GROUPES en ACTIONS

1. Formes réelles des variétés toriques

Intervenants : Giancarlo Lucchini-Arteche (UChile Santiago) et Ronan Terpereau (Lille)

Variétés toriques. Formes réelles. Calculs explicite en cohomologie galoisienne, classification des tores réels et des torseurs sous l’action d’un tore réel.Formes réelles des variétés toriques : résultat principal et exemples.

2. $K3$ surfaces and their automorphisms groups

Intervenants : Yulieth Prieto (PUC Santiago) et Alessandra Sarti (Poitiers)

$K3$ surfaces are important algebraic surfaces due to their remarkable properties. The name was given by A. Weil in 1958 in honour of the mountain K2 in Kashmir and the three renowned mathematicians Kummer, Kähler, and Kodaira. The aim of these lectures is to introduce the basic definitions and properties of K3 surfaces, while also providing many examples. In particular, we will consider quartic surfaces, complete intersections, Kummer surfaces, and elliptic fibrations. We will also discuss the role of their automorphisms group, which is a fundamental tool for understanding their geometry. Furthermore, we will study $K3$ surfaces endowed with an elliptic fibration and an automorphism acting on it, present some general results, and give several applications.

3. Polynomial automorphisms and simple derivations

Intervenants : Ivan Pan et Alvaro Rittatore (CMAT Montevideo)

4. Birational geometry of rational surfaces

Intervenants : Pedro Montero (UTFSM Valparaiso) et Enrica Floris (Toulouse)

A fundamental notion in algebraic geometry is that of a rational variety, which is a geometric object that looks like projective space. In this mini-course we will present basic definitions and examples of rational surfaces alongside with some fundamental classification theorems.

5. Groupes arithmétiques et amalgammes : le Théorème de Nagao

Intervenants : Claudio Bravo (Talca) et Benoit Loisel (Poitiers)

Présentation d'un groupe et groupes libres. Produits libres, amalgammes et énoncé du Théorème de Nagao. Le $p$-arbre de Bruhat-Tits de $\mathrm{(P)SL}_2(\mathbb{Z})$. Interprétation via les fibrés en droites, Théorème de Grothendieck-Birkhoff et preuve du Théorème de Nagao. Généralisation à $\mathrm{SL}_n$: immeubles et le théorème de Soulé