Tropical Methods in the Study of Moduli Spaces of Families of Curves

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The study of curves is a central topic in mathematics, with far-reaching applications to fields from cryptography to mathematical physics. Algebraic curves, which are one-dimensional solutions to systems of polynomial equations, are among the simplest objects in algebraic geometry. Although they have been studied for centuries, many of their basic properties remain unknown. Over the past century, the field has shifted from studying fixed curves to studying curves as they vary in families, or "moduli." This project focuses on outstanding questions in the theory of curves and their moduli, by reducing them to combinatorial questions via a process called tropicalization. The project will also serve to recruit and train a younger generation of mathematicians. This project will use tropical methods to study questions of fundamental importance in algebraic geometry. The principal objects of study, including Hurwitz spaces, moduli spaces of curves, and related combinatorial structures, are of central interest not only in algebraic geometry, but in topology, representation theory, number theory, and mathematical physics. Recent developments in tropical geometry and combinatorics pave a path toward improved understanding of basic geometric properties of moduli spaces. These methods have already been used to explore the Kodaira dimensions of moduli spaces and the Brill-Noether theory of general covers, and this project aims to further develop these results.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.