Tropical geometry studies combinatorial structures that arise as “shadows” or “skeletons” of algebraic varieties. These skeletons were motivated in part by the mirror symmetry conjectures in mathematical string theory, but have now grown to function broadly as a tool for the study of algebraic varieties. Much of the work in the subject in the past decade has focused on the geometry of moduli spaces of abstract and parameterized algebraic curves, their tropical analogues, and the relationship between the two. This has led to new results on the topology of $M_{g,n}$, the geometry of spaces of elliptic curves, and to classical questions about the geometry of Hurwitz spaces. I will give an introduction to these ideas and recent advances.

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- Seminars for 2018