English

The summer school will be dedicated to finite point configurations and rigidity, Erdos problems in discrete geometry and frame theory, the Falconer distance conjecture  in  geometric  measure  theory,  discrete  integrable  systems  and connections between these topics.  Participants will  be introduced to various open problems and possible research projects  in these very active research areas.

Langue officielle de l'école : anglais

One of the fascinating bridges between  classical  mechanics  and quantum physics  is expressed  in the fact that the length spectrum  and the Laplace  spectrum  of a closed Riemannian  manifold determine  each other (at least generically). The main goal of this school  is to introduce  graduate  students  and young  researchers to basic facts on these two spectra, and to the above correspondence.

The length spectrum  generalizes   to the action spectrum  of an autonomous Hamiltonian function.  Floer homofogy filtered  by the action spectrum  can be used to construct  so-called  action selectors,  that are a principal  tool in modern symplectic  geometry and dynamics.

The second goal is to understand  this relevance  of the action spectrum  in symplectic geometry.

A third,  more hypothetical, goal is to see which parts  of the two sides can be related.

Langue officielle de l'école : anglais

Combinatorics is at the center of a variety of areas in pure and applied mathematics. In recent years problems arising in algebra and geometry have been better understood and in many cases fully solved by exploiting their relation to certain combinatorial structures. At this school, we aim to introduce the participants to modern geometric techniques that have been used to solve long-standing conjectures in combinatorics and other areas.

Official language of the school: English. All the main activities and mini-courses will be given in English. Some contributed talks and posters given by students from the region will be in Spanish.

This research school features lectures of the world-leading experts that have driven recent developments on structure and geometry in matrix analysis and computations. The participants of the school will benefit from being trained at the forefront of research on structured matrix and geometry-based algorithms, empowering them to make contributions to the state-of-the-art. Structure is central to matrix computations. lt often reflects properties of the underlying application, such as symmetries, and redundancy introduced during linearization or discretization processes. The exploitation of structure is often mandatory in numerical algorithms, not only to attain efficiency but also to avoid that roundoff and other errors lead to physically meaningless or spurious solutions. During the last decade significant progress has been made in this area in the context of solving polynomial systems, eigenvalue problems, and nonlinear matrix equations as well as in the design of algorithms that build upon the geometry of non li near structures.

Language officielle de l'école : anglais

Mathematical Logic is a subfield of mathematics exploring the applications of formal logic to mathematics. Its inception was motivated by the study of foundations of mathematics and it has found applications in many areas, specially in Theoretical Computer Science. The four pillars of Mathematical Logic are Set Theory, Recursion Theory, Model Theory and Proof Theory. This school intends to cover all such subjects, on different levels and with different applications.  The proposed tree basic courses have the great advantage of requiring no or little prior knowledge. Talking about Set Theory and First Order Logic means establishing a common ground notation for all the formalism that will come from that. And every area in Mathematics can profit from this formal language that will be presented by one of the main active researchers in the field. The four advantage courses have a very strong inter-disciplinary nature, specially in the study of proofs and how they can be automated or analyzed. This is a trend in the area of Mathematical Logic which is interesting to be pursued. Besides the courses, sessions on proof theory and rewriting are planned. This is a flourishing research area, which builds the bridge between pure Mathematics and Theoretical Computer Science. In this setting, the Logic@Natal: School in Mathematical Logic with Applications aims to attend a growing community of undergraduate and graduate students in the area, as well as young researchers, focusing preferentially those coming from Latin America.

Langue officielle de l'Ecole: anglais

The school aims to introduce graduate students and young researchers to the recent connections between p-adic analysis (understood in a large sense) with mathematical physics and computer science. The courses will be focused on active research areas. The tentative list of courses include: (1) introduction to p-adic analysis; (2) introduction to local zeta functions; (3) p-adic models in quantum physics; (4) the p-adic theory of automata; (5) p-adic electrostatics; (6)  Strings amplitudes, local zeta functions, and log-Coulomb gases. Besides lectures, we are also planning sessions devoted to solving exercises and informal meetings to propose and discuss research problems.

Langue officielle de l'école : anglais

The courses are introductory to carry research activity in contemporary trends in Hodge Theory and p-adic Hodge Theory given by main actors in each activity: Hodge structures, Hodge-Tate structures, De Rham Cohomology and Betti Cohomology, Torelli type theorems, Abelian Varieties. p-adic numbers, p-adic Galois representations, absolute Galois group of a p-adic field, continuos representations, Fontaine rings and admisible representations, Hodge-Tate representations, étale cohomology as a p-adic Galois representation, Tate module of an elliptic curve, ring of periods, De Rham representations, good reduction and crystalline representations.

Langue officielle de l'Ecole: anglais

The aim of this CIMPA school will be to familiarise graduate students and young researchers with the field of  enumerative and analytic combinatorics,  and to show its many connections to other areas,  especially computer science.  The courses  range from introductory to  more advanced levels. The introductory courses  will  lay the groundwork  by discussing the  basic  concepts  (such as generating  functions  and  q-series)  and  techniques  (various  enumeration techniques  and  analytic  methods  such  as  singularity  analysis).  The more advanced courses  will  build on these and discuss  various  more specialised topics,  related to  random graphs  and trees,  automata  and the  analysis  of algorithms. These  topics  are  connected  in  various  ways,  not  only methodologically.

Langue officielle de l'école: anglais