Stable logarithmic maps

For the purpose of computing Gromov-Witten invariants, the compactification of the space of relative stable maps with respect to a smooth divisor were introduced and studied using expanded degeneration during the past decade. I will introduce a new compactification using logarithmic structures in the sense of Kato-Fontaine-Illusie. In particular, this covers many interesting cases, such as the target variety with a simple normal crossings divisor, or a simple normal crossings degeneration of a variety with simple normal crossings singularities. This is in part a joint work with Dan Abramovich.