Logic, Group theory, Combinatorics and Ergodic theory

逻辑、群论、组合学和遍历理论

• 批准号: 0600940
• 负责人: Gregory Cherlin
• 金额: $ 68.03万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600940&HistoricalAwards=false
• 关键词: Logic; Group; theory; Combinatorics; Ergodic

Groups of finite Morley rank arise in model theory as a substantialgeneralization of the class of algebraic groups. It has been conjecturedthat the simple groups in this category are all algebraic, and Cherlin isworking on this conjecture with a team of international collaborators.This involves techniques used in the classification of the finite simplegroups as well as some ideas from black box group theory. In graph theory,using a mix of model theoretic and combinatorial techniques, Cherlin andShelah are developing techniques to determine, for a given finite set offorbidden graphs, whether there is a universal graph meeting theconstraints. The ultimate question here is whether the entire problem isalgorithmically decidable. Thomas works on the theory of countable Borelequivalence relations, combining the methods of descriptive set theory withtechniques related to superrigidity. The methods of descriptive set theorycast considerable light on classical classification problems, andconversely powerful methods coming from group theory illuminate and advancethe general theory. Cherlin and Thomas also host a dynamic visitor programat Rutgers, coordinated with annual visits by Shelah.Infinite group theory provides a tool for studying and exploiting thesymmetries of a mathematical model or a physical system. Cherlin and hiscollaborators are aiming at the classification of the groups associatedwith well-behaved algebraic systems, while Simon Thomas approaches thestudy of infinite groups from the point of view of their actions and theanalysis of one action in terms of another. A particularly strong role isplayed here by ideas coming from the theory of dynamical systems. Graphsare the mathematical abstraction of networks in general, and the problemsunder consideration relate to the analysis, preferably by a general(computable) algorithm, of classes of graphs characterized by forbidding afixed set of patterns.

有限的莫利组在模型理论中出现，作为代数群体类别的实质性一般化。 人们已经猜想了这一类别中的简单小组都是代数的，而Cherlin与一个国际合作者团队一起对此进行了猜想。这涉及在有限简单组的分类中使用的技术，以及Black Box Group理论的一些想法。 在图理论中，使用模型理论和组合技术的混合，Cherlin和Shelah正在开发技术，以确定给定有限设置的Offorbidden图，是否有一个通用图形相遇。 这里的最终问题是整个问题是否可以确定。 托马斯（Thomas）致力于可计数的骨质关系理论，将描述性集理论的方法与与超级汇率相关的技术结合在一起。描述性设定理论的方法广播了经典分类问题，以及来自群体理论的强大方法阐明和进步一般理论。 切尔林（Cherlin）和托马斯（Thomas）还拥有一个动态的访客程序，并与谢拉（Shelah）的年度访问协调。Infinite群体理论提供了一种工具，可用于研究和利用数学模型或物理系统的thesymerties。 Cherlin和Hiscollaborator的旨在将与行为良好的代数系统相关的群体的分类，而Simon Thomas从他们的行为的角度和一种行动的角度来看，Simon Thomas接近无限群体的人。 来自动力学系统理论的想法所引起的一个特别强大的角色。 GraphSare一般而言网络的数学抽象，以及与分析有关的问题，最好由一般（可计算的）算法，其图表的类别（由禁止AFIXED AFIXEFIDED模式集）。