### Abstracts

**Classical Hodge Theory**

José Jaime Hernández C.,

CIMAT**Abstract of the course:** Introduction to Hodge structures, their basics results and some archetypical examples. We give some applications to Algebraic Geometry, as well.

**p-adic Geometry**

Miriam Bocardo

University of Guadalajara

Cristhian Garay

CIMAT**Abstract of the course:** We start by giving an introduction to geometric and topological aspects of padic numbers. This can lead naturally to the study of rigid analytic geometry, and Berkovich spaces.

**p-adic Galois representations**

Rogelio Pérez-Buendía

CIMAT**Abstract of the course:** Introduction to the absolute Galois group of a p-adic field, the category of continuous representations, Fontaine ring and admissible representations. This can lead to theconcepts of DeRham, Hodge-Tate and Crystalline representations.

**Étale Cohomology**

Felipe Zaldivar

Autonomous Metropolitan University (UAM)**Abstract of the course:** We give an introduction to Grothendieck topologies, étale and l-adic cohomology. We give the main concepts and results.

**1-motives as Hodge structures of level 1**

Carolina Rivera-Arredondo

University of Milan**Abstract of the course:** We will study the concept of a 1-motive, due to Deligne, and define its Hodge realization, which generalizes the singular homology of an abelian variety endowed with its Hodge structure. We will also see how these objects can be seen as the motives associated to curves, that is, varieties of dimension 1.

**Cristalline representations**

Genaro Hernandez-Mada

University of Sonora**Abstract of the course:** In this course we give an introduction to crystalline representations. We give the definition and basic results, and we give some insight on how they can be used to study the good reduction problem, in some cases.

**A Hodge theoretic geometrization of Jacobi forms of index zero**

Roberto Villaflor

PUC Chile**Abstract of the talk:** In this short talk I will explain how one can recover the algebra of (quasi-)Jacobi forms of index zero from a geometric point of view. This is done by determining their differential equations as unique vector fields on the moduli space of elliptic curves with two marked points and a marked frame of the relative De Rham cohomology bundle with boundary on these points, compatible with its mixed Hodge structure. This is a natural generalization of Movasati's description of the algebra of quasi-modular forms in terms of Ramanujan vector fields on the moduli of elliptic curves with a basis of its De Rham cohomology compatible with its Hodge structure. Based on a joint work with Jin Cao (Tsinghua U) and Hossein Movasati (IMPA).

**Hodge decomposition for the Sobolev space $H^\ell(\lambda^k)$ on a space form of nonpositive sectional curvature**

Carlos Pinilla Suárez**Abstract of the talk:** The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to non-compact complete manifolds in the space of $L^2$ forms. We showed this decomposition with Chan and Czubak in the Sobolev space $H^1$ on a space form of a nonpositive sectional curvature. In this talk we will show the basic tools to prove this decomposition. The decomposition was extended to any Sobolev space $H^\ell$, the necessary tools to make this extension possible will be shown. Some additional results will be discussed, as well.

**All scheduled training sessions will be devoted to problem solving and discussions of contemporary research problems in the area.**