Group Generation: From Finite To Infinite

你曾经对蝴蝶惊人的对称性感到惊奇吗？或者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，我们每天都在大自然、艺术和建筑中遇到对称。因此，对称性在所有科学中都扮演着重要的角色也就不足为奇了。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剧性副作用。正是对称性为理论物理的标准模型提供了语言。对称是在量子计算机时代保持安全的新型密码学的核心。群论是一个致力于发现适用于所有情况下所有对称类型的一般结果的数学领域，从分子的三维形状，到196,883维的物理概念，再到无法简单几何解释的抽象物体。事实上，研究一个物体的一种富有成效的方法是考虑它的所有对称性的集合。就像乐高积木可以分解成乐高积木，分子可以分解成原子一样，一个物体的对称组可以分解成更小的不可分割的“简单组”。20世纪最伟大的数学成就之一是全世界数百名数学家对所有有限单群进行分类的努力。几十年来，数学家们一直感兴趣的是，什么时候可以通过反复组合两个精心选择的对称性来获得一个物体的所有对称性。这被称为“生成”，它产生了令人惊讶的结果，与数学中的联系有关。例如，Liebeck和Shalev证明了有限简单群中的“几乎所有”对称对都会产生整个群。此外，就在去年，Burness， Guralnick和我给出了有限群的完整分类，其中每个对称（除了“什么都不做”的对称）都可以与另一个对称匹配，从而产生整个对称群。然而，这些发展都涉及具有有限数量对称的对象群，但具有无限数量对称的对象在当代数学中非常重要。我的建议是开始一项新的研究计划，将世代的发展推广到无限。更确切地说，我试图研究有限单群的惊人生成性质是否适用于无限单群，如Thompson群和Cantor空间同胚的相关群，以期对有限呈现的无限单群的生成性质形成更深入的理解。此外，通过利用有限（几乎）单群理论的最新发展，我将解决有关有限群生成的开放性问题。我建议在圣安德鲁斯大学进行这项研究，那里有许多有限和无限群体的顶尖研究人员。此外，它还是中国CIRCA的所在地，这是一个数学和计算机科学的联合研究中心。这突出了所提出的工作方案的潜在应用：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，并且知道一个物体的所有对称性都可以由两个产生，这为执行许多这些计算提供了一种有效的方法。

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.