Gromov-Witten theory with expansions

The subject of this talk will be the geometry of the moduli space of stable maps, relative to a normal crossings divisor, and a degeneration formula for the resulting Gromov-Witten theory. The moduli space and the formula together generalize a well-known package, developed by Jun Li, for smooth divisors. The main ingredient is a virtual weak semistable reduction theorem, controlled by the polyhedral geometry of tropical curves and their moduli. I will give an introduction to this circle of ideas, focusing on the very simple case of rational curves in Pn. Time permitting, I will discuss some ongoing work, including logarithmic virtual localization, and reconstruction theorems for Gromov-Witten invariants.