Symmetries and Statistics in Arithmetic

数域是与多项式根相关的代数结构，是代数数论中的重要对象。研究这些代数结构的一个重要组成部分是了解其有趣的不变量，而数域的对称性严重影响着这些不变量的行为。更准确地说，虽然不变量对于单个数字字段可能是神秘和随机的，就像在一个族中一样，但它们的统计量服从由这种对称性决定的某些分布。这个项目的目标是从统计学的角度来研究这些不变量，利用它们的对称性。特别是，本研究将证明具有特殊对称性的数域的不变量的新结果，发现由统计结果激发的算术结构中的新现象，并提出这些不变量的新的统计测量方法。该项目还将为研究生、本科生和博士后提供研究机会，并将促进数论的解析和代数方面之间的深入对话。更具体地说，本项目将使用代数数论和解析数论的工具，研究具有固定伽罗瓦群的全局域的判别式和类群的分布结果。对于判别式的分布，本课题将利用相对不变量和归纳思想证明新的渐近分布、上界和下界。它还将包括对具有不同特征和对称性的全局场的讨论。对于类群的分布，本项目将研究单个数域的极值行为和一组数域的统计行为。它包括研究新的存在型算术问题和由分布问题提出的表示理论问题。另一方面，本课题将提出和研究类数问题的概括和变化。最后，本计画也将探讨多项式根分布的基本问题。这包括给出导致等分布结果的尖锐不等式和构造关于各种泛函的最优根分布。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。

Number fields, algebraic structures associated with the roots of polynomials, are important objects in algebraic number theory. A crucial component in studying these algebraic structures is to understand interesting invariants of it, and the symmetry of the number field heavily influences the behavior of these invariants. More precisely, although the invariants can be mysterious and random for a single number field, as in a family, their statistics obey certain distributions determined by this symmetry. The goal of this project is to study these invariants utilizing their symmetry from a statistical point of view. In particular, this research will prove new results on invariants for number fields with special symmetry, discover new phenomena in arithmetic structures motivated by statistical results, and propose new statistical measurements of these invariants. This project will also include research opportunities for graduates, undergraduates and postdocs, and will facilitate in-depth conversations between the analytic and algebraic sides of number theory. More concretely, this project will investigate distribution results for discriminants and class groups of global fields with a fixed Galois group, using tools from algebraic number theory and analytic number theory. For the distribution of discriminants, this project will prove new asymptotic distribution, upper bound and lower bound using relative invariants and inductive ideas. It will also include discussions of global fields with different characters and symmetry. For the distribution of class groups, this project will study both extremal behavior for a single number field and statistical behavior for a family of number fields. It includes investigating new existence-type arithmetic questions and representation theory questions suggested by distribution problems. On the other hand, this project will propose and study generalizations and variations of the class number problems. Finally, this project will also address fundamental questions on the distribution of roots of polynomials. This includes giving sharp inequalities leading to equidistribution results and constructing optimal root distributions with respect to various functionals.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.