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关于小组理论及其应用的研究计划。 组指定对称类型；例如，所有双侧对称动物都有一个对称群体，与海星或沙美元不同。 群体的重要例子是由几何和代数中的对称性的研究（其中数字系统的对称性捕获的``Galois offers'''）。 通常可以将组用作基本操作的有限序列，例如对rubik的立方体组的面部旋转或作用于量子计算机状态的门。 一个典型的问题是了解在感兴趣的情况下实际上可能出现哪些群体。 另一个是针对特定组的理解，是否可以根据单个元素有效地表达组的所有元素，还是通过固定公式在变化的元素方面表达。 特定群体作为N维空间的对称组的实现是分析这些问题的关键技术方法。该奖项还将支持研究生夏季研究。该项目涉及单独使用字符理论方法或与代数几何形状结合使用，以解决有限简单组的问题。 特别是，当变量是简单组的要素时，可以应用这些工具来调查有关求解方程的问题。 例如，汤普森（Thompson）的猜想，在任何有限的简单组中，在其正方形是整个组的任何有限的简单组中都存在这种类型。 这些方法的关键是观察到，实际上，字符值通常很小。 证明和利用此主题的变化是该项目的主要目标之一。 一类应用程序是研究有限生成的群体（例如紫红色群体）的表示形式的研究。 在不同的方向上，了解哪些GALOIS群体在数字理论中可能出现哪些群体以及他们如何在多项式方程确定的集合上作用，这是该项目的重要目标，实际上，数字理论家的关键目标已有200多年了。奖项反映了NSF的法定任务，并通过使用基金会的智力优点和更广泛的影响审查标准评估值得支持。