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形状以及该形状本身来研究某些方程的解的数量。此类解的数量有一个新的不变性。这种不变性不仅适用于复数解的数量，也适用于普通分数的解，例如二分之一。它是利用 Morel 和 Voevodsky 提出的 A1-同伦理论获得的。此外，数学教育还得到了继续为来自不同背景的有天赋的高中生提供为期一周的暑期数学工作计划的支持。在项目期间的每个夏天，大约八名高中生将研究一个重要的数学问题，根据需要学习背景材料，并以小组形式解决它。他们将由两名高中老师陪同。为该计划的毕业生提供本科生研究经验，以继续数学培训并提供研究指导。 A1 同伦理论由 Morel 和 Voevodksy 在 1990 年代末提出，允许将代数拓扑工具成功引入多项式方程解的研究中。 PI 和合作者正在研究 A1 同伦理论与枚举几何中的经典问题（例如“有多少条线在空间中与四条线相交？”）之间的相互作用。 A1-同伦理论在非常一般的基础方案上，特别是在任何域上都可以很好地发挥作用，从而在以双线性形式评估的非代数闭域上产生枚举结果。该项目搜索与特征类、Gromov-Witten 理论和 zeta 函数相关的结果，并开发这些应用所建议的动机同伦理论工具。该奖项反映了 NSF 的法定使命，并通过使用基金会的评估进行评估，认为值得支持。智力价值和更广泛的影响审查标准。