The moduli space of tropical curves (and its variants) are some of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve (at least in the positive genus case). The classical work of Deligne-Knudsen-Mumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces.

In this talk I am going to give an introduction to these fascinating moduli spaces and report on ongoing work with Renzo Cavalieri, Melody Chan, and Jonathan Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this $2$-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose two different ways of describing the process of tropicalization: one via logarithmic geometry in the sense of Kato-Illusie and the other via non-Archimedean analytic geometry in the sense of Berkovich. Speaker(s): Martin Ulirsch (UM)