Logarithmic Gromov-Witten theory with expansions

Gromov-Witten invariants are intersection numbers on spaces of maps to a smooth and compact variety. They are modeled on counts of algebraic curves in a fixed variety. A basic question in Gromov-Witten theory is to understand how these invariants behave in normal crossings degenerations. Logarithmic Gromov-Witten theory is a natural framework for this, and the foundations of this theory have been laid by Abramovich, Chen, Gross, Siebert, Marcus, and Wise in the last decade. Tropical curves and maps organize the geometry of the relevant logarithmic moduli spaces. In this talk, I will discuss ongoing work that completes a proof of the degeneration formula, which expresses the Gromov-Witten invariants of a variety in terms of the invariants of the torus bundles over the strata of the degeneration.