A dissertation submitted in partial satisfaction of the requirements for

Tropical Geometry by David E Speyer Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bernd Sturmfels, Chair Let K be an algebraically closed field complete with respect to a nonarchimedean valuation v : K∗ → R. The reader should think ofK as the field of Puiseux series, ⋃∞ n=1 C((t )) and v as the map that assigns to a power series the exponent of its lowest degree term. Let X be a subvariety of the torus (K). Then v(X) is (essentially) a polyhedral complex which we term the tropicalization of X and its combinatorial properties reflect the geometry of X . In this thesis, we first discuss structural results about the tropicalization of X for an arbitrary X . We then turn to the special cases of linear spaces, Grassmannians and curves; in each case trying to describe as explicitly as possible the possible combinatorics of the tropicalization. In the case of linear spaces, our investigations lead us to a combinatorial theory of tropical linear spaces. This