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

曲线的研究是数学的中心课题，从密码学到数学物理学都有着深远的应用。代数曲线是多项式方程组的一维解，是代数几何中最简单的对象之一。虽然人们已经研究了几个世纪，但它们的许多基本性质仍然未知。在过去的世纪里，研究领域已经从研究固定曲线转向研究曲线族中的变化，或者说“模量”。“这个项目的重点是突出的问题，在理论的曲线和他们的模量，通过减少他们的组合问题，通过一个过程称为热带化。该项目还将招募和培训年轻一代的数学家。 这个项目将使用热带方法来研究代数几何中的基本重要问题。主要的研究对象，包括Hurwitz空间，模空间的曲线，以及相关的组合结构，是中央的利益，不仅在代数几何，但在拓扑结构，表示论，数论和数学物理。最近在热带几何和组合学的发展铺平了道路，以提高理解的基本几何性质的模空间。这些方法已经被用于探索模量空间的科代拉维数和一般覆盖的Brill-Noether理论，本项目旨在进一步发展这些结果。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.