The birational geometry of tropical compactifications

We study compactifications of subvarieties of algebraic tori using methods from the still developing subject of tropical geometry. Associated to each "tropical" compactification is a polyhedral object called a tropical fan. Techniques developed by Hacking, Keel, and Tevelev [19, 45] relate the polyhedral geometry of the tropical variety to the algebraic geometry of the compactification. We compare these constructions to similar classical constructions. The main results of this thesis involve the application of methods from logarithmic geometry in the sense of Iitaka [22] to these compactifications. We derive a precise formula for the log Kodaira dimension and log irregularity in terms of polyhedral geometry. We then develop a geometrically motivated theory of tropical morphisms and discuss the induced map on tropical fans. Tropical fans with similar structure in this sense are studied, and we show that certain natural operations on a tropical fan correspond to log flops in the sense of birational geometry. These log flops are then studied via the theory of secondary polytopes developed by Gelfand, Kapranov, and Zelevinsky [16] to obtain polyhedral analogues of some results from logarithmic Mori theory.