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Structure and Cohomology in Fusion Systems

融合系统中的结构和上同调

基本信息

批准号：
1902152
负责人：
Justin Lynd
金额：
$ 13.5万
依托单位：
University of Louisiana at Lafayette
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902152&HistoricalAwards=false
关键词：
Structure Cohomology Fusion Systems

项目摘要

The investigator will study problems at the interface between finite group theory and topology. Finite group theory is the study of the symmetry of finite objects by algebraic means. While the project involves fundamental research, group theory is applicable in many of the natural sciences where highly symmetric objects are found, including in biology, chemistry, and physics, and in the study of communication networks and cryptographic schemes. A finite simple group is akin to an atom in that it cannot be broken down into smaller groups. Each finite group of symmetries is made up of simple groups stacked on top one another analogously to the way molecules are built out of atoms. One of the premier mathematical achievements of the20th century was the description and classification of all the finite simple groups (CFSG). Its proof is however long and difficult, currently spanning around 10,000-15,000 pages, and it is desirable to have significantly simpler proofs. The PI will investigate this problem by using strategies that come form topology, a branch of mathematics that studies properties of objects that are invariant under continuous transformations. One bridge between group theory and topology is given by the classifying space of a group. When the group is finite, there is an associated set of prime numbers, and both the group and its classifying space can be studied "one prime at a time". This strategy has been abstracted into the notion of a p-fusion system, the basic object of study.The project arises out of two recent major developments in fusion systems: a program for the classification of simple 2-fusion systems of component type (CSFS) whose ultimate aim is to give a substantially simpler proof of the CFSG, and the recent solution of the existence and uniqueness of centric linking systems, which provide the bridge from fusion systems to topology. Both developments are brought together by their connections with the cohomology of various functors defined on the orbit category and other related categories. The investigator will: (1) investigate fusion systems at the prime 2 whose linking systems support a noninner rigid automorphism by using methods from the CSFS, (2) work to find necessary and sufficient conditions for the existence and uniqueness of centralizers of fusion subsystems through the definition and computation of the cohomology of functors which obstruct internal rigid actions on centric linking systems, and (3) work directly within the nearly-completed CSFS to solve certain outstanding problems, and to generalize others for use in related problems such as in (3). The three interrelated projects aim to gain a better understanding of some aspect of the p-local structure of finite groups and/or the homotopy theory of p-completed classifying spaces by exploiting their interplay. Project (1) uses finite group theoretic methods to provide a better understanding of the group of self-homotopy equivalences of the p-completed classifying space of a finite group, while project (2) provides an application of functor cohomology to the open problem of constructing centralizers. The construction of centralizers is a fundamental problem with applications within the CSFS, e.g. in the defining and understanding of standard subsystems of fusion systems, and it also has potential applications to the open problem of describing maps between different p-completed classifying spaces. Project (3) sits within the CSFS proper, simultaneously supporting the goals in (1) and the completion of the classification program.This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

研究人员将研究有限群理论和拓扑之间的接口问题。有限群论是通过代数方法研究有限对象的对称性。虽然该项目涉及基础研究，但群论适用于许多发现高度对称物体的自然科学，包括生物学、化学和物理学，以及通信网络和密码方案的研究。有限单群类似于原子，因为它不能分解为更小的群。每个有限对称群都是由简单的群相互堆叠而成，类似于由原子构建分子的方式。 20 世纪最重要的数学成就之一是所有有限单群 (CFSG) 的描述和分类。然而，它的证明又长又困难，目前大约有 10,000-15,000 页，并且希望有更简单的证明。 PI 将使用拓扑学中的策略来研究这个问题，拓扑学是数学的一个分支，研究在连续变换下不变的对象的属性。群论和拓扑之间的一座桥梁是由群的分类空间给出的。当群是有限的时，存在一组相关联的素数，并且群及其分类空间都可以“一次一个素数”地研究。该策略已被抽象为 p-fusion 系统的概念，即研究的基本对象。该项目源于融合系统最近的两个重大发展：一个用于组件类型的简单 2-fusion 系统 (CSFS) 分类的程序，其最终目的是给出 CFSG 的简单得多的证明，以及中心链接系统的存在性和唯一性的最新解决方案，它提供了从融合系统到拓扑的桥梁。这两种发展都通过它们与轨道范畴和其他相关范畴上定义的各种函子的上同调联系在一起。研究者将：（1）研究素数 2 处的融合系统，其连接系统通过使用 CSFS 中的方法支持非内刚性自同构，（2）通过定义和计算阻碍中心连接系统内部刚性作用的函子的上同调，努力找到融合子系统集中器的存在和唯一性的必要和充分条件，以及（3）直接在接近完成的 CSFS 中工作解决某些突出问题，并将其他问题推广到相关问题（例如（3））中。这三个相互关联的项目旨在通过利用它们的相互作用来更好地理解有限群的 p 局部结构和/或 p 完备分类空间的同伦理论的某些方面。项目（1）使用有限群理论方法来更好地理解有限群的p完备分类空间的自同伦等价群，而项目（2）则提供了函子上同调在构造集中器的开放问题中的应用。集中器的构建是 CSFS 内应用程序的一个基本问题，例如它不仅可以用于定义和理解融合系统的标准子系统，而且还可以应用于描述不同 p-完备分类空间之间的映射的开放问题。项目 (3) 属于 CSFS 本身，同时支持 (1) 中的目标和分类计划的完成。该项目由代数和数论计划以及刺激竞争性研究既定计划 (EPSCoR) 共同资助。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。