Skeletons, degenerations, and Gromov-Witten theory

This dissertation presents a study of the non--Archimedean and tropical geometry of relative Gromov--Witten theories. We investigate skeletons of analytifications of moduli spaces of maps in three settings -- the Abramovich--Corti--Vistoli space of twisted stable maps between curves, Jun Li's space of relative stable maps from P 1 to P 1, and the Abramovich--Chen--Gross--Siebert space of logarithmic maps to toric varieties and toric stacks. Our main motivation is a program for geometrizing correspondence theorems in tropical geometry, which typically relate an algebrogeometric enumerative problem with a combinatorial one. As applications, we recover new and simplified proofs of a number of important correspondence results in tropical geometry at the level of moduli spaces. We deduce that the tropical rational double ramification cycle, and more generally, tropical rational Gromov--Witten cycles for P 1 , are tropicalizations of their classical counterparts. This answers a question of Bertram, Cavalieri, and Markwig. Our study also reveals. an elementary description of the space of genus 0 logarithmic stable maps to a toric variety X as an explicit toroidal modification of M¯ 0,n x X. Finally, we turn our attention to the tropicalization of stable maps in the case of obstructed geometries. Our main result in this context shows that the every embedded tropical curve in R n may be realized as the tropicalization of a smooth curve together with a logarithmic map to a toric Artin fan, namely the quotient of a toric variety by its dense torus.