### 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.

Miriam Bocardo
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.

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