Tentative Programme (times might change)

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) = ∑^d_j=0 a_jx^j by the number of appropriately defined ”positive roots” of tropicalization of the polynomial p_λ(x) = ∑^d_j=0λ_j,d a_jx^j, where λ _k,d = e^-Δ_dk2 , k = 0,...,d with an explicit Δ_d. This extends Descartes rule’s upper bound from a polynomial to its open neighborhood. We conjecture that one can take Δ_d = 1 and describe an application to the classical Karlin problem on zero-diminishing sequences. Joint work with Jens Forsg˚ard and Boris Shapiro.

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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.

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15h coffee

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15h30 Joseph Tapia (Université de Toulouse III)

La théorie des schémas tropicaux et la géometrie algébrique relative.