Sieve Methods in Group Theory

群论中的筛选方法

• 批准号: 1066427
• 负责人: Alexander Lubotzky
• 金额: $ 20.64万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1066427&HistoricalAwards=false
• 关键词: Sieve; Methods; Group; Theory

The last few years showed a dramatic progress on expander graphs and property 'tau'; these developments led to the 'affine sieve method' showing that for many groups acting on the set of integer lattice points in n-dimensional Euclidean space, each orbit has infinitely many vectors whose entries are all almost primes, i.e., every entry is a product of a bounded number of primes. This is a non-commutative version of classical number theoretic results. We plan to take this direction some steps further and to apply similar methods for developing 'group sieve method' for the study of pure group theoretical problems. This method seems to be suitable for solving problems which are out of reach by the classical group theoretic methods. Applying this method to various groups of important in geometry and topology- e.g., the mapping class group, it is expected to give also some geometric applications. So all together generalized number theoretic methods are expected to have some significant geometric applications.Groups acting on certain mathematical objects as symmetry is in the heart of mathematical research. Most mathematical and physical questions are modeled as group acting on a certain set. This proposal deals with groups on certain geometric and combinatorial objects and is to study properties of these actions. The topics discussed in this proposal involve connections between several areas of research and illustrate the unity of mathematics and its connection with computer science.

过去几年，扩展图和属性“tau”取得了巨大进展；这些发展导致了“仿射筛法”，表明对于作用于 n 维欧几里得空间中的整数格点集的许多群，每个轨道都有无限多个向量，其条目几乎都是素数，即每个条目都是一个乘积有限数量的素数。这是经典数论结果的非交换版本。我们计划在这个方向上更进一步，并应用类似的方法来开发“群筛法”来研究纯群理论问题。这种方法似乎适合解决经典群论方法无法解决的问题。将此方法应用于几何和拓扑中重要的各种组（例如映射类组），预计也能提供一些几何应用。因此，广义数论方法预计将具有一些重要的几何应用。作为对称性作用于某些数学对象的群是数学研究的核心。大多数数学和物理问题都被建模为作用于特定集合的群体。该提案涉及某些几何和组合对象上的组，并研究这些动作的属性。该提案中讨论的主题涉及多个研究领域之间的联系，并说明了数学的统一性及其与计算机科学的联系。