Structure and Cohomology in Fusion Systems

研究人员将研究有限群论和拓扑学之间的交界处的问题。有限群论是用代数方法研究有限对象对称性的一门学科。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。一个有限的单群类似于一个原子，因为它不能被分解成更小的群。每个有限的对称性群都是由一个又一个堆叠在一起的简单群组成的，类似于分子是由原子组成的。20世纪最重要的数学成就之一是对所有有限单群(CFSG)的刻画和分类。然而，它的证明是漫长和困难的，目前大约跨越10,000-15,000页，最好是有明显更简单的证明。PI将通过使用来自拓扑学的策略来研究这个问题，拓扑学是研究对象在连续变换下不变的属性的数学分支。群的分类空间是连接群论和拓扑学的一座桥梁。当群是有限的时，存在一个相联的素数集，群及其分类空间都可以“一次一个素数”地研究。这一策略已被抽象为p-融合系统的概念，p-融合系统是研究的基本对象。该项目源于融合系统中最近的两个主要发展：一个是简单2-融合系统的分类程序，其最终目的是给出一个更简单的CFSG的证明；另一个是最近解决的中心连接系统的存在和唯一性问题，它提供了从融合系统到拓扑的桥梁。这两个发展是通过它们与定义在轨道范畴和其他相关范畴上的各种函子的上同调联系在一起的。研究人员将：(1)利用CSFS的方法研究素数2的融合系统，其链接系统支持非内刚性自同构；(2)通过定义和计算阻碍中心链接系统内部刚性作用的函子的上同调，找到融合子系统中心子存在唯一的充要条件；(3)直接在接近完成的CSFS内工作以解决某些突出问题，并推广其他问题用于相关问题，如(3)。这三个相互关联的项目旨在通过利用它们之间的相互作用来更好地理解有限群的p-局部结构和/或p-完全分类空间的同伦理论。文献(1)利用有限群论的方法对有限群的p-完全分类空间的自同伦等价群有了更好的理解，而文献(2)给出了函子上同调在构造中心化子这一公开问题上的应用。中心子的构造是CSFS中应用的一个基本问题，例如在定义和理解融合系统的标准子系统方面，它也有潜在的应用于描述不同p-完备分类空间之间的映射的公开问题。项目(3)位于CSFS内部，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和既定的激励竞争研究计划(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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.