http://www.math.univ-toulouse.fr/~bbertrand/TGE/1511-Toulouse/programme.html

Tentative Programme (times might change)

Organisateurs : Benoît Bertrand, Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin

Toulouse, 19 novembre 2015

Institut de mathématiques de Toulouse, Salle à déterminer

10h:30 Coffee


11h Dmitry Novikov (Weizmann institute of science)

A tropical analog of Descartes' rule of signs

We investigate relationships between roots of a real univariate polynomial and roots of its tropicalization. We prove a generalization of the Descartes rule of signs, which bounds the number of positive roots of a polynomial p(x) = ∑dj=0 ajxj by the number of appropriately defined ”positive roots” of tropicalization of the polynomial pλ(x) = ∑dj=0λj,d ajxj, where λ k,d = edk2 , k = 0,...,d with an explicit Δd. This extends Descartes rule’s upper bound from a polynomial to its open neighborhood. We conjecture that one can take Δd = 1 and describe an application to the classical Karlin problem on zero-diminishing sequences. Joint work with Jens Forsg˚ard and Boris Shapiro.



14:00 Margarida Melo (Università di Roma Tre / Universidade de Coimbra)

Combinatorial aspects of universal compactified Jacobians over curves with marked points

The study of compactified Jacobians of singular curves has a very deep combinatorial counterpart, which governs their geometry. However, while understanding the combinatorial nature of their universal compactifications may be the key to approaching a number of open questions in the field, there is so far no tropical analogue of this universal object. In the talk, after explaining some basic properties of these (universal) compactifications, I will speculate how the construction of tropical universal Jacobians could be useful to understand several aspects of their classical analogues.


15h coffee


15h30 Joseph Tapia (Université de Toulouse III)

La théorie des schémas tropicaux et la géometrie algébrique relative.

Il s'agit dans le temps imparti d'exposer comment à un schéma en géométrie algébrique relative au sens de Töen-Vaquié ou de Durov on peut lui associer fonctoriellement sa catégorie des théories cohomologiques. Nous montrerons comment pour un schéma tropical après relèvement de celui-ci à la caractéristique nulle via les vecteurs de Witt au sens de J. Borger, on peut construire une telle catégorie.