Tropical and logarithmic moduli theory
This course will be a simultaneous introduction to both tropical and logarithmic moduli theory, with a focus on the moduli spaces of tropical curves and logarithmic curves.
(1) Tropical and non-Archimedean geometry The first part will be an introduction to tropical geometry and its connections to non-Archimedean geometry. Topics to be discussed are Berkovich analytic spaces, the process of tropicalization for subvarieties of algebraic tori and toric varieties, and the principle of faithful tropicalization.
(2) Introduction to tropical moduli spaces This will be an elementary introduction to tropical moduli spaces with many pictures and explicit examples. In particular, I will illustrate the connection between the boundary structure of the moduli space of stable algebraic curves and the moduli space of stable tropical curves via its non-Archimedean skeleton.
(3) The moduli stack of tropical curves In this part, I am going to give an introduction to the theory of stacks with a focus on combinatorial examples. In particular, I intend to give a full construction of the moduli stack of tropical curves and to discuss how it generalizes the set-theoretic moduli space discussed in Part (2).
(4) The moduli space of logarithmic curves and its tropicalization This final part starts with an introduction to logarithmic geometry and its applications to moduli theory, leading up to the construction of the moduli space of logarithmic curves. From there, we can describe the process of tropicalization as a natural smooth and surjective tropicalization morphism over the category of logarithmic schemes (thereby solving the realizability problem for abstract tropical curves).