English

This CIMPA School will introduce participants to selected topics of Mathematics with direct applications in privacy and security issues. The presentations will include the necessary background in order to understand the main concepts, as well as many motivating examples of the application of these topics to particular security and privacy issues, together with open problems in the corresponding areas. The aim is to motivate a group of bright students to continue working in those areas at the doctoral or postdoctoral level. The school will conclude with presentations by researchers and students.

The School will take place at the Faculty of Mathematics and Computer Science of the University of Havana. Founded in 1728, the University of Havana is the oldest and most important university in Cuba. The Faculty of Mathematics and Computer Science has broad experience in the organization of international events, including other CIMPA research schools.

Lattices play a central role in number theory and its applications. The aim of this school is to introduce participants to the ubiquity of lattices in number theory, algebra, arithmetic algebraic geometry, cryptography and coding theory. The theory of lattices will be developed from its very beginning and the basic notions required for the applications in number theory, algebra, arithmetic algebraic geometry will be provided. Appearances of lattices that we intend to cover include: The natural lattices structures of Mordell-Weil groups and unit groups. Lie algebra root lattices. The lattice basis reduction algorithm "LLL", which as many applications to many areas of mathematics and finally the construction of the famous Leech lattice. On the applied side we plan to cover constructions of good error-correcting codes and of good sphere packings via dense lattices.

This Research School is addressed graduate students and young researchers in Geometry, Topology and Mathematical Physics, although undergraduates and senior researchers in other areas may profit of the introductory lectures too, and are warmly invited to join the School.

The main goal is to provide participants with a solid background in Algebraic Topology, and more specifically in Rational Homotopy Theory. This will be used as the starting point and common framework for the presentation of current research topics including operads, string topology, applied topology, and algebraic geometry.

We planned 30 hours of courses, 14 hours of exercises or/and discussion sessions, and 6x20mn-communications. The first and introductory course on the subject will be taught in French, while the other courses will be taught in English.

We have maximized the number of senior speakers in the courses so providing the PhD students with the opportunity of creating connections with well established researchers upon they may build a future research career.

Combinatorics is at the center of many areas in pure and applied mathematics. This summer school will focus on the versatility of combinatorics to shed light on problems in algebra, geometry, optimization, and mathematical physics. We will also discuss how algebraic and geometric perspectives can help us understand purely combinatorial objects.

The main goal of the school is to bring young mathematicians from Colombia and other Latin American countries into close contact with each other and with world experts in various fields of combinatorics. This is part of a long term goal to build a strong regional community of mathematicians in combinatorics and related fields.

Mathematical modeling and computer simulation tools have been growing enormously in the fields of mathematics and physics applied to biophysics, biology, biochemistry and bioengineering. The reasons for this large attention of the mathematical community are multi-faceted. Among them, the mathematical modeling in biology and medicine is one of the most important challenges of mathematics applied to scientific problems. It has been shown to be a substantial tool for the investigation of complex biophysical phenomena, such as cancer. Tumor growth continues and will continue to challenge oncologists. The pace of progress has often been slow, in part because of the time required to evaluate new therapies. To reduce the time to approval, new paradigms for assessing therapeutic efficacy are needed. This requires the intellectual energy of scientists working in the field of mathematics and physics, collaborating closely with biologists and clinicians. This essentially means that the heuristic experimental approach, which is the traditional investigative method in the biological sciences, should be complemented by a mathematical modeling approach. This school is addressed for researchers, doctoral students, students of Master’s degree level and talented undergraduate students to acquire a basic training in that field. This school will cover a wide class of mathematical models and applications in issues related to tumor growth, population dynamics, pattern formation and their implications in developmental cancer biology. The school will constitute a very interesting thematic opening for the young researchers. It will also be a tremendous tool of exchange between world-wide researchers and Cuban/Caribbean researchers in applied mathematics, theoretical biology and medicine.

This research school consists on an introduction to the geometry of hyperbolic groups and their representations. The global objective is to understand how the intrinsic geometry of the hyperbolic group interacts with the geometry of the target group.

When the hyperbolic group is a surface group, several target groups are subject of major deep results: Teichmüller-Thurston Theory (associated to the groups PSL_2(R) and PSL_2(R) ) and the ‘absence’ of geometric meaning due to Goldman for SU(2), just to name a few. Representations of a general hyperbolic group into a semisimple higher rank Lie group is a current topic of research.

The school consists on 5 mini-courses to be held in english: an introductory course, a course on the large scale geometry of hyperbolic groups, Quasi-Fuchsian representations, representations of surface groups on SU(2) and a course on matrix groups over a p-adic field.

The CIMPA research school "Homological Methods, Representation Theory and Cluster Algebras" is intended for PhD and postdoctoral students from Argentina and surrounding South American countries and specifically for students in the region of Mar del Plata. As a sequel to the CIMPA research school "Homological Methods and Representations of Non-Commutative Algebras" held in Mar del Plata in 2006 and to the "South-American Meeting on Representations of Algebras and Related Topics" held in Mar del Plata in 2010, this research school offers courses and research talks on advanced research topics related to the representation theory of finite-dimensional algebras, to Auslander-Reiten theory, to homological algebra and to cluster algebras.

This school is an introduction to subjects of current interest related to moduli, a fundamental aspect of algebraic geometry, with emphasis on moduli of curves. The propose of this school is to introduce young mathematicians to the foundations of the theory, some of its major developments and tools used for its study and also a few glimpses on some more advanced topics. Each course will start with classical material and will take the audience to the frontier of research in the subject. Considering the wide range of topics to be covered, the School will be directed at students and researchers in mathematics and physics interested in the subject.

Number theory and cryptography are important topics between mathematics and computer science. Many industrial products are based on the application of these topics to real word. Number theory and cryptography are deployed to secure sensitive communications, to protect information systems, to secure bank transactions, online shopping, satellite and mobile phone communications, and to ensure storage of medical, personal and military data. In short, number theory and cryptography are frequently used in applications where security of digital data is a concern. The goal of this school is to introduce the students and young researchers to the theory and the practice of the algorithmic number theory with applications to cryptography. Another goal of this school is to promote the development of number theory and cryptography in West and North Africa.

In recent years, toric geometry has experienced a rapid growth with a wealth of new results and connections with other areas of mathematics like arithmetic geometry and number theory, dynamical systems, non­ Archimedean and tropical geometry, mirror symmetry and birational geometry, This adds to the more classical connections to combinatorics, computer algebra and singularity theory among others, and applications to areas of science like mathematical biology and chemistry. At the same time, the concrete and explicit nature of toric varieties makes them a perfect entrance door to higher dimensional algebraic geometry.

The main purpose of this research school is to motivate and train Latin American students and young researchers in these subjects. It will consist of 6 courses of 6h each, all of them delivered by well­ recognized specialists from Argentina, USA, France and Spain. These courses will cover a basic introduction to toric varieties, followed by more advanced courses on the topological aspects, singularities, dynamical systems, non­-Archimidean and tropical geometry, and arithmetic geometry of toric varieties. In addition, we plan a series of 10 survey talks delivered by members of our scientific committee and other invited researchers from Latin America, USA and Spain.