The tropical Grassmannian $TGr_0(3,8)$, Dressian $Dr(3,8)$, and their non-negative parts.

Tropical varieties are polyhedral shadows of algebraic varieties which allow to study the latter with combinatorial methods. In this talk we will focus on tropicalizations of Grassmannians, and in particular the tropical Grassmannian $TGr_0(3,8)$. It might happen that the intersection of finitely many tropical hypersurfaces is not a tropical variety. Such a set is called a tropical prevariety. Examples of those prevarieties are Dressians, which contain tropical Grassmannians and are moduli space of tropical linear spaces. Tropical linear spaces, tropical Grassmannians and Dressians naturally appear in many mathematical areas, and also in computer science, biology, economics and physics. More precisely in the study of shortest paths, phylogenetics, auctions and scattering amplitudes. The later is directly connected to the non-negative part of the tropical Grassmannians. I will intoduce tropical Grassmannians and Dressians and discuss differences between different polyhedral structures on the former with a focus on the case $TGr_0(3,8)$.