Groups and Arithmetic

群与算术

• 批准号: 2401098
• 负责人: Michael Larsen
• 金额: $ 28万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401098&HistoricalAwards=false
• 关键词: Groups; Arithmetic

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.

该奖项将支持PI关于群论及其应用的研究计划。 组指定对称类型;例如，所有左右对称的动物都共享一个对称群，这与海星或沙元的对称群不同。 群的重要例子来自于对几何和代数中对称性的研究（其中数制的对称性被“伽罗瓦群”捕获）。 群通常可以有效地表示为基本运算的有限序列， 就像魔方组的面旋转，或者作用于量子计算机状态的门。 一个典型的问题是理解哪些群体实际上会出现在感兴趣的情况下。 另一个是理解，对于特定的组，是否该组的所有元素可以有效地表示为一个单一的元素或由一个固定的公式表示为不同的元素。 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暑期研究。该项目涉及单独使用特征理论方法或与代数几何相结合，以解决有关有限简单群的问题。 特别是，这些工具可以应用于调查有关解决方程的问题时，变量是一个简单的组的元素。 例如，汤普森猜想，断言存在，在任何有限的简单群的共轭类的平方是整个群体，是这种类型。 这些方法的一个关键是观察到，在实践中，字符值通常非常小。 证明和利用这一主题的变化是该项目的主要目标之一。 一类的应用是研究的代表品种的非线性生成的群体，例如Fuchsian群体。 在另一个方向，理解伽罗瓦群可以出现在数论中，以及它们如何作用于由多项式方程确定的集合，是这个项目的一个重要目标，事实上，这是200多年来数论学家的一个关键目标。这个奖项反映了NSF的法定使命，并被认为值得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来支持。