Vladimir Berkovich: F1-geometry

I'll talk on work in progress on algebraic and analytic geometry over the field of one element F1. This work originates in non-Archimedean analytic geometry as a result of a search for appropriate framework for so called skeletons of analytic spaces and formal schemes.
I'll explain the notion of a scheme over F1 and its relation to the notion of a logarithmic scheme.


Antoine Ducros: Models of p-adic curves and Berkovich's theory

if X is a (smooth, projective) p-adic curve, I will explain how its models are encoded in the associated analytic space, and how recover and/or reinterpret in that setting several classical notions: copntractions and blowing up, stable, semi-stable, and minimal models, phenomena which are specific to low genera (0 and 1), Weierstraß models of elliptic curves, and so on; I plan to give a lot of examples.


Jean-Marc Fontaine: Factorization of analytic functions in mixed characteristic

I'll describe a mixed characteristic analogue of the classical factorizations theorems for analytic functions of one variable over a field complete with respect to a non trivial absolute value (joint work with Laurent Fargues).


Walter Gubler: A guide to tropicalizations over valued fields

Tropicalizations form a bridge between algebraic and convex geometry. In this talk, we generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. As a basic tool, we will use Berkovich spaces as analytifications.


Urs Hartl: The Hodge conjecture over function fields

Over a global function field one has a neutral Tannakian category of mixed motives, namely the uniformizable mixed t-motives defined by G. Anderson.  R. Pink clarified the concept of Hodge structures in equal characteristic and defined Hodge realizations of t-motives, by using the theory of σ-bundles on the rigid analytic punctured open unit disc.  We prove the analog of the famous Hodge conjecture in this situation, namely that the Hodge realization functor induces an isomorphism of the motivic Galois group of a t-motive onto its Hodge group.


Jérôme Poineau: Topology and sequences in Berkovich spaces

Berkovich spaces over the p-adics are metrizable, but this property may be lost when working over bigger fields. However, we will show that many topological properties, like openness or compactness, can still be characterized by sequences. Our proof makes an essential use of extension of scalars and we will have a close look at points that lift canonically in such an extension.


Amaury Thuillier: On the homotopy type of non archimedean analytic spaces

Recently, E. Hrushovski and F. Loeser developped a model theoretic framework to study Berkovich spaces and proved strong results on the homotopy type of Berkovich spaces coming from algebraic varieties. I will explain how to establish a slightly weaker form of these results using the methods initiated by Berkovich in his study of the problem, namely alterations and toroidal deformations.