Seminar in Algebraic Geometry and Number Theory

Abstract: Tropical geometry can be considered as a toolbox meant for solving problems in algebraic geometry. In numerous applications, tropical varieties appear as degeneration of families of algebraic subvarieties (of some ambient space). The process of associating a tropical variety to a family of algebraic varieties is usually referred to as tropicalization. One might want to understand what the tropical variety tells about the variation of the moduli of the algebraic subvarieties along the family. Doing so, one might be able to emancipate tropicalization from any ambient space and define tropicalization of families of abstract algebraic objects inside the relevant moduli space. Eventually, one might be able to construct a tropical compactification of the latter moduli space.
In this talk, I want to describe how the above program can be carried for the moduli space of curves. As much as times permits, I also want to discuss one approach in arbitrary dimension. The talk is intended to a broad audience and is partially based on a joint work with M. Melo, J. Rau and F. Viviani.