We prove that the moduli space of stable logarithmic maps from logarithmic curves to a fixed target logarithmic scheme is a proper algebraic stack when the target scheme is projective with fine and saturated logarithmic structure. This was previously known only with further restrictions on the logarithmic structure of the target.

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: J. Li's original approach, B. Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

Introduction 1 2. Toric varieties and toroidal embeddings 5 3. Logarithmic structures 9 4. Kato fans and resolution of singularities 13 5. Artin fans 21 6. Algebraic applications of Artin fans 27 7. Skeletons and tropicalization 32 8. Analytification of Artin fans 42 9. Where we are, where we want to go 45 References 47

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. We also describe the attendant deformation theory and virtual fundamental class of this moduli space. This recovers the double ramification cycle, as well as variants associated to differentials. Contents

We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of [CMW12] where the same identity was proved over on the locus of rational tails curves.