Groups and Arithmetic Geometry

首席研究员的研究探索了代数中的三个核心概念之间的联系：群、域和簇。群是对称类型；例如，所有两侧对称的动物共享一个对称群，这与海星的对称群不同。域是可以加、减、乘或除的数字系统；例如，通常的实数形成域，但可以表示为分数的有理数形成数学家特别感兴趣的更小的域。变数是由多个变量组成的联立方程组。PI正在试图理解这三个概念之间的深层联系。例如，作为所有有限群的基础的单群基本上总是可以表示为有限域上的各种点。变种决定了场的对称性群在数学上已经有200多年的历史了，当时高斯给出了一个指南针和正17边的结构。这个项目还为研究生提供了在这些问题上进行研究的培训机会。更具体地说，这个项目涉及到使用群论来描述由簇或自同构形产生的伽罗瓦表示的像。这类问题通常可以通过不变量理论中的“逆问题”来解决，即从关于其表示范畴的数据中识别一个代数群。群论和代数几何可以用来分析Galois扩张上交换变种的Mordell-Weil群上的Galois作用。在不同的方向上，该项目涉及使用代数几何，单独或与特征标论方法相结合，来解决关于有限单群的问题。特别是，这些工具可以用于调查由自由组或相关对象(如表面组)中的元素定义的单词图。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.