Motivic Homotopy Theory and Applications to Enumerative Geometry

本征同伦理论及其在枚举几何中的应用

• 批准号: 2103838
• 负责人: Kirsten Wickelgren
• 金额: $ 29.29万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing 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=2103838&HistoricalAwards=false
• 关键词: Motivic; Homotopy; Theory; Applications; Enumerative

This project studies the number of solutions to certain equations, using the shape of spaces associated to these equations, as well as this shape itself. There is a new invariance property of the number of such solutions. This invariance not only applies to the number of solutions in the complex numbers, but also solutions which are ordinary fractions, such as one half. It is obtained by using A1-homotopy theory due to Morel and Voevodsky. Mathematics education is furthermore supported by continuing a program of week-long summer math jobs for gifted high school students from diverse backgrounds. During each of the summers of the project period, approximately eight high school students will work on an important mathematical problem, learning the background material as necessary, and solving it as a group. They will be accompanied by two high school teachers. A Research Experience for Undergraduates aimed at the graduates of the program is provided to continue mathematical training and provide research mentorship. A1-homotopy theory was introduced by Morel and Voevodksy in the late 1990's and allows the successful import of tools from algebraic topology into the study of solutions to polynomial equations. The PI and collaborators are studying the interaction between A1-homotopy theory and classical questions from enumerative geometry such as "How many lines meet four lines in space?" A1-homotopy theory functions well over very general base schemes and in particular over any field, resulting in enumerative results over non-algebraically closed fields valued in bilinear forms. This project searches for such results connected with characteristic classes, Gromov--Witten theory, and zeta functions and develops tools in motivic homotopy theory suggested by these applications.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.

该项目使用与这些方程相关的空间以及此形状本身研究某些方程式的解决方案数量。此类解决方案数量有一个新的不变性属性。这种不变性不仅适用于复数中解决方案的数量，而且还适用于普通分数的解决方案，例如一半。它是通过使用Morel和Voevodsky引起的A1-HOMOTOPY理论获得的。此外，通过继续为来自不同背景的有天赋的高中生开设一项为期一周的夏季数学工作的计划，数学教育得到了支持。在项目期间的每个夏季，大约八名高中生都会解决一个重要的数学问题，根据需要学习背景材料并将其作为一个小组解决。他们将由两名高中老师陪同。提供针对该计划毕业生的本科生的研究经验，以继续数学培训并提供研究指导。 莫雷尔（Morel）和Voevodksy在1990年代后期引入了A1-HOMOTOPY理论，并允许成功从代数拓扑导入工具到对多项式方程的解决方案研究。 PI和合作者正在研究A1-Homotopicy理论与列举几何学的经典问题之间的相互作用，例如“有多少线在太空中遇到四行？” A1-HOMOTOPY理论在非常一般的基础方案（尤其是在任何领域）上的发挥作用很好，从而在双线性形式的价值的非代数闭合场上产生了枚举结果。该项目搜索与特征类别，格罗莫夫（Gromov）的理论以及Zeta功能相关的结果，并开发了这些应用程序提出的动机同型理论中的工具。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来支持的。