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

置换群、完全不连通的局部紧群以及局部同构关系。

基本信息

批准号：
EP/V036874/1
负责人：
Simon Smith
金额：
$ 56.33万
依托单位：
University of Lincoln
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV036874%2F1
关键词：
Permutation groups totally disconnected locally

项目摘要

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.

群论是对称性理论。它与所有纯数学和物理学（例如广义相对论中的Noether定理，量子力学中的Wigner定理），化学（例如晶体学）和计算机科学（例如计算机图形学和AI中的线性代数，密码学中的代数数论）的基本应用有着深刻的联系。群论的发展促进了数学和自然科学的突破。这些发展是从少数几个自然视角之一来看待群体的。一个这样的观点是把一个群看作一个置换群--对象的对称性。另一种方法是将组视为拓扑对象，研究组的“形状”（这里的“形状”是拓扑的，可以拉伸和弯曲，但不能切割）。一类重要的拓扑群是那些非离散的、局部紧的（它们具有非平凡的“形状”，在局部水平上看起来像我们周围的空间）和紧生成的（这些群可以从局部片段中“构建”出来，就像我们周围的空间）。从历史上看，对这些局部紧群的研究导致了物理学的重大突破，以及数学新领域的发展，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。希尔伯特第五问题在1950年代初的解决导致了对连通情况的广泛理解。直到20世纪90年代乔治威利斯的变革性工作，人们才认为理解完全不相关的案例（以下简称tdlc）是不可能的。目前，对复生成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“简单的片段”。tdlc理论的一个中心焦点是理解这些“简单的片断”，因为它们是理解所有由局部紧群生成的紧群结构的关键，在群论中，两个本质相同的群被称为同构。我们已经知道，我们不能指望用同构关系来理解这些“简单的片断”--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，其中两个群是局部同构的，如果它们有同构的“局部”（即紧开）子群。为了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，目前还没有取得进展-对局部同构的了解太少，也没有通用的工具。拟议的研究旨在解决这一问题，通过利用置换群和tdlc理论之间的有用的相互作用。我们的想法是将问题转移到置换群和作用于图的群的语言中，在那里有许多新的和强大的工具可用（一些是最近开发的），解决问题，然后将解决方案翻译回紧生成的tdlc群的语言，这个建议将导致更深入地理解局部紧群。这将影响到许多使用局部紧群的数学和物理领域。该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。这些无限图是大的有限图族的极限情况，由许多较小图的副本构建。这些大型有限图广泛用于计算机科学和科学建模。这种不断增长的理解可能有一天会导致计算和科学建模中更有效的算法。