Permutation groups, totally disconnected locally compact groups, and the local isomorphism relation.

群论是对称论。它与所有纯数学以及物理学(例如广义相对论中的Noether定理、量子力学中的Wigner定理)、化学(例如结晶学)和计算机科学(例如计算机图形学和人工智能中的线性代数、密码学中的代数数论)中的基本应用程序有着深刻的联系。群论的发展促进了数学和科学的突破。这些发展是从少数几个自然视角中的一个来看群体的。一种这样的观点是将一个群视为一个置换群--一个对象的对称性。另一种是将组视为一个拓扑对象，研究组的“形状”(这里的“形状”是拓扑的，可以拉伸和弯曲，但不能剪切)。这些拓扑群中的一类很重要，它们是非离散的、局部紧凑的(这些拓扑群具有非平凡的“形状”，在局域层面上看起来就像我们周围的空间)，并且是紧凑生成的(这些群可以用局部碎片“构建”出来，同样也像我们周围的空间一样)。在历史上，对这些局部紧群的研究导致了物理学上的重要突破，以及新的数学领域的发展，如抽象调和分析。局部紧群的研究分为两种情况：连通情况和完全不连通情况。20世纪50年代初希尔伯特第五问题的解决使人们对关联案件有了更广泛的理解。直到20世纪90年代乔治·威利斯做出变革性的工作之前，人们一直认为不可能理解这个完全不相关的案例(以下称为TDLC)。目前，紧生成TDLC群的研究是一个重要的研究领域。我们现在知道这些群与对称群(即置换群)密切相关；它们有一个几何，它们的几何和拓扑之间的相互作用限制了它们的结构；它们可以被“分解”成“简单的碎片”。TDLC理论的一个中心焦点是理解这些“简单片”，因为它们是理解所有紧生成的局部紧群的关键。在群论中，本质相同的两个群被称为同构。已经知道，我们不能指望用同构关系来理解这些“简单的片段”--这些群太复杂了。然而，人们认为它们可以用“局部同构”关系来理解，即如果两个群具有同构的“局部”(即紧开子群)，则它们是局部同构的。要使用局部同构来理解这些“简单片段”，我们首先需要知道有多少不同的(直到局部同构)“简单片段”。这被认为是一个非常困难和重要的问题。目前，无法取得任何进展--对局部同构知之甚少，也没有可用的通用工具。这项拟议的研究试图通过利用置换群和TDLC理论之间的有用相互作用来解决这个问题。我们的想法是将问题转化为置换群和作用于图的群的语言，其中有许多新颖而强大的工具可用(有些是最近开发的)，解决问题，然后将解转换回紧生成的TDLC群的语言。这一建议将导致对局部紧群的更深层次的理解。这将影响使用局部紧群的数学和物理的许多领域。这一建议也将增加我们对高度对称无限图的对称性和结构的理解。这些无限图是由较小图的许多副本构建而成的大型有限图族的极限情况。这些大型有限图在计算机科学和科学建模中被广泛使用。这种加深的理解有朝一日可能会在计算和科学建模方面带来更有效的算法。

Group theory is the theory of symmetry. It has deep links to all of pure maths, and fundamental applications in physics (e.g. Noether's theorems in general relativity, Wigner's Theorem in quantum mechanics), chemistry (e.g. crystallography) and computer science (e.g. linear algebra in computer graphics and AI, algebraic number theory in cryptography). Developments in group theory precipitate breakthroughs in maths and the sciences. These developments follow from looking at groups from one of a handful of natural perspectives. One such perspective is to view a group as a permutation group - the symmetries of an object. Another is to view a group as a topological object, where the "shape" of the group is studied (the "shape" here is topological and can be stretched and bent, but not cut). An important class of these topological groups are those that are non-discrete and locally compact (these have a nontrivial "shape" that on a local level looks like the space around us) and compactly generated (these groups can be "built" out of local pieces, again like the space around us). Historically, research into these locally compact groups led to important breakthroughs in physics, as well as the development of new areas of mathematics, like abstract harmonic analysis.The study of locally compact groups breaks into two cases: the connected case and the totally disconnected case. The solution of Hilbert's Fifth problem in the early 1950s led to a broad understanding of the connected case. Understanding the totally disconnected case (henceforth, tdlc) was considered impossible until transformative work by George Willis in the 1990s. Today the study of compactly generated tdlc groups is an important area of research. We now know these groups are strongly related to groups of symmetries (i.e. permutation groups); that they have a geometry, and the interplay between their geometry and topology restricts their structure; and they can be "decomposed" into "simple pieces". A central focus of tdlc theory is to understand these "simple pieces", since they hold the key to understanding the structure of all compactly generated locally compact groups.In group theory, two groups that are essentially the same are said to be isomorphic. It is known already that we cannot hope to understand these "simple pieces" using the isomorphism relation - the groups are too complicated. However, it is thought that they could be understood using the "local isomorphism" relation, where two groups are locally isomorphic if they have isomorphic "local" (i.e. compact open) subgroups. To understand these "simple pieces" using local isomorphisms, we need as a first step to know how many different (up to local isomorphism) "simple pieces" there are. This is considered to be a very hard and important problem.At present, no progress can be made - too little is known about local isomorphisms, and there are no general tools available. The proposed research seeks to address this, by exploiting a useful interplay between permutation groups and tdlc theory. The idea is to move the problem into the language of permutation groups and groups acting on graphs, where there are many novel and powerful tools available (some developed recently), solve the problem, and then translate the solution back into the language of compactly generated tdlc groups.This proposal will lead to a deeper understanding of locally compact groups. This will impact the many areas of maths and physics where locally compact groups are used. The proposal will also increase our understanding of the symmetries and structure of highly-symmetric infinite graphs. These infinite graphs are limiting cases of families of large finite graphs, built from many copies of a smaller graph. These large finite graphs are used extensively in computer science and scientific modelling. This increased understanding could one day lead to more efficient algorithms in computing and scientific modelling.