Higher rank tropical and polyhedral geometry

In this talk, we will introduce some ideas for the tropicalization of varieties with respect to valuations of higher rank. We will show that these spaces have the structure of iterated fibrations of usual tropical varieties, and we will explain how to understand these fibrations in the case of hypersurfaces. Moreover, we will outline a theory of polyhedral geometry over the ordered ring of real numbers extended with a nilpotent infinitesimal R[x]/(x^n), which will allow us to endow higher rank tropical varieties with the structure of a polyhedral complex over this ring.