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.

研究数学中复杂对象的一种方法是计算对象的更简单的不变量，它提供有关对象的信息。与多项式方程组相关的对象（称为代数簇）是许多数学领域的基础，可以模拟许多现实世界的现象。该项目研究代数簇产生的不变量，特别是在考虑大族时某些不变量的分布方式。例如，如果不变量是整数，人们可能会问对于“随机”对象来说，不变量为零的可能性有多大。这样的结果可以转化为对给定多项式方程的解的更好理解。除了拟议的研究之外，PI还将组织一系列外展活动，包括针对学龄女孩的课后和周末活动、针对研究生的研讨会以及针对学生和博士后的区域研讨会。代数和解析数论、代数几何和表示论的交叉点，重点研究算术统计的分布。人们可能不仅需要理论结果，还需要启发式预测，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、启发式和数据——涉及数域类群、椭圆曲线的等级以及代数几何对象的其他不变量的问题。 PI 打算使用经典代数几何、李理论、随机矩阵理论和解析数论的筛选技术来开展所提出的研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。