About Hrushovski and Loeser's work on the homotopy type of Berkovich spaces

Those are the notes of the two talks I gave in april 2013 in St-John (US Virgin Islands) during the Simons Symposium on non-Archimedean and tropical geometry. They essentially consist of a survey of Hrushovski and Loeser's work on the homotopy type of Berkovich spaces; the last section explains how the author has used their work for studying pre-image of skeleta.Comment: 31 pages. This text will appear in the Proceedings Book of the Simons Symposium on non-Archimedean and tropical geometry (april 2013, US Virgin Islands). I've taken into account the remarks and suggestion of the referee

Mixed Crop Livestock Farming Incorporating Agroforestry Orchards Facing the New Cap

In the context of the new CAP, decoupling subsidies from production should incite farmers to reorganize their production systems, particularly through diversification opportunities. In this paper we focus our analysis on the conditions that could permit the development of extensive orchards by modelling mixed crop livestock farms, which incorporate orchards. A mathematical programming model is built to simulate various intensification levels characterizing different technical pathways within the different farm activities (cattle breeding, forage fields, arboriculture). This model also enables us to take into account some environmental indicators related to these pathways. Moreover, the method illustrates technical complementarities existing within the diversified systems, thanks to the joint production phenomena introduced into our analysis. We show how these complementarities can be integrated into the farmer's decision criteria.decoupling, diversification, agroforestry orchards, joint production, mathematical programming, Agribusiness, C61, D24, Q12, Q21,

Triangulation et cohomologie \'{e}tale sur une courbe analytique

Let $k$ be a non-archimedean complete valued field and let X be a smooth Berkovich analytic $k$-curve. Let $F$ be a finite locally constant \'{e}tale sheaf on $k$ whose torsion is prime to the residue characteristic. We denote by $|X|$ the underlying topological space and by $\pi$ the canonical map from the \'{e}tale site to $|X|$. In this text we define a triangulation of $X$, we show that it always exists and use it to compute $H^{0}(|X|,R^{q}\pi\_{*}F)$ and $H^{1}(|X|,R^{q}\pi\_{*}F)$. If $X$ is the analytification of an algebraic curve we give sufficient conditions so that those groups are isomorphic to their algebraic counterparts ; if the cohomology of $k$ has a dualizing sheaf in some degree $d$ (e.g $k$ is $p$-adic, or $k=C((t))$) then we prove a duality theorem between $H^{0}(|X|,R^{q}\pi\_{*}F)$ and $H^{1}\_ {c}(|X|,R^{d+1}\pi\_{*}G)$ where $G$ is the tensor product of the dual sheaf of $F$ with the dualizing sheaf and the sheaf of $n$-th roots of unity

Les espaces de Berkovich sont excellents

In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of non-archimedean complete fields (it includes the case of the finite separable extensions, and also the case of any complete extension of a perfect complete non-archimedean field) and show that the usual commutative algebra properties (Rm, Sm, Gorenstein, Cohen-Macaulay, Complete Intersection) are stable under analytically separable ground field extensions; we also establish a GAGA principle with respect to those properties for any finitely generated scheme over an affinoid algebra. A second part of the paper deals with more global geometric notions : we define, show the existence and establish basic properties of the irreducible components of analytic space ; we define, show the existence and establish basic properties of its normalization ; and we study the behaviour of connectedness and irreducibility with respect to base change.Comment: This is the (almost) definitive version of the paper, which is going to appear in "Annales de l'institut Fourier