Tropical and non-Archimedean geometry (2)

Logarithmic structures and the tropical geometry of moduli spaces While $\overline{M}_{0,n}$ admits a well-understood embedding into a toric variety, this is not true for the Deligne-Knudsen-Mumford moduli stacks $\overline{\mathcal{M}}_{g,n}$ of higher genus curves. In this talk I will explain how logarithmic geometry in the sense of K. Kato provides us with a much more general framework for tropicalization that naturally applies to these moduli spaces. The final goal will be a theorem of Abramovich-Caporaso-Payne which gives a modular interpretation for the tropicalization of the algebraic moduli spaces. Given time I will also talk about ongoing work with R. Cavalieri, M. Chan, and J. Wise, where we propose a stack-theoretic generalization of this result.