Tropical Brill-Noether Theory

The study of curves is a central topic in algebraic geometry. Brill-Noether theory aims to study the geometry of a curve by examining all of its maps to projective space, or equivalently all of its linear series. This proposal focuses on outstanding problems in the theory of curves and their moduli, and an approach to them using tropical Brill-Noether theory. Intellectual Merit: This proposal uses tropical methods to study problems 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. Although these moduli spaces have been studied extensively by generations of mathematicians, many of their basic geometric properties remain unknown. Recent developments in tropical geometry and combinatorics pave a path forward. 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 will further develop these results. At the same time, this project will push toward a deeper understanding of the combinatorics and geometry underlying this recent progress. Broader Impacts: In conjunction with this research program, the primary educational component of this proposal concerns the Math Lab at the University of Kentucky. The lab serves as a central hub for undergraduate research in the UK math department, where the PI has served simultaneously as a project mentor and assistant director of communications since the lab''s inception. Additional activities with significant broader impacts include mentoring postdocs, supervising PhD students, and organizing conferences and summer schools.