will be devoted to PARI/GP and SageMath, two efficient computer algebra systems which are open-source and widely used by the community.

The school aims at promoting and developing these new research areas for teachers and researchers in Haiti, attracting new students and developing scientific collaborations with other countries (France, USA, the Caribbean,...), including co-directions of Ph.D. students.

The aim of this is school to introduce the participants to the arithmetic and computational aspects of the theory of elliptic curves.

We will develop the theory of elliptic curves from its very beginning also providing an introductory course on algebraic curves and the Riemann Roch theorem. Topics that will be covered include: basic geometric and arithmetic results for elliptic curves over number felds and over fnite felds, the Mordell- Weil theorem for elliptic curves, Galois representations attached to elliptic curves, and the Birch and Swinnerton Dyer Conjecture.

On the computational side we will cover applications of elliptic curves such as the discrete logarithm problem, elliptic curves cryptosystems and the construction of elliptic curve over fnite felds with a prescribed order. 

The school is intended for students in mathematics, physics and engineering, graduate or undergraduate in the late years of their bachelor. The participants will be offered geometric notions of some mathematical theories useful for understanding problems related to mechanics and control theory. The programme of the school consists in 7 courses of 6 hours each, lectured by experts on the subjects, and is divided in three parts. The first part of the school is devoted to introduce the students to the tools needed from the geometry of smooth manifolds and their relation to mechanics. The connection of optimal control theory with Lie group theory and an introduction to integrable Hamiltonian systems make the second part. The last part of the school is thought to further develop the past content with discrete mechanics and its application to numerical simulations, non-holonomic systems and Poisson geometry. / La escuela está orientada a estudiantes de matemáticas, física e ingenierías, graduados o en los últimos años de su carrera. A los participantes se les ofrecerá nociones geométricas de algunas teorías matemáticas útiles en la comprensión de problemas relacionados con la mecánica y la teoría de control. El programa de la escuela consiste en 7 cursos de 6 horas cada uno, impartidos por expertos en las temáticas, y está dividido en tres partes. La primera parte de la escuela está destinada a introducir a los estudiantes en las herramientas que se precisan de la geometría de variedades diferenciables y su relación con la mecánica. La conexión de la teoría de control óptimo con los grupos de Lie y una introducción a los sistemas hamiltonianos integrables constituyen la segunda parte. La última parte de la escuela está pensada para seguir desarrollando los contenidos anteriores con la mecánica discreta y su aplicación a la simulación numérica, sistemas no-holónomos y geometría de Poisson. 

The objective is to train young scientists and introducing them to recent development of analysis and geometry on complex homogeneous domains.