Interactions of Logic with Group Theory and Combinatorics

逻辑与群论和组合学的相互作用

• 批准号: 0100794
• 负责人: Gregory Cherlin
• 金额: $ 42.5万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100794&HistoricalAwards=false
• 关键词: Interactions; Logic; Group; Theory; Combinatorics

Cherlin and Thomas will pursue interactions of the techniques of logic with problems in algebra and combinatorics. Cherlin will work with groups of finite Morley rank using methods modeled heavily on finite group theory, aiming particularly at an approach to the odd characteristic case compatible with the existence of bad fields, and on problems in graph theory susceptible to model theoretic analysis (universal graphs and problems of wqo). Thomas will work on Borel equivalence relations, particularly with those associated with natural classification problems in algebra, which may well provide the examples needed to settle some problems presently open in full generality, as well as providing information on the relative difficulty (according to a very robust system of measurement) of the algebraic problems, some very classical and open, for what can now seen to be essential reasons. Thomas will also pursue his work on the automorphism tower problem, using set theoretic techniques.Mathematical logic provides tools of great generality which can be applied to various areas of mathematics. In combinatorial contexts the model theoretic point of view provides methods that can be used to handle specific problems very uniformly, rather than on the case by case basis sometimes encountered in the literature. Descriptive set theory provides methods for analyzing the relative difficulty of both solved and unsolved problems in algebra, and in particular provides concrete information as to how detailed an answer one may usefully seek in a classification problem, making it possible to distinguish dead ends from fruitful lines of inquiry on an a priori basis.

Cherlin和Thomas将与代数和组合学中的问题进行逻辑技术的相互作用。 Cherlin将使用以有限的群体理论建模的方法与有限的Morley等级的组合作，尤其是针对与奇怪的特征案例相兼容的方法，以及图表理论中的问题易于模型理论分析（通用图和WQO的问题）。托马斯（Thomas）将致力于鲍勒（Borel）等效关系，尤其是与代数中的自然分类问题相关的人，这很可能提供了解决目前完全普遍打开的一些问题所需的示例，以及提供有关代数问题的相对困难（根据非常强大的测量系统）的信息，这些问题是非常经典和开放的，现在可以看出的是必不可少的。托马斯还将使用设定的理论技术来从事自动型塔问题的工作。数学逻辑提供了可以应用于数学的各个领域的伟大通用工具。在组合上下文中，模型理论的观点提供了可用于非常统一地处理特定问题的方法，而不是根据有时在文献中遇到的情况。 描述性集理论提供了分析代数中解决和未解决问题的相对难度的方法，特别是提供了有关在分类问题中可能有用的答案的具体信息，从而使死者与先验基础上的富有成果的询问区分开来。