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

研究曲线族模空间的热带方法

• 批准号: 2054135
• 负责人: David Jensen
• 金额: $ 29.29万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054135&HistoricalAwards=false
• 关键词: Tropical; Methods; Study; Moduli; Spaces

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.

曲线研究是数学的一个中心课题，在从密码学到​​数学物理等领域有着深远的应用。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。尽管人们对它们的研究已经有几个世纪了，但它们的许多基本特性仍然未知。在过去的一个世纪中，该领域已经从研究固定曲线转向研究曲线，因为它们在族或“模数”中有所不同。该项目重点关注曲线及其模理论中的突出问题，通过称为热带化的过程将它们简化为组合问题。该项目还将用于招募和培训年轻一代的数学家。 该项目将使用热带方法来研究代数几何中具有根本重要性的问题。主要研究对象，包括赫尔维茨空间、曲线模空间和相关组合结构，不仅在代数几何中具有中心意义，而且在拓扑、表示论、数论和数学物理中也具有中心意义。热带几何和组合学的最新发展为更好地理解模空间的基本几何特性铺平了道路。这些方法已被用于探索模空间的 Kodaira 维数和一般覆盖的 Brill-Noether 理论，该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智力价值和更广泛的影响审查标准。