Tropical Brill-Noether Theory

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

The study of curves is a central topic in mathematics. In the past, mathematicians thought about a curve as living in a fixed ambient space. In the 20th century, however, much of mathematics underwent a fundamental shift, with objects defined in terms of their intrinsic properties, without reference to an ambient space. From this point of view, it is interesting to ask about all the different ways that a given curve can be mapped into a target space. The study of this, and related, questions is known as Brill-Noether theory. The scientific goal of this project is to advance research in Brill-Noether theory to study problems of fundamental importance in algebraic geometry. In conjunction with this research program, this award supports undergraduate research through the Math Lab at the University of Kentucky. The lab serves as a central hub for undergraduate research in the University of Kentucky Math Department, where the principal investigator has served simultaneously as a project mentor and assistant director of communications since the lab’s inception in 2017. This award will also support the newly-created annual workshop of the Kentucky-Ohio Algebra Alliance (KOALA). The goals of this conference are to establish a community of mathematicians in Kentucky and Ohio working in combinatorial algebraic geometry, and to expose graduate students and junior researchers at both institutions to the many techniques and outstanding problems in combinatorial 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. 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.