Logic, Group Theory, Combinatorics and Ergodic Theory

分类的概念是数学研究的许多分支的基础：人们寻找许多广泛变化的主题共同的深层底层结构，特别注意底层的对称性。现代数学逻辑技术，结合其他数学分支的更专业的工具，使分析这种分类的范围和局限性成为可能，并揭示了丰富的附加结构。数学逻辑的技术也为组合问题提供了一个广阔的视角。在这里，我们必须首先将无法被整齐分类的“混沌”问题分离出来，并识别出更容易处理的问题的子类，最终寻找一种算法来在完全通用的水平上执行这部分分析。本研究项目探讨了这类分类问题。特别是，可计算性问题是程序的重要组成部分。该研究是跨学科的，在数理逻辑与经典数学之间建立了新的关系。这个奖支持一个小组研究项目。利用拓扑动力学、组合群论、模型论和代数的方法，两位研究者将研究分析、代数和组合学中的各种分类问题。Simon Thomas将把这些方法与现代描述集理论结合起来，分析代数中的分类问题，特别是那些与表示理论和Vershik意义上的“不变随机子群”相关的问题，而Gregory Cherlin将与访问者saharron Shelah共同研究具有禁止子图的普遍图的存在性以及相关的结构Ramsey理论。以及所有度量齐次图的分类和有限置换群的类似问题。这个项目的另一个组成部分是一个活跃的访客项目，由希拉在罗格斯大学居住期间指导。

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.