The University of Michigan Combinatorics Seminar
I will give a brief sketch of a rapidly emerging technique from algebraic geometry known as tropical geometry. I will then discuss what happens when these ideas are applied to linear spaces. We will give a detailed description of the case of tropical lines in terms of the "space of trees" investigated by Billera, Holmes and Vogtman. In higher dimensions, we give a partial description in terms of decompositions of hypersimplices into "matroidal polytopes". Time permitting, I will discuss the "positively oriented" versions of these theories, which appear to be strongly related to the theory of cluster algebras.