Tame Groups, Universal Graphs, Automorphism Towers, and Cofinalities of Infinite Groups

Cherlin建议与一个由8名研究人员组成的团队合作，使用有限群经验所建议的方法，对有限Morley阶的连通简单驯服群进行分类：预计所有此类群都是代数的。此外，Cherlin与Shih合作，试图将模型论方法引入到万能图的研究中，并展示了模型论代数闭包算子在这类问题中的作用。托马斯将利用强迫、PCF理论、大基数以及特征标理论，结合自同构塔和有限群的无限类比的余定性，研究集合论和群论的相互作用。一个突出的问题是无限对称群的余定性与Blass不变量GroupWise密度的关系。矩阵群在现代数学中的作用是公认的，在纯数学和应用数学中都占有中心地位。据推测，在对许多明显无关的一般类型结构的详细分析中，这些基团也起着主导作用。本项目的目标之一是在所谓的“温顺情况”中确认这一点，这种情况更容易获得。另一个单独的目标是证明标准的逻辑思想(模型理论)可以为组合学(图论)中存在的问题提供新的认识。人们希望图论家自己也能在这种情况下采用这些方法。托马斯的工作揭示了代数和集合论问题之间以前未曾被怀疑的关系，并应该导致新的“强迫”工具，这些工具是现代集合论最强大的基础工具之一。该提案的主旨是展示在一个领域中自然产生的技术如何能够卓有成效地推广到其他主题领域。

Cherlin proposes to collaborate with a team of eight researchers on the classification of connected simple tame groups of finite Morley rank, using methods suggested by experience with finite groups: it is anticipated that all such groups are algebraic. In addition Cherlin, working with Shi, seeks to introduce model theoretic methods into the study of universal graphs, and to show the role of the model theoretic algebraic closure operator for this class of problems. Thomas will study interactions of set theory and group theory in connection with automorphism towers and the cofinalities of infinite analogs of finite groups, making use of forcing, pcf theory, and large cardinals, as well as character theory. An outstanding problem is the relationship of the cofinality of the infinite symmetric group to Blass' invariant, the groupwise density. The role of matrix groups in modern mathematics is well established and occupies a central place in both pure and applied mathematics. It has been conjectured that these groups also play a predominant role in the detailed analysis of many apparently unrelated structures of general type. One of the goals of the present project is to confirm this in the so-called ``tame case'', which is more immediately accessible. A separate goal is to demonstrate that standard ideas of logic (model theory) can cast new light on existing problems in combinatorics (graph theory). It is hoped that graph theorists will themselves adopt these methods in such cases. The work of Thomas uncovers previously unsuspected relations between algebraic and set theoretical issues, and should lead to new ``forcing'' tools, which are among the most powerful foundational tools of modern set theory. The general thrust of the proposal is to show how techniques arising naturally in one area can be transported fruitfully to other subject areas.