PROGRAMME DU MERCREDI 5 NOVEMBRE (Amphi Lebesgue)
14h-15h : Erwan Brugallé : Welschinger-Witt invariants
This talk will address the problem of enumerating with quadratic forms rational curves in complex surfaces or symplectic 4-manifolds. The real Abramovich-Bertram formula for Welschinger invariants of real symplectic rational 4-manifolds allows one to encode them in what is known as a Witt invariant over any field. It turns out that these Weslchinger-Witt invariants recover the quadratic Gromov-Witten invariants, recently defined by Kass-Levine-Solomon-Wickelgren, in the case of rational del Pezzo surface of degree at teast 6 (and conjecturally of degree at least 3). As a consequence, quadratic Gromov-Witten invariants of these rational surfaces over any field are determined by the two special fields $C$ and $R$. This is a joint work with Johannes Rau and Kirsten Wickelgren.
15h15-16h15 : Samuel Lerbet I
16h45-17h45 : Aurore Boitrel : Automorphism groups of real rational Del Pezzo surfaces of degree 4
Del Pezzo surfaces and their automorphism groups play a key role in the classification up to conjugacy of subgroups of the Cremona group of the plane. Over an algebraically closed field, they are completely classified together with their automorphism groups. In this talk, we will focus on real rational Del Pezzo surfaces of degree 4. Unlike larger degrees, the degree 4 case involves an infinite moduli space of surfaces, already over the complex numbers. We will explain how studying the action of the Galois group on the conic bundle structures enables us to give a complete description of their automorphism groups by generators in terms of automorphisms and birational automorphisms.
PROGRAMME DU JEUDI 6 NOVEMBRE (Amphi Lebesgue le matin, salle 16 l'après-midi)
9h15-10h15 : Felipe Espreafico Guerlerman
10h45-11h45 : Samuel Lerbet II
14h30-15h30 : Andrea Fanelli
16h-17h : Enzo Pasquereau : Combinatorial patchworking in codimension 2 and more
Combinatorial patchworking is a powerful method used for constructing real algebraic hypersurfaces with controlled topology. I will discuss generalization of this method to higher codimension using real phase structure. In codimension 2, we give explicit patchworking rules (based on triangulations, sign distributions, and edge orientations) similar to Viro's original formulation for hypersurface. As an application, we obtain families of maximal T-curves in real projective 3-space. For higher codimension, we derive new bounds on the number of connected components and prove non-existence of maximal T-curves (for codimension >3) and of high codimension T-surfaces.
PROGRAMME DU VENDREDI 7 NOVEMBRE (amphi Lebesgue)
9h15-10h15 : Marie-Françoise Roy : Nombre d'enroulement algébrique
Il y a de nombreuses preuves du théorème fondamental de l'algèbre mais jusqu'à récemment, la seule preuve algébrique était celle de Laplace de 1790. Michael Eisermann en a proposé en 2012 une preuve basée sur un résultat fondateur de la géométrie algébrique réelle, le théorème de Sturm et ses variantes: le nombre d'enroulement se calcule ainsi par une méthode algébrique qui fonctionne pour tout corps réel clos. Un raffinement de la méthode d'Eisermann permet même de compter les racines complexes dans un rectangle si elles sont sur un côté ou en un sommet, ce qui est interdit par les méthodes analytiques. De plus il est possible, si on utilise des sous-résultants pour calculer les restes successifs, de comparer la preuve de Laplace et la preuve par le nombre d'enroulement algébrique d'un point de vue quantitatif et la preuve par le nombre d'enroulement algébrique est nettement meilleure.