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Fusion Systems and Localities

聚变系统和地点

基本信息

批准号：
EP/R010048/1
负责人：
Ellen Henke
金额：
$ 12.86万
依托单位：
University of Aberdeen
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR010048%2F1
关键词：
Fusion Systems Localities

项目摘要

The abstract algebraic concept of a group is central in contemporary mathematics, since it allows us to handle objects of a diverse mathematical origin in a uniform way. Groups arise for example from symmetries of other mathematical or geometrical objects. On the other hand, a given group leads again to new algebraic, topological and geometric structures. In particular, to every group one associates a topological space, called the classifying space. This leads to many different ways of studying groups. However, it is a common feature of different approaches that they focus on properties related to a fixed prime p. Saturated fusion systems provide a new algebraic framework which allows us to discover more and more connections between the different approaches related to a prime. There are however still many open questions remaining. For the prime 2, it is an important long-term goal to classify the simplest building blocks of saturated fusion systems. In this project, we will overcome some major conceptual difficulties arising on the way to such a classification result for fusion systems. This will feed into a simplified proof of the classification of finite simple groups, which is an important theorem classifying the simplest building blocks of finite groups. The new algebraic theory we develop will also allow us to study structure-preserving maps between classifying spaces of groups "at a prime p". We expect that this will open new ways for research in homotopy theory and representation theory.

群的抽象代数概念是当代数学的核心，因为它允许我们以统一的方式处理不同数学起源的对象。例如，群产生于其他数学或几何对象的对称性。另一方面，一个给定的群又导致新的代数、拓扑和几何结构。特别是，每一个群都有一个拓扑空间，称为分类空间。这导致了许多不同的研究群体的方法。然而，这是一个共同的特点，不同的方法，他们专注于一个固定的素数p.饱和融合系统相关的属性提供了一个新的代数框架，使我们能够发现越来越多的不同方法之间的连接有关的一个素数。然而，仍然存在许多悬而未决的问题。对于素数2，对饱和融合系统的最简单构建块进行分类是一个重要的长期目标。在这个项目中，我们将克服一些主要的概念上的困难，融合系统的方式，这样的分类结果。这将提供给有限单群分类的简化证明，这是分类有限群的最简单构建块的重要定理。新的代数理论，我们开发也将使我们能够研究结构保持映射之间的分类空间的群体“在一个素数p”。我们期望这将为同伦理论和表示论的研究开辟新的途径。