Group Generation: From Finite To Infinite

群生成：从有限到无限

• 批准号: EP/X011879/1
• 负责人: Scott Harper
• 金额: $ 35.59万
• 依托单位: University of St Andrews
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX011879%2F1
• 关键词: Group; Generation; Finite; Infinite

Have you ever marvelled at the stunning symmetry of a butterfly? Or been frustrated by a photograph taken slightly off-centre? As humans we're attracted to symmetry and we encounter it every day in nature, art and architecture. It's no surprise, therefore, that symmetry plays a fundamental role across all sciences. For instance, it's the mathematical theory of symmetry that explains the tragic side-effects of the drug thalidomide seen in the 1960s. It's symmetry that provides the language for the Standard Model of theoretical physics. It's symmetry that's at the heart of novel cryptography that remains secure in an era of quantum computers. Group theory is the area of mathematics dedicated to discovering general results that apply to all types of symmetry in all contexts, from the three-dimensional shapes of molecules, to concepts from physics in 196,883 dimensions, to abstract objects admitting no simple geometric interpretation. Indeed, one fruitful way to study an object is to consider the group of all its symmetries. Just as Lego constructions can be broken into Lego bricks and as molecules can be broken into atoms, the group of symmetries of an object can be broken into smaller indivisible "simple groups". One of the greatest mathematical achievements of the twentieth century was the effort of hundreds of mathematicians across the world to classify all the finite simple groups. For decades, mathematicians have been interested in when one can obtain all an object's symmetries by repeatedly combining two well-chosen symmetries. This is called "generation", and it has yielded surprising results, with links across mathematics. For example, Liebeck and Shalev proved that "almost all" pairs of symmetries in a finite simple group generate the entire group. Moreover, just last year, Burness, Guralnick and I gave a complete classification of the finite groups where every symmetry (other than the "do nothing" symmetry) can be matched with another with which it generates the entire group of symmetries.However, these developments all concern groups of objects with a finite number of symmetries, but objects with infinitely many symmetries are very important in contemporary mathematics. My proposal is to begin a new programme of research to generalise developments on generation to the infinite.More precisely, I seek to investigate whether the startling generation properties of the finite simple groups hold for the infinite simple groups such as Thompson groups and related groups of homeomorphisms of Cantor space, with a view to forming a deeper understanding of the generation properties of finitely presented infinite simple groups. In addition, by exploiting recent developments in the theory of finite (almost) simple groups, I will address open questions regarding the generation of finite groups.I propose carrying out this research at the University St Andrews, which is home to a number of leading researchers in both finite and infinite groups. Moreover, it hosts CIRCA, a research centre joint between mathematics and computer science. This highlights potential applications of the proposed programme of work: from cryptographers to chemists, researchers carry out computer calculations involving symmetry, and knowing that all the symmetries of an object can be generated by just two provides an efficient way to carry out many of these computations.

你是否曾为蝴蝶惊人的对称性而感到困惑？或者是因为一张稍微偏离中心的照片而感到沮丧？作为人类，我们被对称所吸引，我们每天都在自然、艺术和建筑中遇到它。因此，对称性在所有科学中起着基础性的作用也就不足为奇了。例如，对称性的数学理论解释了20世纪60年代沙利度胺药物的悲惨副作用。对称性为理论物理学的标准模型提供了语言。对称性是新型密码学的核心，在量子计算机时代仍然安全。群论是数学领域，致力于发现适用于所有背景下所有类型对称性的一般结果，从分子的三维形状，到196，883维的物理概念，再到不允许简单几何解释的抽象对象。事实上，研究一个物体的一个富有成效的方法是考虑它的所有对称性的群。正如乐高积木可以分解成乐高积木，分子可以分解成原子，物体的对称群可以分解成更小的不可分割的“简单群”。二十世纪最伟大的数学成就之一是全世界数百名数学家努力对所有有限单群进行分类。几十年来，数学家们一直对何时可以通过反复组合两个精心选择的对称性来获得物体的所有对称性感兴趣。这就是所谓的“生成”，它产生了令人惊讶的结果，与数学的联系。例如，Liebeck和Shalev证明了有限单群中的“几乎所有”对称对生成整个群。此外，就在去年，Burness，Guralnick和我给出了一个完整的分类有限群，每一个对称（除了“不做任何事情”的对称）可以与另一个匹配，它产生整个群的对称。然而，这些发展都涉及群体的对象有限数量的对称，但对象无限多的对称是非常重要的当代数学。我的建议是开始一个新的研究计划，以概括的发展代infinite.More确切地说，我试图调查是否惊人的生成属性的有限简单的群体举行的无限简单的群体，如汤普森群体和相关群体同胚的康托空间，以期形成一个更深入的了解的生成属性的无限简单的群体。此外，利用最近的发展，在理论上的有限（几乎）简单的群体，我将解决悬而未决的问题，关于生成的有限groups.I建议进行这项研究在圣安德鲁斯大学，这是家里的一些领先的研究人员在有限和无限的群体。此外，它还拥有CIRCA，这是一个数学和计算机科学之间的研究中心。这突出了拟议的工作计划的潜在应用：从密码学家到化学家，研究人员进行涉及对称性的计算机计算，并且知道一个对象的所有对称性可以由两个生成，这提供了一种有效的方法来执行许多这些计算。

项目成果

Totally deranged elements of almost simple groups and invariable generating sets

几乎简单群和不变生成集的完全混乱的元素

• DOI: 10.48550/arxiv.2304.10213
• 发表时间: 2023
• 作者: Harper S

The maximal size of a minimal generating set

最小发电机组的最大尺寸

• DOI: 10.1017/fms.2023.71
• 发表时间: 2023
• 期刊: Forum of Mathematics, Sigma
• 作者: Harper S