Groups and Arithmetic

该奖项将支持该协会关于群论及其应用的研究计划。群指定对称类型；例如，所有两侧对称的动物共享一个对称群，这与海星或沙币的对称群不同。群的重要例子来自于几何和代数中对称性的研究(数系的对称性被“伽罗华群”捕捉到)。群通常可以有用地表示为基本运算的有限序列，比如魔方群的面旋转，或者作用于量子计算机状态的门。一个典型的问题是了解哪些群体实际上可能在感兴趣的情况下出现。另一个问题是，对于特定的群体，理解群体的所有元素是可以用单个元素有效地表达，还是可以用一个固定的公式来表达不同的元素。将特定群实现为n维空间的对称群是分析这些问题的关键技术方法。该奖项还将支持研究生的暑期研究。该项目涉及单独使用特征标论方法或与代数几何结合使用，以解决有关有限单群的问题。具体地说，当变量是简单群的元素时，这些工具可以用来研究关于求解方程的问题。例如，汤普森的猜想，断言在任何有限的单群中，存在一个平方是整个群的共轭类，就是这种类型。这些方法的一个关键是注意到，在实践中，字符值通常令人惊讶地小。证明和利用这一主题的变化是该项目的主要目标之一。一类应用是研究有限生成群的表示簇，例如Fuchsian群。在不同的方向上，了解在数论中哪些伽罗瓦群可以出现，以及它们如何作用于由多项式方程确定的集合，是该项目的重要目标，实际上也是200多年来数学家的关键目标。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award will support the PI's research program concerning group theory and its applications. Groups specify symmetry types; for instance, all bilaterally symmetric animals share a symmetry group, which is different from that of a starfish or of a sand dollar. Important examples of groups arise from the study of symmetry in geometry and in algebra (where symmetries of number systems are captured by ``Galois groups''). Groups can often be usefully expressed as finite sequences of basic operations, like face-rotations for the Rubik's cube group, or gates acting on the state of a quantum computer. One typical problem is understanding which groups can actually arise in situations of interest. Another is understanding, for particular groups, whether all the elements of the group can be expressed efficiently in terms of a single element or by a fixed formula in terms of varying elements. The realization of a particular group as the symmetry group of n-dimensional space is a key technical method to analyze these problems. The award will also support graduate student summer research. The project involves using character-theoretic methods alone or in combination with algebraic geometry, to solve problems about finite simple groups. In particular, these tools can be applied to investigate questions about solving equations when the variables are elements of a simple group. For instance, Thompson's Conjecture, asserting the existence, in any finite simple group of a conjugacy class whose square is the whole group, is of this type. A key to these methods is the observation that, in practice, character values are usually surprisingly small. Proving and exploiting variations on this theme is one of the main goals of the project. One class of applications is to the study of representation varieties of finitely generated groups, for instance Fuchsian groups. In a different direction, understanding which Galois groups can arise in number theory and how they can act on sets determined by polynomial equations, is an important goal of this project and, indeed, a key goal of number theorists for more than 200 years.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.