Organisateurs : Benoît Bertrand, Arthur Renaudineau

Toulouse, 4 et 5 Juillet 2018

Institut de mathématiques de Toulouse, Salle 207


Wednesday July 4th

10:00 Welcome Coffee

11:00 Kristin Shaw : Vanishing Theorems for Tropical Homology

12:40 Lunch

14:00 Matilde Manzaroli : Courbes algébriques réelles dans les surfaces de del Pezzo réelles minimales

La classification des types d’isotopie réalisés par les courbes algébriques réelles d’un degré fixé dans le plan projectif réel est un sujet classique qui a connu un essor considérable depuis les années 1970. En revanche, en dehors de quelques études concernant les surfaces de Hirzebruch et les surfaces de degré au plus 3 dans $\mathbb R\mathbb P^3$, à peu près rien n’est connu dans le cas de surfaces ambiantes plus générales. Mon exposé propose de présenter l’étude des types d’isotopie réalisés par les courbes algébriques réelles dans les surfaces réelles minimales de del Pezzo de degré 2 et 1 pour lesquelles le groupe de Picard réel est engendré par la classe anti-canonique. Je parlerais surtout de la première surface qui est un revêtement double de $\mathbb C\mathbb P^2$ ramifié le long d'une quartique réelle maximale, et dont la partie réelle est composée de quatre sphères disjointes.

15:10 Vladimiro Benedetti : Orbital degeneracy loci
(joint work with Sara Angela Filippini, Laurent Manivel, Fabio Tanturri)

In this talk I will introduce orbital degeneracy loci. These can be seen as generalisations of zero loci of sections of vector bundles. By choosing an orbit in a representation of an algebraic group, the construction allows to construct interesting Fano varieties and varieties with trivial canonical bundle. I will explain what kind of tools we have to study these loci, namely how to construct a desingularisation, by exhibiting explicit examples of such varieties.

16:10 Coffee Break

16:40 Christoph Goldner: Counting Tropical Rational Curves with Cross-Ratio Constraints.

We want to enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is a paper of Ilya Tyomkin, where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. For that we will first talk about tropical cross-ratios and introduce degenerated tropical cross-ratios. In a special case combinatorial objects, so-called cross-ratio floor diagrams, are introduced which can be used to determine the enumerative numbers we are looking for. If there is enough time left, we will deal with the general case, which is provided by a lattice path algorithm that produces all tropical curves satisfying such degenerated conditions explicitly.

Thursday July 5th

9:45 Jens Forsgård: Hyperfields

Abstract: A hyperfield is the algebraic structure which arises from the quotient of a field with a subgroup of its multiplicative group. Hyperfields provide an alternative approach to the foundations of tropical geometry as promoted by Viro. We will survey the recent discussions and development in Hyperfield theory, including the relationship with matroids, functorial properties, and the relationship with toric geometry and amoebas.

10:45 Coffee Break

11:15 Lionel Lang: On the fundamental group of univariate polynomials with fixed support.

It is a classical fact that the fundamental group of the space of univariate polynomials of degree d (with distinct roots!) is Artin's braid group on d strands. Consider now the space of polynomials with a fixed support $A \subset \mathbb N$. Apart from the case of sparse trinomials studied by Libgober, not much is known about their fundamental group. For instance, the image of the monodromy map valued in the permutation group of the roots is not known, except when $A$ generates the lattice $\mathbb Z$ (see [Esterov]). In this context, the consideration of univariate phase-tropical polynomials provides us with interesting collections of braids. It allows us to determine the image of the monodromy map for any support $A$ and gives some criteria on $A$ for the fundamental group to be the full Artin's braid group. This is a joint work in progress with A. Esterov.

12:40 Lunch

14:00 Johannes Rau : Lower bounds on real double Hurwitz numbers
(joint with Boulos El Hilany and Maksim Karev)

We use tropical methods to show that real double Hurwitz numbers are logarithmically equivalent to classical double Hurwitz numbers for certain asymptotics. In time permits, we discuss ideas to upgrade this to a "refined" polynomial count.