Topology of tropical varieties

Resumen:

It has been known for a long time that a tropical curve in $\mathbb R^2$ of degree $d$ has genus at most $\frac{(d-1)(d-2)}{2}$. In this talk I will explain how to construct a plane tropical cubic curve of arbitrary genus. In particular, I will resolve the apparent contradiction of the last two sentences.  More generally, I will talk about (upper and lower) bounds on Betti numbers of tropical varieties of $\mathbb R^n$ (and if time permits on tropical Hodge numbers). Generalizing what is written above for cubics, I will show that there is no finite upper bound on the total Betti numbers of projective tropical varieties of degree $d$ and dimension $m$.  This is a joint work with B. Bertrand and L. Lopez de Medrano.

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