CAREER: Theory, Heuristics, and Data for Arithmetic Invariants

职业：算术不变量的理论、启发式和数据

• 批准号: 2309115
• 负责人: Wei Ho
• 金额: $ 40万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-12-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309115&HistoricalAwards=false
• 关键词: CAREER; Theory; Heuristics; Data; Arithmetic

One way to study complicated objects in mathematics is to compute simpler invariants of the object, which provide information about the object. Objects associated to sets of polynomial equations, called algebraic varieties, are fundamental in many areas of mathematics and model numerous real-world phenomena. This project studies invariants arising from algebraic varieties, especially how certain invariants are distributed when considering large families of them. For example, if the invariant is an integer, one may ask how likely the invariant is, say, zero for a "random" object. Such results then translate into a better understanding of the solutions of the given polynomial equations. Along with the research proposed, the PI will organize a range of outreach activities, including after-school and weekend activities for school-age girls, workshops for graduate students, and regional workshops for students and postdocs.The research in this project is at the intersection of algebraic and analytic number theory, algebraic geometry, and representation theory, and focuses on distributions of arithmetic statistics. One may ask for not only theoretical results but also heuristic predictions, as well as computational data to predict or verify conjectures. The PI will study all three aspects--theoretical results, heuristics, and data--concerning questions about class groups of number fields, ranks of elliptic curves, and other invariants of algebro-geometric objects. The PI intends to use methods from classical algebraic geometry, Lie theory, random matrix theory, and sieve techniques from analytic number theory to pursue the proposed research.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.

研究数学中复杂物体的一种方法是计算对象的简单不变性，该对象提供有关对象的信息。在许多数学领域，与多项式方程相关的对象（称为代数品种）至关重要，并模拟了许多现实现象。该项目研究由代数品种引起的不变性，尤其是在考虑大型家庭时如何分发某些不变的人。例如，如果不变是一个整数，则可能会询问“随机”对象不变的可能性为零。然后，这些结果可以更好地理解给定多项式方程的解决方案。 Along with the research proposed, the PI will organize a range of outreach activities, including after-school and weekend activities for school-age girls, workshops for graduate students, and regional workshops for students and postdocs.The research in this project is at the intersection of algebraic and analytic number theory, algebraic geometry, and representation theory, and focuses on distributions of arithmetic statistics.人们可能不仅要求理论结果，还要求启发式预测，以及用于预测或验证猜想的计算数据。 PI将研究所有三个方面 - 理论结果，启发式方法和数据 - 对数字字段组，椭圆曲线等级以及代数几何对象的其他不变式的问题进行了解决。 PI打算使用经典代数几何学，谎言理论，随机矩阵理论以及分析数理论的筛分技术来追求拟议研究的方法。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智力和更广泛影响的评估来通过评估来获得支持的。