Group Generation: From Finite To Infinite

你有没有惊叹过蝴蝶惊人的对称性？或者被一张略微偏离中心的照片所挫败？作为人类，我们被对称性所吸引，我们在自然界、艺术和建筑中每天都会遇到这种对称性。因此，对称性在所有科学中发挥着基础性作用也就不足为奇了。例如，正是数学对称理论解释了20世纪60年代出现的沙利度胺药物的悲剧性副作用。对称性为理论物理的标准模型提供了语言。对称性是新密码学的核心，在量子计算机时代仍然是安全的。群论是一个数学领域，致力于发现适用于所有环境中所有对称类型的一般结果，从分子的三维形状，到196883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，研究一个物体的一种卓有成效的方法是考虑它的所有对称性的群。就像乐高积木可以被分解成乐高积木，分子可以被分解成原子一样，物体的对称性群可以被分解成更小的、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以百计的数学家努力对所有有限单群进行分类。几十年来，数学家们一直对什么时候可以通过反复组合两个精心选择的对称来获得一个物体的所有对称性感兴趣。这就是所谓的“世代”，它产生了令人惊讶的结果，并与数学联系在一起。例如，Liebeck和Shalev证明了有限单群中“几乎所有”对称对都生成整个群。此外，就在去年，Burness，Guralnick和我给出了有限群的一个完整的分类，其中每个对称(不做任何事情的对称除外)都可以与另一个对称匹配，从而生成整个对称群。然而，这些发展都涉及具有有限个对称的对象群，但是具有无限多个对称的对象在当代数学中是非常重要的。我的建议是开始一个新的研究计划，将生成的发展推广到无穷大。更准确地说，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，如康托空间的Thompson群和相关的同胚群，以期形成对有限表示无限单群的生成性质的更深层次的理解。此外，通过利用有限(几乎)单群理论的最新发展，我将解决关于有限群生成的公开问题。我建议在圣安德鲁斯大学进行这项研究，该大学是有限群和无限群的许多领先研究人员的所在地。此外，它还拥有一个集数学和计算机科学于一身的研究中心--Circa。这突显了拟议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，知道一个物体的所有对称性可以由两个对称性产生，这为进行许多此类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.