Groups and Arithmetic Geometry

首席研究员的研究探索了代数中三个核心概念：群、场和变量之间的联系。群是对称类型；例如，所有两侧对称的动物都有一个对称群，这与海星的对称群不同。字段是可以加、减、乘或除的数字系统；例如，通常的实数形成一个域，但可以表示为分数的有理数形成一个较小的域，这是数论学家特别感兴趣的。变分是由若干变量组成的联立方程组。PI正试图理解这三个概念之间的深层联系。例如，所有有限群的构成单元——简单群，基本上总是可以表示为有限域上的各种点。当高斯给出了一个指南针和正17边形的直线结构时，对称性群在数学中已经引起了200多年的兴趣。这个项目也为研究生提供研究训练的机会，让他们与PI一起研究这些问题。更具体地说，这个项目涉及使用群论来描述伽罗瓦表示的图像，这些图像要么来自变种，要么来自自同构形式。这类问题通常可以通过不变量理论中所谓的“逆问题”来解决，即从其表示范畴的数据中识别代数群。群论和代数几何可以共同应用于伽罗瓦扩展上的阿贝尔变异体的Mordell-Weil群上的伽罗瓦作用。在另一个方向上，该项目涉及使用代数几何，单独或结合特征理论方法，解决有限简单群的问题。特别是，这些工具可以用于研究由自由组中的元素或相关对象（如表面组）定义的词映射。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。

The principal investigator's research explores the connections between three of the central concepts in algebra: groups, fields, and varieties. Groups are symmetry types; for instance, all bilaterally symmetric animals share a symmetry group, which is different from the symmetry group of a starfish. Fields are systems of numbers which can be added, subtracted, multiplied, or divided; for instance, the usual real numbers form a field, but the rational numbers, which can be expressed as fractions, form a smaller field of particular interest to number theorists. Varieties are systems of simultaneous equations in a number of variables. The PI is trying to understand the deep connections between these three concepts. For instance, simple groups, the building blocks for all finite groups, can almost always be expressed essentially as the points of a variety over a finite field. Varieties determine fields whose symmetry groups have been of interest in mathematics for more than 200 years, when Gauss gave a compass and straightedge construction for the regular 17-gon. This project also provides research training opportunities for graduate students working with the PI on these questions.More specifically, this project involves using group theory to describe the images of Galois representations arising either from varieties or from automorphic forms. Problems of this kind can often be approached via by what might be termed the "inverse problem" in invariant theory, recognizing an algebraic group from data about its representation category. Group theory and algebraic geometry can be applied together to analyze the Galois action on Mordell-Weil groups of abelian varieties over Galois extensions. In a different direction, the project involves using algebraic geometry, alone or in combination with character-theoretic methods, to solve problems about finite simple groups. In particular, these tools can be applied to investigate word maps, defined by elements in free groups or related objects such as surface groups.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.