Algorithms for Arithmetic Groups

算术群的算法

• 批准号: 1720146
• 负责人: Alexander Hulpke
• 金额: $ 8万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720146&HistoricalAwards=false
• 关键词: Algorithms; Arithmetic; Groups

The objects, called arithmetic groups, considered in this project are sets of matrices (representing symmetries of a space) whose entries can be considered as integral coordinates in a geometrical object. They lie at the intersection of Algebra, Geometry and Number Theory. If a number of such matrices are given, we consider the subset that can be formed by combining these matrices that is applying the symmetries in turn. It is called the subgroup (generated by the matrices). A natural question now arises on whether finitely many translated copies of such a given subgroup amount to the whole set, in this case the subgroup is called arithmetic. This question has come up in many concrete examples brought up by researchers in number theory or geometry but so far the only solution approach have been ad-hoc methods for small cases. The PI will design new algorithms for answering this question on a computer and provide an open-source implementation of this algorithm. The project will also train a graduate student in the development of mathematical software.The PI will develop new algorithms for calculations with classes of arithmetic groups, in particular developing methods that may determine whether a subgroup given by generating matrices is itself arithmetic. Practical implementations of these algorithms will be made available as a package for the computer algebra system GAP. By utilizing state-of-the art techniques for finite matrix groups, developed only in the last years, this project will make significant progress on algorithmic questions for infinite integral matrix groups, extending current methods that have been mostly restricted to the case of finite groups. The approach envisioned will intermesh calculations with finitely presented groups and with matrix groups in a novel way. Several subtasks require the development of new algorithms for finite groups, as well as for coset enumeration, and thus contribute to the algorithmic theory of finite and finitely presented groups. The software developed as part of the project will be of use to investigators in the many areas encountering arithmetic groups -- geometry, number theory, theoretical physics, and others. The project will contribute to the development of the open-source computer algebra system GAP (and thus also to the system SAGE) that is used by a multitude of researchers world-wide, has been cited in over 2000 refereed publications, and also has seen use as a tool in undergraduate abstract algebra classes. Integrating research and education this project will contribute to the training of a graduate student in developing mathematical software and utilizing standard open-source software development tools.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.

在这个项目中考虑的对象，称为算术群，是一组矩阵（表示空间的对称性），其元素可以被认为是几何对象中的整数坐标。它们位于代数、几何和数论的交叉点。 如果给定了许多这样的矩阵，我们考虑可以通过依次应用对称性组合这些矩阵来形成的子集。 它被称为子群（由矩阵生成）。 一个自然的问题现在出现在是否有许多翻译副本这样一个给定的子组总数的整个集合，在这种情况下，子组被称为算术。这个问题已经出现在许多具体的例子中提出的研究人员在数论或几何，但到目前为止，唯一的解决办法一直是特设的方法，为小的情况。 PI将设计新的算法来在计算机上回答这个问题，并提供该算法的开源实现。 该项目还将培训一名研究生开发数学软件。PI将开发用于计算算术群类的新算法，特别是开发可以确定通过生成矩阵给出的子群本身是否是算术的方法。这些算法的实际实现将作为计算机代数系统GAP的包提供。 通过利用最先进的技术有限矩阵群，仅在过去几年中开发的，这个项目将取得重大进展的算法问题的无限积分矩阵群，扩展目前的方法，主要是限制在有限群的情况下。 所设想的方法将以一种新的方式将计算与非线性呈现的组和矩阵组交织在一起。有几个子任务需要发展新的算法，为有限的群体，以及陪集枚举，从而有助于算法理论的有限和非线性提出的群体。 作为该项目的一部分开发的软件将用于在许多领域遇到算术组的调查-几何，数论，理论物理等。 该项目将有助于开发开放源代码计算机代数系统GAP（因此也是系统SAGE），该系统被世界各地的众多研究人员使用，已在2000多份参考出版物中引用，并已被用作本科抽象代数课程的工具。 整合研究和教育这个项目将有助于培养一个研究生在开发数学软件和使用标准的开放源码软件开发工具。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

项目成果

The Perfect Groups of Order up to Two Million

完美的订单量高达 200 万组

• DOI: 10.1090/mcom/3684
• 发表时间: 2021
• 期刊: Mathematics of computation
• 影响因子: 2
• 作者: Hulpke, Alexander

The strong approximation theorem and computing with linear groups

强逼近定理与线性群计算

• DOI: 10.1016/j.jalgebra.2019.04.011
• 发表时间: 2019
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Detinko, A.S.;Flannery, D.L.;Hulpke, A.
• 通讯作者: Hulpke, A.

Constructive Membership Tests in Some Infinite Matrix Groups

某些无限矩阵群中的建设性隶属度检验

• DOI: 10.1145/3208976.3208983
• 发表时间: 2018
• 期刊: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation
• 作者: Hulpke, Alexander

Universal covers of finite groups

有限群的通用覆盖

• DOI: 10.1016/j.jalgebra.2020.10.032
• 发表时间: 2021
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Dietrich, Heiko;Hulpke, Alexander
• 通讯作者: Hulpke, Alexander

Experimenting with Symplectic Hypergeometric Monodromy Groups

辛超几何单峰群的实验

• DOI: 10.1080/10586458.2020.1780516
• 发表时间: 2020
• 期刊: Experimental Mathematics
• 影响因子: 0.5
• 作者: Detinko, A. S.;Flannery, D. L.;Hulpke, A.
• 通讯作者: Hulpke, A.