Logic, Group Theory, Combinatorics and Ergodic Theory

逻辑、群论、组合学和遍历理论

• 批准号: 1362974
• 负责人: Gregory Cherlin
• 金额: $ 36.2万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362974&HistoricalAwards=false
• 关键词: Logic; Group; Theory; Combinatorics; Ergodic

The notion of classification is fundamental to many branches of mathematical research: one looks for the deep underlying structure common to many widely varying topics, with particular attention to underlying symmetries. Techniques of modern mathematical logic, in combination with more specialized tools from other branches of mathematics, make it possible to analyze the scope and limitations of such classifications and reveal a wealth of additional structure. The techniques of mathematical logic also provide a broad perspective on combinatorial problems. Here one must first separate the "chaotic" problems which cannot be neatly categorized and identify the subclass of more tractable problems, looking ultimately for an algorithm to carry out this part of the analysis at a completely general level. This research project investigates such classification problems. In particular, issues of computability are an essential component of the program. The research is interdisciplinary, establishing new relationships between mathematical logic and classical mathematics.This award supports a group research project. Using methods of topological dynamics, combinatorial group theory, model theory, and algebra, the two investigators will investigate a variety of classification problems in analysis, algebra, and combinatorics. Simon Thomas will combine these methods with modern descriptive set theory to analyze classification problems in algebra, particularly those associated with representation theory and the subject of "invariant random subgroups" in the sense of Vershik, while Gregory Cherlin will pursue joint work with visitor Saharon Shelah on the existence of universal graphs with forbidden subgraphs and the associated structural Ramsey theory, as well as the classification of all metrically homogeneous graphs and analogous problems for finite permutation groups. A further component of the project is an active visitor's program under the direction of Shelah when he is in residence at Rutgers.

分类的概念对数学研究的许多分支至关重要：人们寻找许多广泛不同主题共有的深层基础结构，特别关注潜在的对称性。 现代数学逻辑的技术与其他数学分支的更专业的工具结合在一起，使得分析此类分类的范围和局限性并揭示了许多其他结构。 数学逻辑的技术还为组合问题提供了广泛的观点。在这里，必须首先将无法整齐分类的“混乱”问题分开，并确定更易于处理的问题的子类，最终寻找算法以完全一般的层面进行分析。 该研究项目调查了此类分类问题。 特别是，计算性问题是该程序的重​​要组成部分。 这项研究是跨学科的，建立了数学逻辑和经典数学之间的新关系。该奖项支持小组研究项目。 使用拓扑动力学，组合群体理论，模型理论和代数的方法，两个研究者将研究分析，代数和组合学方面的各种分类问题。西蒙·托马斯（Simon Thomas）将这些方法与现代描述性设置理论相结合，以分析代数中的分类问题，尤其是与表示理论相关的分类问题和“不变的随机亚组”的主题，从VERSHIK的意义上讲，而Gregory Cherlin将与访客Sharah在范围内与范围相关的范围内的图形和相关的构图的共同介绍，而Gregory Cherlin则将与访客的sharah shelah进行联合工作，并将其构成相关的构图，并将其与Forb the Interbienty rementimally Ramentimention进行了分类。有限置换组的均匀图和类似问题。该项目的另一个组成部分是在谢拉（Shelah）居住在罗格斯（Rutgers）的指导下的一个活跃的访客计划。