CAREER: Theory, Heuristics, and Data for Arithmetic Invariants

在数学中研究复杂对象的一种方法是计算对象的简单不变量，它提供了关于对象的信息。与多项式方程集相关的对象，称为代数变量，是许多数学领域的基础，并模拟了许多现实世界的现象。本课题研究由代数变量引起的不变量，特别是当考虑它们的大族时，某些不变量是如何分布的。例如，如果不变量是一个整数，人们可能会问，对于一个“随机”对象，不变量是零的可能性有多大。这样的结果转化为对给定多项式方程的解的更好理解。与此同时，公团还将为适龄女性举办课外和周末活动、为研究生举办研讨会、为学生和博士后举办地区研讨会等。该项目的研究是在代数和解析数论，代数几何，和表示理论的交叉点，并侧重于算术统计的分布。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——关于数域的类群、椭圆曲线的秩和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、李论、随机矩阵理论和解析数论中的筛选技术来进行拟议的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.